# Least Common Multiple calculator

Least common multiple (LCM) calculator. LCM is also known as least common denominator (LCD).

## LCM calculator

 First number: Second number: Least common multiple (lcm): Greatest common divisor (gcd):

## LCM example

Find the least common multiple number for numbers 8 and 12:

The multiples of 8 are:

8, 16, 24, 32, 40, ...

The multiples of 12 are:

12, 24, 36, 48, ...

So the least common multiple number is 24:

lcm = 24

# Math Calculators

Here we discuss list of online math calculators that help us for doing calculation easily.

The Sum (Summation) Calculator is used to calculate the total summation of any set of numbers. In mathematics, summation is the addition of a sequence of any kind of numbers, called addends or summands; the result is their sum or total.

Addition and sum calculator helps you add a long number with decimal values. It supports very long numbers with decimal values calculation.

#### How to use Addition and sum calculator?

For using addition and sum calculator you need to follow below steps.

2. Enter the two numbers which you want to do an addition.

3. Click on "Calculate" to get a result.

In Mathematics, addition is one of the arithmetic operations. It gives the sum of the numbers. In other words, the process of adding the numbers together is called addition. The symbol used to represent the addition is “+”. The result obtained from the addition process is called the sum. The numbers which are added together are called addends.

For example,

3 + 5 = 8

Here,

3 and 5 are called addends

“+” Is called an arithmetic operator

8 is called the sum.

The following are the rules of addition operation:

1. Positive Number + Positive Number = (Add) Positive Number

2. Negative Number + Negative Number = (Add) Negative Number

3. Negative Number + Positive Number = (Subtract) Take the sign of the number with the largest absolute value

### 2. Arccos Calculator

An online calculator to calculate the inverse cosine function arccos(x) in radians and degrees.Use arccos calculator to easily calculate the arccosine of a number. Supports input of either decimal numbers (e.g. 0.5, -0.5) or fractions (e.g. 1/2, -1/2).

Arccos definition

The arccosine function is the inverse function of cos(x).

arccos(x) = cos-1(x)

For example, If the cosine of 60° is 0.5:

cos(60°) = 0.5

Then the arccos of 0.5 is 60°:

arccos(0.5) = cos-1(0.5) = 60°

How to use the arccos (x) calculator?

Enter x as a real number, within the domain of arccos function such that -1 ≤ x ≤ 1 and the number of decimal places desired then press "enter". Two answers are displayed: one in radians and the second in degrees.

What is the inverse of cosine (arccos)?

Arccos is the inverse of a trigonometric function - specifically, the inverse of the cosine function. However, as trigonometric functions are periodic, then, in a strict sense, they cannot be inverted. We can deal with that problem by choosing an interval in which the basic function is monotonic. You can pick many different ranges, but for cosine the common choice is [0,π]. This range is called the set of principal values.

Arccos(x) is the most commonly used notation, as cos-1x may be misleading - remember that inverse cosine is not the same as the reciprocal of the function (in other words, raising to the power -1):

cos-1x ≠ 1/cos(x)

Arccos function :- The arccos (arcus cosine, arccosine) is one of the inverse trigonometric functions (antitrigonometric functions, arcus functions) and is the inverse of the cosine function. It is sometimes written as cos-1(x), but this notation should be avoided as it can be confused with an exponent notation (power of, raised to the power of). The arccos is used to obtain an angle from the cosine trigonometric ratio, which is the ratio between the side adjacent to the angle and the hypotenuse in a right triangle.

The function spans from -1 to 1, and so do the results from our arccos calculator. The range of the angle values is usually between 0° and 180°. There are a number of arccos rules, like that cos(arccos(x)) = x, or that arccosα + arccosβ = arccos(αβ - √((1-α2)(1-β2)), as well as sine of the arccosine: sin(arccos(x)) = √(1-x2), which can help you in trigonometry calculus.

### 3. Arcsin Calculator

Arcsin is the inverse of the sine. It is normally represented by arcsin(θ) or sin-1(θ).Use arcsine calculator to easily calculate the arcsine of a number. Online arcsine calculation tool with output in degrees or radians. Supports input of decimal numbers (0.5, 6, -1, etc.) and fractions (1/4, 2/3, 4/3, 1/3 etc.).

Arcsine definition

The arcsine function is the inverse function of y = sin(x).

arcsin(y) = sin-1(y) = x + 2kπ

For every

k = {...,-2,-1,0,1,2,...}

For example, If the sine of 30° is 0.5:

sin(30°) = 0.5

Then the arcsine of 0.5 is 30°:

arcsin(0.5) = sin-1(0.5) = 30°

How to use the arcsin (x) calculator?

Enter x as a real number, within the domain of arcsin such that -1 ≤ x ≤ 1, and the number of decimal places desired then press "enter". Two answers are displayed: one in radians and the second in degrees.

What is arcsine?

Arcsine is an inverse of the sine function. In other words, it helps to find the angle of a triangle which has a know value of sine. As sine's codomain for real numbers is [−1, 1] , we can only calculate arcsine for numbers in that interval.

Sine is a periodic function, so there are multiple numbers that have the same sine value. For example sin(0) = 0, but also sin(π) = 0, sin(2π) = 0, sin(-π) = 0 and sin(-326π) = 0. Therefore, if somebody wants to calculate arcsin(0), the answer can be 0, 2π (360°) or -π (-180°), to name a few options! All of them are correct, but we usually only give one number called the principal value.

Arcsin(x) is the most common notation, as sin-1x may lead to confusion (because sin-1x ≠ 1/sin(x) ). The abbreviation asin in usually used in computer programming languages.

Arcsine function :- The arcsine is one of the inverse trigonometric functions (antitrigonometric functions) and is the inverse of the sine function. It is sometimes written as sin-1(x), but this notation should be avoided as it can be confused with an exponent notation (power of, raised to the power of). The arcsine is used to obtain an angle from the sine trigonometric ratio, which is the ratio between the side opposite to the angle and the longest side of the triangle.

The function spans from -1 to 1, and so do the results from our arcsin calculator. The range of the angle values is usually between -90° and 90°. There are a number of arcsin rules, like that sin(arcsin(x)) = x, or that arcsinα + arcsinβ = arcsin(α√(1-β2) + β√(1-α2)), as well as cosine of the arcsine: cos(arcsin(x)) = sin(arccos(x)) = √(1-x2), which can help you in trigonometry calculus.

### 4. Arctan Calculator

The arctan is the inverse of the tangent. It is normally represented by arctan(θ) or tan-1(θ).Use arctan calculator to easily calculate the arctan of a given number. Online arctangent calculation tool to compute the arcus tangens function in degrees or radians. Supports input of decimal numbers (0.5, 6, -1, etc.) and fractions (1/3, 3/4, 1/6, -4/3 etc.).

Arctangent definition

The arctangent function is the inverse function of y = tan(x).

arctan(y) = tan-1(y) = x+ kπ

For every

k = {...,-2,-1,0,1,2,...}

For example, If the tangent of 45° is 1:

tan(45°) = 1

Then the arctangent of 1 is 45°:

arctan(1) = tan-1(1) = 45°

How to Use the Arctan Calculator?

The procedure to use the arctan calculator is as follows:

1. Enter the tangent value in the respective input field

2. Now click the button “Calculate Arctan” to get the angle

3. Finally, the angle in both degree and radian measure will be displayed in the output field

What is arctan?

Arctangent is the inverse of the tangent function. Simply speaking, we use arctan when we want to find an angle for which we know the tangent value.

However, in the strictest sense, because the tangent is a periodic trigonometric function, it doesn't have an inverse function. Still, we can define an inverse function if we restrict the domain to the interval where the function is monotonic. Using the tan-1x convention may lead to confusion about the difference between arctangent and cotangent. It turns out that arctan and cot are really separate things:

1. cot(x) = 1/tan(x), so cotangent is basically the reciprocal of a tangent, or, in other words, the multiplicative inverse

2. arctan(x) is the angle whose tangent is x

We hope that now you do not doubt that arctan and cotan are different. To avoid any further misunderstandings, you may want to use the arctan(x) rather than tan-1x notation

Arctan function :- The arctan (a.k.a. arcus tangens) is one of the inverse trigonometric functions (antitrigonometric functions) and is the inverse of the tangent function. It is sometimes written as tan-1(x), but this notation should be avoided as it can cause confusion with an exponent notation. The arctan is used to obtain an angle from the tangent trigonometric ratio, which is the ratio between the side opposite to the angle and the adjacent side of the triangle.

The function spans all real numbers (-∞ - +∞) and so do the results from our calculator. The range of the angle values is usually between -90° and 90°. There are a number of arctan rules, like that tan(arctan(x)) = x, or that arctanα + arctanβ = arctan((α + β) / (1-αβ)), as well as sine of the arctangent: sin(arctan(x)) = x / √(1+x2), which can help you in trigonometry calculations.

### 5. Cosine Calculator

The cosine calculator is a twin tool of our sine calculator - add to them the tangent tool and you'll have a pack of the most popular trigonometric functions. Simply type the angle - in degrees or radians - and you'll find the cosine value instantly. The cosine calculator helps to find out the cosine value for a specific angle in degrees or radians.

Cosine definition

In a right triangle ABC the sine of α, sin(α) is defined as the ratio betwween the side adjacent to angle α and the side opposite to the right angle (hypotenuse):

cos α = b / c

Example

b = 3"

c = 5"

cos α = b / c = 3 / 5 = 0.6

How to Use the Cosine Calculator?

In order to calculate cos(x) on the calculator:

1. Enter the input angle.

2. Select angle type of degrees (°) or radians (rad) in the combo box.

3. Press the = button to calculate the result.

What is cosine?

Cosine is one of the most basic trigonometric functions. It may be defined on the basis of right triangle or unit circle, in analogical way as the sine is defined:

The cosine of an angle is the length of the adjacent side divided by the length of the hypotenuse.

cos(α) = adjacent / hypotenuse = b / c

Right triangle: illustration of the cosine definition. Adjacent side over a hypotenuse. The cosine function may be defined that way: as the sine of the complementary angle - the other non-right angle. The abbreviation of cosine is cos, e.g. cos(30°).

Cosine function :-The cosine is a trigonometric function of an angle, usually defined for acute angles within a right-angled triangle as the ratio of the length of the adjacent side to the hypotenuse. It is the complement to the sine. In the illustration below, cos(α) = b/c and cos(β) = a/c.

Since cos(α) = b/c, from this definition it follows that the cosine of any angle is always less than or equal to one, and it can take negative values. The cosine of a 90-degree angle is equal to zero, since in order to calculate it we would need a triangle with two 90-degree angles, which is the definition of a straight line. As the third side of the triangle does not exist (length is 0), the cosine equals zero (0 divided by the length of the hypotenuse equals 0). You can use this cosine calculator to verify this.

A commonly used law in trigonometry which is trivially derived from the cosine definition is the law of cosines: c2 = a2 + b2 - 2ab·cosγ

Important properties of a cosine function:

1. Range (codomain) of a cosine is -1 ≤ cos(α) ≤ 1

2. Cosine period is equal to 2π

3. It's an even function (while sine is odd!), which means that cos(-α) = cos(α)

4. Cosine definition is essential to understand the law of cosines - a very useful law to solve any triangle.

### 6. Exponent Calculator

The exponent calculator will calculate the value of any base raised to any power. An exponent is a way to represent how many times a number, known as the base, is multiplied by itself. It is represented as a small number in the upper right hand corner of the base. For example: x² means you multiply x by itself two times, which is x * x. If the exponent is 3, in the example 5³, then the result is 5 * 5 * 5.

Exponent Definition

An exponent is the small number located in the upper, right-hand position of an exponential expression (baseexponent), which indicates the power to which the base of the expression is raised.

The following exponential expression shows how the exponent is used in raising the base:

43 = 4 x 4 x 4 = 64

4 x 4 = 16

4 x 16 = 64

What is an exponent?

Exponentiation is a mathematical operation, written as an, involving the base a and an exponent n. In the case where n is a positive integer, exponentiation corresponds to repeated multiplication of the base, n times.

an = a × a × ... × a

n times

The calculator above accepts negative bases, but does not compute imaginary numbers. It also does not accept fractions, but can be used to compute fractional exponents, as long as the exponents are input in their decimal form.

Basic exponent laws and rules

The exponent formula is:

a n = a×a×...×a

n times

The base a is raised to the power of n, is equal to n times multiplication of a.

For example:

25 = 2×2×2×2×2 = 32

Multiplying exponents

an ⋅ am = an+m

Example: 23 ⋅ 24 = 2(3+4) = 27 = 128

an ⋅ bn = (a ⋅ b) n

Example: 32 ⋅ 42 = (3⋅4)2 = 122 = 144

Dividing exponents

an / am = an-m

Example: 25 / 23 = 2(5-3) = 22 = 4

an / bn = (a / b)n

Example: 82 / 22 = (8/2)2 = 42 = 16

Power of exponent

(an)m = an⋅m

Example: (23)4 = 2(3 ⋅ 4) = 212 = 4096

m√(an) = an/m

Example: 2√(26) = 2(6 / 2) = 23 = 8

Negative exponent

a -n = 1 / a n

Example: 2-3 = 1 / 23 = 1 / 8 = 0.125

Zero exponent

a 0 = 1

Example: 40 = 1

below is an another example of an argument for a0=1 using one of the previously mentioned exponent laws.

If an × am = a(n+m)

Then an × a0 = a(n+0) = an

Thus, the only way for an to remain unchanged by multiplication, and this exponent law to remain true, is for a0 to be 1.

When an exponent is a fraction where the numerator is 1, the nth root of the base is taken.

It is also possible to compute exponents with negative bases. They follow much the same rules as exponents with positive bases. Exponents with negative bases raised to positive integers are equal to their positive counterparts in magnitude, but vary based on sign. If the exponent is an even, positive integer, the values will be equal regardless of a positive or negative base. If the exponent is an odd, positive integer, the result will again have the same magnitude, but will be negative. While the rules for fractional exponents with negative bases are the same, they involve the use of imaginary numbers since it is not possible to take any root of a negative number.

### 7. Factorial Calculator

Instead of calculating a factorial one digit at a time, use this calculator to calculate the factorial n! of a number n. A factorial is a function that multiplies a number by every number below it. For example 5!= 5*4*3*2*1=120. The function is used, among other things, to find the number of way “n” objects can be arranged.

Factorial definition

The factorial, symbolized by an exclamation mark (!), is a quantity defined for all integers greater than or equal to 0.

The factorial formula is:

n! = 1⋅2⋅3⋅4⋅...⋅n

For example:

3! = 1⋅2⋅3 = 6

4! = 1⋅2⋅3⋅4 = 24

5! = 1⋅2⋅3⋅4⋅5 = 120

In mathematics, there are n! ways to arrange n objects in sequence. "The factorial n! gives the number of ways in which n objects can be permuted." For example:

1) 2 factorial is 2! = 2 x 1 = 2

There are 2 different ways to arrange the numbers 1 through 2. {1,2,} and {2,1}.

2) 4 factorial is 4! = 4 x 3 x 2 x 1 = 24

There are 24 different ways to arrange the numbers 1 through 4. {1,2,3,4}, {2,1,3,4}, {2,3,1,4}, {2,3,4,1}, {1,3,2,4}, etc.

3) 5 factorial is 5! = 5 x 4 x 3 x 2 x 1 = 120

4) 0 factorial is a definition: 0! = 1. There is exactly 1 way to arrange 0 objects.

what is a factorial?

When you saw an exclamation point in maths for the first time, you probably got shocked or even thought that there was some kind of mistake or typo. But the reality is different: this exclamation point in maths is called the factorial or n-factorial. The factorial is a reasonably unknown operator that can, in fact, be viewed more as an abbreviation than an actual operator, at least at the beginning.

It is important not to confuse the factorial with the prime factorization of a number, which is a way of obtaining the prime numbers that, when multiplied together, give your number. Prime factorization has its uses in maths and is arguably more well known than the n-factorial. Part of the reason for the popularity of prime factorization is its usefulness when calculating Greatest Common Factor (GCM) and the Least Common Multiplier (LCM), but we digress.

To understand what the factorial does or means, we should start with an example. We could choose any number n and calculate its n-factorial value, but it's best to choose a relatively small number, so let's use 5-factorial.

5! = 5 * 4 * 3 * 2 * 1 = 120

From this example, you can see that it is not rocket science, and you might even take a guess at the factorial formula. You can also understand why this exclamation point in maths can be regarded as an abbreviation since it is not a new operation but rather a collection of multiplications. In short, and somewhat informally, we can define the factorial as the multiplication of all the positive integers smaller than and equal to the given number.

Playing a bit with this, we can see that for 5-factorial, we can relate it to the 4-factorial in a straightforward way:

5! = 5 * (4 * 3 * 2 * 1) = 5 * 4!

This kind of relationship between n-factorials with a different n is the basis of the mathematical formula that defines the factorial operation, as we will see in the next section.

The factorial operation is not used everywhere in maths, but it is essential in statistics and probability problems. In those cases, especially when one has to deal with permutations or combinatorics, the n-factorial appears almost all the time. In the following sections of our factorial calculator, we will see real-world examples of problems that require the usage of factorials and the factorial formula.

### 8. Natural Logarithm Calculator

The Natural Log Calculator is used to calculate the natural logarithm of a number x, which is generally written as ln(x) or loge(x).The natural log calculator determines the logarithm to the base of a famous mathematical constant, e, an irrational number with an approximate value of e = 2.71828. But, what is the natural logarithm, ln x, of a given number x? This is the power the number e has to be raised to in order to result in a given number x.

Natural logarithm definition

The natural logarithm is the logarithm to the base e (Euler's number, approximately equal to 2.718281828). It is generally written as ln(x), loge(x) or sometimes, if the base of e is implicit, as simply log(x). It is often used in mathematical analysis, physics, chemistry, statistics, economics, and some engineering fields.

The natural logarithm of x is the base e logarithm of x:

ln x = loge x = y

ln(x) = y

x: is real number, x>0

Natural logarithm symbol is ln

ln(x) = y

ln(x) is equivalent of loge(x)

How to use the natural logarithm calculator

Like all other logarithms, the natural logarithm of x returns the power, or exponent, to which a given base e must be raised to yield back the number x. It is easier to understand this notion when the base is an integer, for example 2 or 3:

log2 16 = 4 since 24 = 16

log3 81 = 4 since 34 = 81.

In the case of the natural logarithm, this is somewhat less intuitive, because it's base, e, is not an integer. But, since the value of e is between values of 2 and 3, we understand that e4 has to be somewhere between 24 = 16 and 34 = 81.

It turns out that e4 = 54.498. This equality can be stated in terms of the natural logarithm in the following way, which you can check by using the ln calculator:

ln 54.498 = 4

Here are some examples of Natural Logarithm Examples

ln(2) = loge(2) = 0.6931

ln(3) = loge(3) = 1.0986

ln(4) = loge(4) = 1.3862

ln(5) = loge(5) = 1.609

ln(6) = loge(6) = 1.7917

ln(10) = loge(10) = 2.3025

Check out the correctness of the above results by using our natural log calculator and the exponent calculator.

### 9. Logarithm Calculator

The Log (Logarithm) Calculator is used to calculate the logarithm logbx for a base b and a number x. Logarithm (LOG) calculator is an online math calculator that calculates the log value for the positive real number with respect to the given or natural base values (positive, not equal to 1). Using this calculator, we will understand methods of how to find the logarithm of any number with respect to the given base.

It is necessary to follow the next steps:

1.Enter the number and the base of logarithm. These values must be positive real numbers or parameter. The base of logarithm can not be 1.

2. Press the "CALCULATE" button to make the computation.

3. Logarithm calculator will give the logarithm of the positive real number number with the positive base not equal to 1.

Logarithm definition

The logarithm of a number x with respect to base b is the exponent to which b has to be raised to yield x. In other words, the logarithm of y to base b is the solution y of the following equation:

by = x

And for any x and b, there is:

x = logbbx

The logarithm to base b = 10 is called the common logarithm and has many applications in science and engineering. The natural logarithm has the constant e (approximately equal to 2.718281828) as its base. The binary logarithm uses base b = 2 and is prominent in computer science.

What is Log?

The logarithm, or log, is the inverse of the mathematical operation of exponentiation. This means that the log of a number is the number that a fixed base has to be raised to in order to yield the number. Conventionally, log implies that base 10 is being used, though the base can technically be anything. When the base is e, ln is usually written, rather than loge. log2, the binary logarithm, is another base that is typically used with logarithms. If for example:

x = by; then y = logbx; where b is the base

Each of the mentioned bases are typically used in different applications. Base 10 is commonly used in science and engineering, base e in math and physics, and base 2 in computer science.

Logarithm rules

Logarithm product rule

logb(x × y) = logb(x) + logb(y)

Logarithm quotient rule

logb(x / y) = logb(x) - logb(y)

Logarithm power rule

logb(x y) = y × logb(x)

Logarithm base switch rule

logb(c) = 1 / logc(b)

Logarithm change of base rule

logb(x) = logc(x) / logc(b)

### 10. Percentage Calculator

Percentage calculator is a tool that lets you do a simple calculation: what percent of X is Y? The tool is pretty straightforward. All you need to do is fill in two fields, and the third one will be calculated for you automatically. This method will allow you to answer the question of how to find a percentage of two numbers.

Percentage definition

A percentage (%) is a fraction expressed as a part of one hundred, instead of any other denominator. The word comes from the Latin per cent, meaning ‘out of one hundred’.

In mathematics, a percentage is a number or ratio that represents a fraction of 100. It is often denoted by the symbol "%" or simply as "percent" or "pct." For example, 35% is equivalent to the decimal 0.35, or the fraction 35/100.

How to Calculate Percentages

There are many formulas for percentage problems. You can think of the most basic as X/Y = P x 100. The formulas below are all mathematical variations of this formula.

Let's explore the three basic percentage problems. X and Y are numbers and P is the percentage:

1. Find P percent of X

2. Find what percent of X is Y

3. Find X if P percent of it is Y

1. How to calculate percentage of a number. Use the percentage formula: P% * X = Y

Example: What is 10% of 150?

1. Convert the problem to an equation using the percentage formula: P% * X = Y

2. P is 10%, X is 150, so the equation is 10% * 150 = Y

3. Convert 10% to a decimal by removing the percent sign and dividing by 100: 10/100 = 0.10

4. Substitute 0.10 for 10% in the equation: 10% * 150 = Y becomes 0.10 * 150 = Y

5. Do the math: 0.10 * 150 = 15

6. Y = 15

7 So 10% of 150 is 15

Check your answer with the original question: What is 10% of 150? Multiply 0.10 * 150 = 15

2. How to find what percent of X is Y. Use the percentage formula: Y/X = P%

Example: What percent of 60 is 12?

1. Convert the problem to an equation using the percentage formula: Y/X = P%

X is 60, Y is 12, so the equation is 12/60 = P%

2. Do the math: 12/60 = 0.20

The result will always be in decimal form, not percentage form. You need to multiply the result by 100 to get the percentage.

3. Converting 0.20 to a percent: 0.20 * 100 = 20%

4. So 20% of 60 is 12.

Check your answer with the original question: What percent of 60 is 12? 12/60 = 0.20, and multiplying by 100 to get percentage, 0.20 * 100 = 20%

3. How to find X if P percent of it is Y. Use the percentage formula Y/P% = X

Example: 25 is 20% of what number?

1. Convert the problem to an equation using the percentage formula: Y/P% = X

2. Y is 25, P% is 20, so the equation is 25/20% = X

3. Convert the percentage to a decimal by dividing by 100.

4. Converting 20% to a decimal: 20/100 = 0.20

5. Substitute 0.20 for 20% in the equation: 25/0.20 = X

6. Do the math: 25/0.20 = X

7. X = 125

So 25 is 20% of 125

Check your answer with the original question: 25 is 20% of what number? 25/0.20 = 125

This online calculator is a quadratic equation solver that will solve a second-order polynomial equation such as ax2 + bx + c = 0 for x, where a ≠ 0, using the quadratic formula.

The calculator solution will show work using the quadratic formula to solve the entered equation for real and complex roots. Calculator determines whether the discriminant (b2−4ac) is less than, greater than or equal to 0.

When b2−4ac=0 there is one real root.

When b2−4ac>0 there are two real roots.

When b2−4ac<0 there are two complex roots.

A quadratic is a polynomial of degree two. The quadratic formula is the solution of a second degree polynomial equation of the following form:

Ax² + Bx + C = 0

If you can rewrite your equation in this form, it means that it can be solved with the quadratic formula. A solution to this equation is also called a root of an equation.

The quadratic formula is as follows:

x = (-B ± √Δ)/2A

where:

Δ = B² - 4AC

Using this formula, you can find the solutions to any quadratic equation. Note that there are three possible options for obtaining a result:

1. The quadratic equation has two unique roots when Δ > 0. Then, the first solution of the quadratic formula is x₁ = (-B + √Δ)/2A, and the second is x₂ = (-B - √Δ)/2A.

2. The quadratic equation has only one root when Δ = 0. The solution is equal to x = -B/2A. It is sometimes called a repeated or double root.

3. The quadratic equation has no real solutions for Δ < 0.

You can also graph the function y = Ax² + Bx + C. It's shape is a parabola, and the roots of the quadratic equation are the x-intercepts of this function.

### 12. Radicals and Roots Calculator

This calculator will find the given root of real numbers.If you want to find the radical numbers of any nᵗʰ degree radical, then this radical calculator is just for you.

Examples:

The 4th root of 81, or 81 radical 3, is written as 4√81=±3.

The 3rd root, or cube root, of 8 is written as 3√8=2

The 3rd root of -27, or -27 radical 3, or the cube root of -27 is written as 3√−27=−3.

The 5th root of 1024, or 1024 radical 5, is written as 5√1024=4.

In general, the radical function is a root of a number x for which we use the radical symbol ⁿ√, with the superscript n describing the degree of the root. The radical definition is:

rⁿ = x,

and therefore:

r = x1/n,

where r is the result of the radical. So, in other words, radicalization is the opposite operation to exponentiation, for which the exponent is a fraction. To make writing the radicals easier, we use the radical notation:

r = ⁿ√x,

which means the same thing as the above equation.

It turns out that you cannot calculate the root of any number. What are the conditions for radical numbers then?

In general, the number you want to root should be positive, although it is possible to find the answer when the degree is odd, e.g., when n = 3 (³√x - cube root), as ³√8 = 2 and -2.

Secondly, the degree of the radical is usually a positive integer number. Technically, you can use fractions, but in that case, the exponentiation is a better way to present the computation.

n - should be a positive integer; and

x - should be any positive number.

How to use the radical calculator?

The radical calculator consists of three fields that should be self-explanatory. Fill in any two boxes and we will estimate the third one automatically:

1. Number - that's the number for which you'd like to find the radical. We decided not to allow entering negative values here even if the radical degree is odd. That's because, in most cases, you would obtain multiple answers (like in the quadratic formula), including complex numbers, which we wanted to avoid in this calculator. If you insist on negative numbers, then check our root calculator (a very similar tool to this radical calculator, but which allows negative numbers) or complex root calculator dedicated to finding radicals of any number.

2. Radical degree (n) - as we mentioned before, n should be a positive integer. However, since you can treat a radical function as an exponent, technically, it's possible to calculate the radical for other degrees. You can input non-integer values as well as negative numbers if you want so.

3. Radical (nᵗʰ radical) - the nᵗʰ radical is probably the reason you came here. We display a more advanced result under this field, i.e., with step-by-step radical simplification.

### 13. Sine Calculator

Sine Calculator is a free online tool that displays the sine angle of the right triangle. Online sine calculator tool makes the calculation faster and it displays the sine angle in a fraction of seconds.

In order to calculate sin(x) on the calculator:

1. Enter the input angle.

2. Select angle type of degrees (°) or radians (rad) in the combo box.

3. Press the = button to calculate the result.

What is Sine function?

Sine is one of the three most common trigonometric functions (others are cosine and tangent, as well as secant, cosecant and cotangent). The abbreviation of sine is sin e.g. sin(30°). Most common and well known sine definition is based on right-angled triangle. Let's start with nomenclature of triangle sides which will be useful in further steps. As the picture on the right shows, we can name the sides of a right triangle as:

1. adjacent side - the shorter adjacent side of the angle of interest (in this case, angle α). This side is adjacent to both the angle of interest and the right angle.

2. opposite side - merely the side opposite to the angle of interest.

3. hypotenuse - the opposite side to the right angle, it's always the longest side in the right triangle.

### 14. Square Root Calculator

Square root calculator helps you find the square root of any positive number.In some situations, you don't need to know the exact result of the square root. If this is the case, our square root calculator is the best option to estimate the value of every square root you desired. For example, let's say you want to know whether 4√5 is greater than 9. From the calculator, you know that √5 ≈ 2.23607, so 4√5 ≈ 4 * 2.23607 = 8.94428. It is very close to the 9, but it isn't greater than it! The square root calculator gives the final value with relatively high accuracy.

Square root definition

In mathematics, the traditional operations on numbers are addition, subtraction, multiplication, and division. Nonetheless, we sometimes add to this list some more advanced operations and manipulations: square roots, exponentiation, logarithmic functions and even trigonometric functions (e.g., sine and cosine). In this article, we will focus on the square root definition only.

The square root of a given number x is every number y whose square y² = y*y yields the original number x. Therefore, the square root formula can be expressed as:

√x = y ⟺ x = y²,

where ⟺ is a mathematical symbol that means if and only if. Each positive real number always has two square roots - the first is positive and second is negative. However, for many practical purposes, we usually use the positive one. The only number that has one square root is zero. It is because √0 = 0 and zero is neither positive nor negative.

There is also another common notation of square roots that could be more convenient in many complex calculations. This alternative square root formula states that the square root of a number is a number raised to the exponent of the fraction one half:

√x = x^(1/2) = x^(0.5)

In geometric interpretation, the square root of a given area of a square gives the length of its side. That's why √ has word square in its name. A similar situation is with the cube root ∛. If you take the cube root of the volume of a cube, you get the length of its edges. While square roots are used when considering surface areas, cube roots are useful to determine quantities that relate to the volume, e.g., density.

### 15. Tangent Calculator

This tangent calculator will help you to find the tangent of any angle you want. Just enter the angle in degrees or radians, and the tangent will appear in no time.

Tangent function

The tangent is a trigonometric function, defined as the ratio of the length of the side opposite to the angle to the length of the adjacent side, in a right-angled triangle. It is called "tangent" since it can be represented as a line segment tangent to a circle.

In the graph above, tan(α) = a/b and tan(β) = b/a. A tangent of an angle α is also equal to the ratio between its sine and cosine, so tanα = sinα / cosα. Following from the definition, the function results in an undefined value at certain angles, like 90°, 270°, 460°, and so on.

How to calculate the tangent of an angle?

Tangent calculator accepts input in degrees or radians, so assuming the angle is known, just type it in and press "calculate". Easy as that.

If the angle is unknown, but the lengths of the opposite and adjacent side in a right-angled triangle are known, then the tangent can be calculated from these two measurements. For example, if a = 15 and b = 20, then tan(α) = 15 / 20 = 0.75.

Applications of the tangent function:-

The tangent function is used in measuring height of objects located at known distances and has application in flight path and altitude gain calculations. In engineering, it is used to calculate forces of supporting structures, like roof beams. They are also used in robotics to calculate robot arm kinematics.

Law of tangents

The law of tangents describes the relationship between the tangent of two angles of a triangle and the opposite sides lengths. Specifically, it states that:

(a - b) / (a + b) = tan(0.5(α - β)) / tan(0.5(α + β))

Although the law of tangents is not so popular as law of sines or law of cosines, it may be useful when we have given:

two sides and the angle between them;

two angles and a side.

Tangent calculator - example of use

All you need to do is type one value into the calculator - the angle, in radians or degrees. To change between units, click on the unit name and choose from a drop-down list. Then, you'll see the result immediately - tangent value of your angle of interest. Remember that tan may be undefined. This situation occurs when cos(α) is 0, because we can never divide by zero (other explanation: the lines will be parallel, so they'll never cross each other and they won't form the point C).

Adding fractions calculator helps you evaluate the sum of up to five fractions in the blink of an eye. In maths, fractions represent a part of a whole thing. Mathematically, a fraction can be written in the form of a/b, where a is the numerator and b is the numerator. While performing the comparison of fractions, we may come across different types of fractions such as like and unlike fractions, etc. If the denominators of fractions are different, then they are said to be unlike fractions.

A fraction is a number formed from the ratio of two numbers (A and B). These numbers are typically integers (whole numbers), so that:

fraction = A/B.

It turns out you can convert any decimal to a fraction. As a result, you can add as many decimals as you like together by treating them as if they were fractions. Whenever we want to add two fractions with a common denominator, let's say 2/7 and 3/7, we need to add numerators, while the denominator remains the same:

2/7 + 3/7 = (2+3)/7 = 5/7.

When the numbers have different denominators, the addition of fractions is a bit more challenging, as you can't just add the numerators like before. The trick is to use common denominators. Let's see how it works. Let's say we want to add 1/2 and 1/3:

1) 1/2 + 1/3 = …

2) Find the common denominator. To do so, we can estimate the least common multiple (LCM) of 2 and 3. LCM(2,3) = 6

3) Expand each fraction so that the denominator is this LCM, in this case 6: 1/2 = 3/6, 1/3 = 2/6

4) As we know how to add fractions with the same denominator, you can just add these fractions normally: … = 3/6 + 2/6 = 5/6

There are also other equivalent fractions to this result, such as 10/12, 15/18, to name a few. However, it's convenient to present the result in its simplest form.

Addition of fractions in practice - how to use adding fractions calculator?

Imagine a story - you are at a party with some of your friends. And there it comes - the hunger! What's even worse, you've just realized there is nothing left in the fridge.

A solution is simple - you are going to order a pizza, or two, or even more. You have to decide! The point is your favorite pizzeria sells pizzas as a whole, but you have an innovative method that can cut pizzas into 6, 8, or 12 slices. Everyone wants a certain fraction of a pizza: five of you want 4 of 6-slice pizzas, four of you prefer 3 of the 8-slice pizzas, and the remaining three will be happy with 6 of 12-slice ones. The main question is: How many pizzas should we order?

You can always evaluate it by hand, but why don't you try our adding fractions calculator and saving your time! Input the following values:

5*4 and 6 for the first number

4*3 and 8 for the second number

3*6 and 12 for the third one

The outcome is 19/3, or 6 1/3, as a mixed number. It means that six pizzas won't be enough, so you'd better order seven! Moreover, you can choose the step by step solution to see all the calculations with explanations. You are all welcome to read it while enjoying your delicious meal!

### 17. Antilog Calculator

The antilog calculator determines the value of an antilogarithm with an arbitrary base.In order to calculate the inverse function log-1(y) on the calculator, enter the base b (10 is the default value, enter e for e constant), enter the logarithm value y and press the = or calculate button. The Antilog Calculator is used to calculate the antilogarithm antilogb(y) for a base b and a logarithm y.

The antilogarithm is simply the exponentiation! To calculate an antilog of any number y, you need to raise the logarithm base b (usually 10, sometimes e), to the power of y:

x = logb-1(y) = by

Since log and antilog are inverse functions, then this means that x = by = blogbx, and y = logbx = logb(by).

How do you find the antilog of a number?

1. Decide on the base of your calculation. Regular numbers are base 10.

2. Choose the number you wish to find the antilog of. In our example we will choose 2.

3. Raise the number to the base, bx. In our example the calculation is 102.

What are the graphical characteristics of an antilog?

The antilog formula for base 10, y = 10x, will have the following graphical characteristics. As x approaches minus infinity, y will tend towards 0, but never actually touch the x axis. As x approaches infinity, y will also tend towards infinity, but at an exponential pace. The y-intercept is at y = 1. These graphical properties hold true for any positive, non-zero base.

How do you remove log and antilog?

As log and antilog are inverse functions, you should use one to remove the other. For example, to remove a log, raise both sides of the formula to the base of the log (the antilog). To remove antilog, find the log of both sides with a base equal to the base of the indices.

### 18. Average Calculator

The Average Calculator is used to calculate the average value of any set of numbers.The term average has a number of different meanings. Most generally, it is a single number that is used to represent a collection of numbers. In the context of mathematics, "average" refers to the mean, specifically, the arithmetic mean. It is a relatively simple statistical concept that is widely used in many areas.

The average (arithmetic mean) is equal to the sum of the n numbers divided by n:

Average = sum / count = (a1+a2+...+an) / n

Example

The average of 1,2,5 is:

Average = (1+2+5) / 3 = 2.667

How to Calculate Average?

The average of a set of numbers is simply the sum of the numbers divided by the total number of values in the set. For example, suppose we want the average of 24,55, 17, 87 and 100. Simply find the sum of the numbers: 24 + 55 + 17 + 87 + 100 = 283 and divide by 5to get 56.6. A simple problem such as this one can be done by hand without too much trouble, but, for more complex numbers involving many decimal places, it is more convenient to use this calculator. Note that the mean rating calculator does a similar math - it calculates an average rating given the number of votes with values from 1 to 5.

What Does the Term ‘Average’ Mean?

When people describe the ‘average’ of a group of numbers, they often refer to the arithmetic mean. This is one out of 3 different types of average, which include median and mode.

In conversational terms, most people just say ‘average’ when they’re really referring to the mean. Arithmetic mean and average are synonymous words which are used interchangeably.

It’s calculated by adding the numbers in a set and dividing it by the total number in the set—which is what most people do when they’re finding the average. See the example below.

Mean

Set: 8, 12, 9, 7, 13, 10

Mean = (8 + 12 + 9 + 7 + 13 + 10) / 6

= 59 / 6

= 9.83

The average or arithmetic mean in this example is 9.83.

Median

The median, on the other hand, is another type of average that represents the middle number in an ordered sequence of numbers. This works by ordering a sequence of numbers (in ascending order) then determining the number which occurs at the middle of the set. See the example below.

Average Median

Set: 22, 26, 29, 33, 39, 40, 42, 47, 53

In this example, 39 is the median or middle value in the set.

Mode

The mode is basically the most frequent value that repeats itself in a set of values. For instance, if your set has 21, 9, 14, 3, 11, 33, 5, 9, 16, 21, 5, 9, what is the mode?

The answer is 9 because this value is repeated 3 times.

In statistics, mean, median, and mode are all terms used to measure central tendency in a sample data.

Is average better than mode?

There is no easy answer to whether the average is better than the mode - it depends entirely on the data set in front of you. If the data is normally distributed, has no outliers, then you should probably use the average, as it will present you with the most representative value. The mode, however, is more robust, and will present the most common value, regardless of any outliers. The mode should always be used with categorical data - that is, data with distinct groups - as the groups are not continuous.

What is better, average or median?

Whether you should use the average or the median will depend on the data you are analysing. If the data is normally distributed, has no outliers, then you should probably use the average, although the value will be quite similar to that for the median. If the data is heavily skewed, the median should be used as it is less effected by outliers.

### 19. Base Calculator

This calculator is concerned only with changing numbers into different bases and no attempt is made to explain what these numbers are and how they work.Enter the value that you wish to calculate the base number or you can easy convert one base to another base on base converter.

The base calculator supports to add, subtract, divide, & multiply any base numeral system. It supports 36 different base numbers, including binary, decimal, octal, hex, base 16, base 20 & more. Our number base calculator concerned to perform addition, subtraction, multiplication, & division operations.

To perform binary addition there are some rules that need to follow. Below we have listed out addition rule. If we get carried we need forward it to next digit.

0+0=0

1+0=1

0+1=1

1+1=0, 1 is carried forward to the next digit.

Binary Subtraction:

In the same way for subtraction we have 0-0=0

1-0=1

0-1=1, -1 is carried forward to the next digit.

1-1=0

Binary multiplication:

1x0=0

0x1=0

1x1=1

How to use base calculator?

Open Base calculator.

Select the type of number you need to perform operations.

Enter the two numbers and select the operator.

Click on calculate to get the result.

### 20. Binary Calculator

The Binary Converter is used to convert numbers from binary to decimal, octal, hexadecimal and other bases.It can add, subtract, multiply, or divide two binary numbers. It can operate on very large integers and very small fractional values — and combinations of both.

Binary Numeral System

In mathematics and computer science, binary is a positional numeral system with a base of 2. It represents numeric values using two symbols, 0 and 1. The binary numeral system is used in computer science because it is convenient to represent the two digits 0 and 1 in electrical circuits.

The binary system is a numerical system that functions virtually identically to the decimal number system that people are likely more familiar with. While the decimal number system uses the number 10 as its base, the binary system uses 2. Furthermore, although the decimal system uses the digits 0 through 9, the binary system uses only 0 and 1, and each digit is referred to as a bit. Apart from these differences, operations such as addition, subtraction, multiplication, and division are all computed following the same rules as the decimal system.

Almost all modern technology and computers use the binary system due to its ease of implementation in digital circuitry using logic gates. It is much simpler to design hardware that only needs to detect two states, on and off (or true/false, present/absent, etc.). Using a decimal system would require hardware that can detect 10 states for the digits 0 through 9, and is more complicated.

### 21. Convolution Calculator

convolution calculator combines two data sequence into a single data sequence. Enter the data sequences into its appropriate position and click on calculate to get unique single data sequence.Convolution is a mathematical operation, which applies on two values say X and H and gives a third value as an output say Y.

In convolution, we do point to point multiplication of input functions and gets our output function.

Convolution function:

A mathematical function performs on two functions to produce the third function. Convolution is a combination of result function and computing process.

If convolution applied on two functions x(n), h(n) then the result function is y(n), then y(n) = x(n)*h(n).

Convolution Formulas:

The sequence y(n) is equal to the convolution of sequences x(n) and h(n):

y(n) = x(n)*h(n) = ∑ x(k)h(n-k)

For finite sequences x(n) with M values and h(n) with N values.

y(n) = ∑ x(n+k)h(N-1-K) where n = 0 .. M+N-2

How to use the convolution calculator?

To use our calculator you need to follow below steps.

1. Open convolution calculator.

2. Enter the two data sequence or functions.

3. Click on calculate to get the third sequence

### 22. Dividing Fractions

This calculator divides two fractions. It accepts proper, improper, mixed fractions and whole number inputs. If they exist, the solutions and answers are provided in simplified, mixed and whole formats.

How to dividing fractions calculator is used?

Enter fractions and press the = button.

Enter simple fractions with slash (/).

For example: 1/2 ÷ 1/3

Enter mixed numbers with space.

For example: 21/2 ÷ 11/3

What is Meant by Dividing Fractions?

In mathematics, the fraction represents the parts of the whole. It is defined as the ratio of numerator value to the denominator value. The number above the slash “/” symbol is called the numerator, and the below the slash symbol is called the denominator. The fraction is of different types, namely proper fraction, improper fraction, mixed fraction, like fraction, unlike fractions and so on. If a/b and c/d are the two fractions, then the division of two fractions is given by

Where a, b, c and d are integers.

Dividing Mixed Numbers using the Dividing Fractions Formula

1. Convert the mixed numbers to improper fractions

2. Use the algebraic formula for division of fractions: a/b ÷ c/d = ad / bc

3. Reduce fractions and simplify if possible

How to Divide Fractions

1. Rewrite the equation as in "Keep, Change, Flip"

2. Keep the first fraction

3. Change the division sign to multiplication

4. Flip the second fraction by switching the top and bottom numbers

5. Multiply all numerators together

6. Multiply all denominators together

7. Reduce the result to lowest terms

### 23. Division Calculator

Division Calculator is a free online tool that displays the quotient and remainder for the division operation. Online division calculator tool makes the calculation faster, and it displays the result of the division process in a fraction of seconds.

How to use Online division calculator?

Divide 2 numbers and find the quotient.

Enter dividend and divisor numbers and press the = button to get the division result:

What Are the Parts of Division

For the division sentence 487 ÷ 32 = 15 R 7

487 is the dividend

32 is the divisor

15 is the quotient part of the answer

7 is the remainder part of the answer

What is Meant by the Division?

In Mathematics, the basic arithmetic operations are addition, subtraction, multiplication and division. For any mathematical operation, it needs operands and the operator. In the division process, the two operands are dividend and divisor. The operator is a division (÷). A dividend is a number which is being divided. A divisor is a number that divides the dividend. The result is called the quotient. Also, a remainder is a part of the dividend which is left over after the division operation. For example, 11/2

11/2 = 2(5) + 1

Here,

11 is the dividend

2 is the divisor

### 24. Exponential Growth and Decay

The Exponential Growth/ Decay Calculator is used to solve exponential growth/decay problems. It will calculate any one of the values from the other three in the exponential decay model equation. One of the most prevalent applications of exponential functions involves growth and decay models. Exponential growth and decay show up in a host of natural applications. From population growth and continuously compounded interest to radioactive decay and Newton’s law of cooling, exponential functions are ubiquitous in nature.

Exponential growth/decay formula

x(t) = x0 × (1 + r)t

x(t) is the value at time t.

x0 is the initial value at time t=0.

r is the growth rate when r > 0 or decay rate when r < 0, in percent.

t is the time in discrete intervals and selected time units.

How to calculate exponential growth

Consider the following problem: the population of a small city at the beginning of 2019 was 10,000 people. It was noticed that the population of the city grows at a steady rate of 5% annually. What should you do to calculate the projected population size in the year 2030? From the given data, we can conclude the initial population value, x0, equals 10,000. Also, we have the growth rate of r = 5%.

Therefore, the exponential growth formula we should use is:

x(t) = 10,000 * (1 + 0.05)t = 10,000 * 1.05t.

Here t is the number of years passed since 2019. In our case, for the year 2030, we should use t = 11, since this is the difference in the number of years between 2030 and the initial year 2019. Finally, we get:

x(11) = 10,000 * 1.0511 = 17,103.

So, the projected number of inhabitants of our small city in the year 2030 is around 17,103.

Example on how to use the formula for exponential decay

Radioactive decay is a well-known example of where the exponential decay formula is used. For a given initial quantity of radioactive substance, you may write down the law which governs its decay over time. But, maybe a more fun example is to measure how much coffee remains in your body at 10 pm if you drank a cup of coffee with x0 = 95 mg of caffeine at noon.

We will use the fact that the half-life of caffeine in the human body is approximately six hours. Half-life is defined as the time needed a given quantity to reduce to half of its initial value. So, in this example we have

x(6) = 1/2 * 95 mg = 47.5 mg.

Here, it will be easier to use the alternative notation for the exponential growth formula:

x(t) = 95 * ek*t.

### 25. Fraction Calculator

Fraction Calculator is a free online tool that displays the arithmetic operation for the given fraction. Online fraction calculator tool makes the calculation faster, and it displays the arithmetic operations in a fraction of seconds.The calculator performs basic and advanced operations with fractions, expressions with fractions combined with integers, decimals, and mixed numbers.

How to Use the Fraction Calculator?

The procedure to use the fraction calculator is as follows:

1. Enter the fractions and the arithmetic operator in the respective input field

2. Now click the button “Submit” to get the result

3. Finally, the result for the arithmetic operation of fractions will be displayed in the new window

What is Meant by Fraction?

In maths, a fraction represents a part of the whole number. It consists of a numerator and denominator where numerator represents the number of equal parts and denominator represents the total amount that makes up a whole.

For example, 5/6 is a fraction where 5 is the numerator and 6 is the denominator. Fractions can be added, subtracted, multiplied and divided like all other numbers.

By using the algebraic formula for addition, subtraction, multiplication and division of fractions, this calculator will add, subtract, multiply or divide fractions and gets you an answer in a reduced fraction form.

Subtraction:

Multiplication:

(a/b) . (c/d) = ac/bd

Division:

How to Manually Calculate Fractions

Find the lowest common denominator by multiplying each denominator by the other.

Multiply each numerator by the same numbers the denominators were multiplied by.

Reduce result to most simplified number.

How to Subtract Fractions

Find the lowest common denominator by multiplying each denominator by the other.

Multiply each numerator by the same numbers the denominators were multiplied by.

Add the second numerator from the first.

Reduce result to most simplified number.

How to Multiply Fractions

Multiply the numbers across the top together.

Multiply the numbers across the bottom together.

Reduce result to most simplified number.

How to Divide Fractions

Turn the second fraction upside down to create the reciprocal number.

Multiply the fractions together (just like in the multiplication section above).

Reduce result to most simplified number.

### 26. Greatest Common Factor

The GCF calculator evaluates the Greatest Common Factor between two to six different numbers. The greatest common factor (GCF or GCD or HCF) of a set of whole numbers is the largest positive integer that divides evenly into all numbers with zero remainder. For example, for the set of numbers 18, 30 and 42 the GCF = 6.

What is GCF?

The Greatest Common Factor definition is the largest integer factor that is present between a set of numbers. It is also known as the Greatest Common Divisor, Greatest Common Denominator (GCD), Highest Common Factor (HCF), or Highest Common Divisor (HCD). This is important in certain applications of mathematics such as simplifying polynomials where often it's essential to pull out common factors. Next, we need to know how to find the GCF.

How to Find the Greatest Common Factor

There are various methods which help you to find GCF. Some of them are child's play, while others are more complex. It's worth knowing all of them so you can decide which you prefer:

Using the list of factors,

Prime factorization of numbers,

Euclidean algorithm,

Binary algorithm (Stein's algorithm),

Using multiple properties of GCF (including Least Common Multiple, LCM).

The good news is that you can estimate the GCD with simple math operations, without roots or logarithms! For most cases they are just subtraction, multiplication, or division.

Prime Factorization Method

There are multiple ways to find the greatest common factor of given integers. One of these involves computing the prime factorizations of each integer, determining which factors they have in common, and multiplying these factors to find the GCD. Refer to the example below.

EX: GCF(16, 88, 104)

16 = 2 × 2 × 2 × 2

88 = 2 × 2 × 2 × 11

104 = 2 × 2 × 2 × 13

GCF(16, 88, 104) = 2 × 2 × 2 = 8

Prime factorization is only efficient for smaller integer values. Larger values would make the prime factorization of each and the determination of the common factors, far more tedious.

Euclidean Algorithm

Another method used to determine the GCF involves using the Euclidean algorithm. This method is a far more efficient method than the use of prime factorization. The Euclidean algorithm uses a division algorithm combined with the observation that the GCD of two integers can also divide their difference. The algorithm is as follows:

GCF(a, a) = a

GCF(a, b) = GCF(a-b, b), when a > b

GCF(a, b) = GCF(a, b-a), when b > a

In practice:

1. Given two positive integers, a and b, where a is larger than b, subtract the smaller number b from the larger number a, to arrive at the result c.

2. Continue subtracting b from a until the result c is smaller than b.

3. Use b as the new large number, and subtract the final result c, repeating the same process as in Step 2 until the remainder is 0.

4. Once the remainder is 0, the GCF is the remainder from the step preceding the zero result.

EX: GCF(268442, 178296)

268442 - 178296 = 90146

178296 - 90146 = 88150

90146 - 88150 = 1996

88150 - 1996 × 44 = 326

1996 - 326 × 6 = 40

326 - 40 × 8 = 6

6 - 4 = 2

4 - 2 × 2 = 0

From the example above, it can be seen that GCF(268442, 178296) = 2. If more integers were present, the same process would be performed to find the GCF of the subsequent integer and the GCF of the previous two integers. Referring to the previous example, if instead the desired value were GCF(268442, 178296, 66888), after having found that GCF(268442, 178296) is 2, the next step would be to calculate GCF(66888, 2). In this particular case, it is clear that the GCF would also be 2, yielding the result of GCF(268442, 178296, 66888) = 2.

### 27. GPA Calculator

The calculator can be used to determine the minimum GPA required in future courses to raise GPA to a desired level or maintain the GPA above a certain level.

GPA calculation :- The GPA is calculated as a weighted average of the grades, when the number of credit/hours is the weight and the numeric grade is taken from the GPA table.

The GPA is equal to the sum of the product of the credit hours weight (w) times the grade (g):

GPA = w1×g1+ w2×g2+ w3×g3 + ... + wn×gn

The credit hours weight (wi) is equal to the credit hours of the class divided by the sum of the credit hours of all the classes:

wi= ci / (c1+c2+c3+...+cn)

Letter grade and the numerical equivalents used for this calculator

Grade point average (GPA) is a commonly used indicator of an individual's academic achievement in school. It is the average of the grades attained in each course, taking course credit into consideration. Grading systems vary in different countries, or even schools. This calculator accepts letter grades as well as numerical inputs. These letter grades are translated into numerical values as shown below.

P (pass), NP (not pass), I (incomplete), W (withdrawal) will be ignored.

Most schools, colleges, and universities in the United States use a grading system based on the letters above, though E is sometimes used instead of F. Grading systems do differ however based on what constitutes an A or B, and some do not include grades such as an A+ or a B-. Others may attribute more weight to certain courses, and thus whatever grade is attained in the course will have a larger effect on overall GPA.

Guidelines for raising GPA

There is no sure formula for raising a person's GPA, and strategies that work for one person may not work for another. However, there are some common guidelines and study habits that can be helpful when trying to raise GPA. The guidelines below are mostly anecdotal and are not intended as fail-safe ways to raise one's GPA, but are generally good habits that can have positive effects on learning, which may in turn increase GPA.

Actively attending classes:

Classes are being paid for likely either by a student or their parent, and not attending classes is both a financial loss, as well as a loss in potential education. While a student may decide that attending a particular class is not beneficial to their learning, or not a good use of their time, even if the professor is largely ineffective, there is usually valuable information that can be obtained simply by attending class. Not attending class for example, could result in negative effects on a student's GPA if for some reason the student misses information about a change in exam location or material.

Furthermore, while it may be true that professors largely repeat notes in class that are often later posted to a website, skipping classes can result in missed opportunities. Questions from students in class, as well as the explanations that may follow can provide seemingly inconsequential bits of information that can in fact make a large difference on tests. This is because interaction with the professor and other students can increase a person's depth of knowledge on a subject, or may provide the small tip necessary to solidify a student's understanding of a topic.

In addition, attending class, particularly if the class is smaller in size, can allow the professor to link a name, a face, and a grade, particularly if the student actively participates. Professors that see attentive and involved students are more inclined to be understanding of any potential issues that may arise such as emergencies resulting in missed due dates. Along with this, active participation is more likely to engage a student's mind in regards to the subject matter than reading online notes or a textbook, and points of confusion can also be clarified on the spot. These can in turn affect a person's grade and overall GPA.

Planning:

Every student has his or her own learning style. Some like to work for hours at a time to complete an assignment, while others may take many breaks. There is no ideal strategy, and how a person approaches learning is highly dependent on learning style, as well as adhering to a study strategy that complements their schedule and desires. The method that maximizes the value of the time spent is likely the most effective for improving learning, and subsequently, GPA.

Organization of work that needs to be done, as well as notes taken is also important. It is as important to be able to find relevant information as it is to take notes in class. Notes are most valuable when they can be used to supplement learning. Professors present large amounts of information during the course of a lecture, not all of which a student may have time to process. It is important to practice taking notes in a manner that enables the student to look back and learn (or look up) the information.

Time management is also an important aspect of planning. There are only 24 hours in a day, not all of which a person can use effectively. While learning is important, taking more courses or activities than a person can handle can be detrimental both to learning, as well as to average GPA. Once all courses have been selected, budgeting and scheduling time for each course can help to put the amount work and time necessary into perspective. While the amount of work necessary for a number of courses may initially seem daunting, planning how and when to approach the work for each course may help reduce stress and improve efficiency once the work is quantified (or could help a person realize that they are tackling more than they can handle).

Reviewing work regularly, in terms of studying, is another aspect of time management. A substantial amount of information is covered in a course by the time of the final exam, and reviewing some of the information regularly over a period of time is often more effective than attempting to memorize all of the information right before an exam. Learning the information through periodic review can ultimately save a person more time, and potentially position them to perform better on an exam, and thereby improve GPA.

### 28. Least Common Multiple

LCM calculator uses two or more numbers and calculates the least common multiple, i.e. the smallest positive integer which is divisible by each one of these numbers. It is an online mathematical tool specially programmed to find out the LCM of two or more integers.

It is necessary to follow the next steps:

1. Enter two or more numbers in the box. These numbers must be integers and may be separated by commas. The values can be copied from a text document or a spreadsheet;

2. Press the "GENERATE WORK" button to make the computation;

3. LCM calculator will give the least common multiple of two or more numbers.

The Least Common Multiple (LCM) is also referred to as the Lowest Common Multiple (LCM) and Least Common Divisor (LCD). For two integers a and b, denoted LCM(a,b), the LCM is the smallest positive integer that is evenly divisible by both a and b. For example, LCM(2,3) = 6 and LCM(6,10) = 30. The LCM of two or more numbers is the smallest number that is evenly divisible by all numbers in the set.

What is the Least Common Multiple (LCM)?

In mathematics, the least common multiple, also known as the lowest common multiple of two (or more) integers a and b, is the smallest positive integer that is divisible by both. It is commonly denoted as LCM(a, b).

Brute Force Method

There are multiple ways to find a least common multiple. The most basic is simply using a "brute force" method that lists out each integer's multiples.

EX: Find LCM(18, 26)

18: 18, 36, 54, 72, 90, 108, 126, 144, 162, 180, 198, 216, 234

26: 52, 78, 104, 130, 156, 182, 208, 234

As can be seen, this method can be fairly tedious, and is far from ideal.

Prime Factorization Method

A more systematic way to find the LCM of some given integers is to use prime factorization. Prime factorization involves breaking down each of the numbers being compared into its product of prime numbers. The LCM is then determined by multiplying the highest power of each prime number together. Note that computing the LCM this way, while more efficient than using the "brute force" method, is still limited to smaller numbers. Refer to the example below for clarification on how to use prime factorization to determine the LCM:

EX: Find LCM(21, 14, 38)

21 = 3 × 7

14 = 2 × 7

38 = 2 × 19

The LCM is therefore:

3 × 7 × 2 × 19 = 798

Greatest Common Divisor Method

A third viable method for finding the LCM of some given integers is using the greatest common divisor. This is also frequently referred to as the greatest common factor (GCF), among other names. Refer to the link for details on how to determine the greatest common divisor. Given LCM(a, b), the procedure for finding the LCM using GCF is to divide the product of the numbers a and b by their GCF, i.e. (a × b)/GCF(a,b). When trying to determine the LCM of more than two numbers, for example LCM(a, b, c) find the LCM of a and b where the result will be q. Then find the LCM of c and q. The result will be the LCM of all three numbers. Using the previous example:

EX: Find LCM(21, 14, 38)

GCF(14, 38) = 2

LCM(14, 38) = 38 × 14/2= 266

GCF(266, 21) = 7

LCM(266, 21) = 266 × 21/7 = 798

LCM(21, 14, 38) = 798

Note that it is not important which LCM is calculated first as long as all the numbers are used, and the method is followed accurately. Depending on the particular situation, each method has its own merits, and the user can decide which method to pursue at their own discretion.

### 29. Multiplication Calculator

Multiplication Calculator is a free online tool that displays the product of two numbers. Online multiplication calculator tool makes the calculation faster and it displays the multiplication of two numbers in a fraction of seconds.

How to Use the Multiplication Calculator?

The procedure to use the multiplication calculator is as follows:

1. Enter two numbers in the input field

2. Now click the button “Multiply” to get the product

3. Finally, the product of two numbers will be displayed in the output field

What is Meant by Multiplication?

In Mathematics, multiplication is one of the arithmetic operations, that will display the product of two numbers. In this process, the numbers are multiplied together to produce the result. In other words, multiplication is the repeated addition process. Multiplication operation is denoted by the notations, such as Asterisk(*), cross(×), dot(.). The result of the multiplication process is called the product, and the numbers which are multiplied are called multiplier and multiplicand. It is noted that, when a number is multiplied by 0, the product obtained is 0.

For example 10 × 2 = 20

Here

10 is multiplier

2 is multiplicand

20 is a product

Example: using the multiplication calculator

Let's find 2020 times 12 with the multiply calculator. At the top of our tool, we see the formula:

result = a₁ * a₂.

This means that to calculate 2020 * 12, we need to input:

a₁ = 2020 and a₂ = 12.

The moment we give the second number, the multiplication calculator spits out the answer in the Result field.

result = 2020 * 12 = 24240

However, say that you'd like to multiply the result further by 1.3 (remember that our tool also works as a multiplying decimals calculator).

We could just clear out the fields and write the answer from above into one of the factors, i.e., input a₁ = 24240 and a₂ = 1.3. Alternatively, we can simply select many numbers under Multiply..., which lets us find the product of multiplication for more numbers. If we do so, we'll get the option to input a₁, a₂, a₃, and so on up to a₁₀ (note how initially only a₁ and a₂ are there, but more variables appear once you start filling the fields). It's then enough to input:

a₁ = 2020, a₂ = 12, a₃ = 1.3,

result = 2020 * 12 * 1.3 = 31512.

Well, this multiply calculator sure saves a lot of time.

### 30. Multiplying Fractions

A 'Multiplying Fractions Calculator' is a free online tool that multiplies fractions and displays their product.Use multiplying fractions calculator to multiply up to 5 fractions. You can multiply simple fractions by whole numbers, mixed numbers, improper numbers, or negative fractions.

Multiplying fractions calculator online.

Enter fractions and press the = button.

Enter simple fractions with slash (/).

For example: 1/2 × 1/3

Enter mixed numbers with space.

For example: 2 1/2 × 1 1/3

How to multiply fractions?

Multiplying fractions is not rocket science, believe us. You only need to know how to multiply numbers and how to simplify fractions!

Let's say you have two fractions, 3/5 and 5/8. How to multiply these fractions?

3/5 * 5/8 = ?

Multiply the numerator by the other numerator and the denominator by the other denominator:

3/5 * 5/8 = (3 * 5)/(5 * 8)= 15/40

If you had more fractions than two, the process would be analogical: multiply all numerators together to make a new numberator, and do the same with the denominators to get a new denominator.

If needed, simplify a fraction:

15/40 = 3/8

To make a reduction, find greatest common factor of the numerator and denominator, and divide them by that value.

Multiplying a fraction by a whole number

Let's analyze another example: the multiplication of the fraction 7/8 by 13, which is a whole number, of course.

Pick the type of fraction from the drop-down list. Choose mixed form, as you'll need the whole fraction part.

Enter the fractions:

The first numerator is 7, and the denominator is 8 (the whole part is left empty)

In the whole part of second fraction, enter 13 and leave the fractional part empty

The multiplying fractions calculator displays the result: 91/8, which is 11 3/8 in its mixed number form.

### 31. Percent Error Calculator

The Percent Error Calculator (% error calculator) will calculate the percentage error between an experimental measured value and a theoretical actual value.

Percentage Error Formula

Percentage Error=(|E−T|)/|T|×100

Where E is the experimental value and T is the theoretical value. This formula is similar to percentage change.

The absolute error is equal to the absolute value of the difference between the exact value and the approximated value:

ε = | Vexact - Vapprox |

The percent error is equal to the 100% times the absolute error divided by the exact value:

δ = 100% × | Vexact - Vapprox | / | Vexact |

What is the percent error?

Sometimes, we try to estimate the value of a particular quantity by making repeated measurements. For example, we can try to find out what is the gravitational acceleration by measuring the time of free fall from a specific height.

The measured acceleration will probably deviate a bit from the actual acceleration of 9.80665 m/s². To check what is the relative - or percent - error, you need to know two values: the real value and observed (measured) one.

How to calculate percent error: an example

To make it easier for you to understand what percent error is, we'll analyze a simple example. Let's assume that we want to find out what is the relative error between the measured and true values of the speed of sound.

1. Check what the real value of the speed of sound is. In the air at 20°C, it is 343 m/s

2. Take your measurement. You found it was equal to 329 m/s.

3. Subtract the observed value from the true value:

343 - 329 = 14 m/s

4. 14 m/s is the absolute error. To calculate the percent error, you need to divide it by the true value:

percent error = 14 / 343 *100% = 4.082%

5. The percent error, otherwise known as the relative error, is equal to 4.082%.

Percent error, margin of error, and standard error

There are a lot of terms similar to percent error. To ensure you don't get confused and mix them up, we provide a short explanation of each of them here.

Percent error, or relative error, is precisely what we described above - the relative difference between the observed and true value. You can apply the aforementioned percent error formula to calculate it.

Standard error is a value that can be calculated for a set of data. It is a statistical term that describes the accuracy with which a sample represents the whole population.

Margin of error, on the other hand, is closely related to the confidence interval. It can be calculated by multiplying the standard error with the z-score corresponding to a certain confidence level.

### 32. Percentage Change

The Percentage Change Calculator (% change calculator) will quantify the change from one number to another and express the change as an increase or decrease.

This is a % change calculator. From 10 apples to 20 apples is a 100% increase (change) in the number of apples. Percentage change calculator determines the difference in values and the percent change from the original value to the new value.

This calculator will be most commonly used when there is an “old” and “new” number or an “initial” and “final” value. A positive change is expressed as an increase amount of the percentage value while a negative change is expressed as a decrease amount of the absolute value of the percentage value.

Percentage Change Formula

Percentage change equals the change in value divided by the absolute value of the original value, multiplied by 100.

The percent change formula is as follows:

Percentage Change=ΔV / |V1|×100

=(V2−V1) / |V1|×100

or

=(new_value - original_value) / |original_value| * 100

The two straight lines surrounding a number or expression (in this case original value) indicate the absolute value, or modulus. It means that if the value inside the straight lines is negative, we have to turn it into a positive one. The easiest way to do this is by erasing the minus before it. If the value inside the straight lines is positive, we don't need to do anything, it stays positive. After the absolute value is found, we can erase the straight lines or turn them into a bracket as they may serve this function as well.

How to calculate the percent change?

Percentage change differs from percent increase and percent decrease in the sense that we can see both directions of the change. For example, the percent increase calculator calculates the amount of increase, in which we would say, "x percent increase". The percent decrease calculator calculates the amount of decrease, in which we would say, "x percent decrease". The percent change calculator would yield a result in which we would say, "x percent increase or decrease".

To calculate percent change we need to take the difference between the starting value and final value, divide by the absolute value of the starting value, and multiply the result by 100. The section below will go into more detail and show how to solve this problem.

Percentage change formula – mathematical examples

Let's do a few examples together to get a good grasp on how to find percent change. In the first case, let's suppose that you have a change in value from 60 to 72 and you want to know the percent change.

1. Firstly, you need to input 60 as the original value and 72 as the new value into the formula.

2. Secondly, you have to subtract 60 from 72. As a result, you get 12.

3. Next, you should get the absolute value of 60. As 60 is a positive number, you don't need to do anything. You can erase the straight lines surrounding 60.

4. Now, you can divide 12 by 60. After this division, you get 0.2.

5. The last thing to do is to multiply the 0.2 by 100. As a result, you get 20 %. The whole calculations look like this:

[(72 – 60) / |60|] * 100 = (12 / |60|) * 100 = (12 / 60) * 100 = 0.2 * 100 = 20 %

You can check your result using the percentage change calculator.

### 33. Percentage Increase/Decrease

The Percentage increase/decrease Calculator finds the increase/ decrease from one value to another in terms of a percentage.Enter starting value and final value to find percentage increase and decrease.

Percentage increase/decrease calculation

The percentage increase/decrease from old value (Vold) to new value (Vnew) is equal to the old and new values difference divided by the old value times 100%:

percentage increase/decrease = (Vnew - Vold) / Vold × 100%

Example 1

Price percentage increase from old value of \$1000 to new value of \$1200 is caluclated by:

percentage increase = (\$1200 - \$1000) / \$1000 × 100%

= 0.2 × 100% = 20%

Example 2

Price percentage decrease from old value of \$1000 to new value of \$800 is caluclated by:

percentage decrease = (\$800 - \$1000) / \$1000 × 100%

= -0.2 × 100% = -20%

Difference and final value calculation

The difference d is equal to the initial value V0 times the percentage increase/decrease p divided by 100:

d = V0 × p / 100

The final value V1 is equal to the initial value V0 plus the difference d:

V1 = V0 + d

How to calculate percent increase

The concept of percent increase is basically the amount of increase from the original number to the final number in terms of 100 parts of the original. An increase of 5 percent would indicate that, if you split the original value into 100 parts, that value has increases by an additional 5 parts. So if the original value increased by 14 percent, the value would increase by 14 for every 100 units, 28 by every 200 units and so on.

Percent increase formula

The percent increase formula is as follows:

Percent increase = [(new value - original value)/original value] * 100

An example using the formula is as follows. Suppose a \$1,250 investment increased in value to \$1,445 dollars in one year. What is the percent increase of the investment? To answer this, us the following steps:

Identify the original value and the new value.

Input the values into the formula.

Subtract the original value from the new value, then divide the result by the original value.

Multiply the result by 100. The answer is the percent increase.

Working out the problem by hand we get:

[(1,445 - 1,250)/1,250] * 100

(195/1,250) * 100

0.156 * 100

15.6 percent increase.

The percentage growth calculator is a great tool to check simple problems. It can even be used to solve more complex problems that involve percent increase.

Calculating percent decrease

If you want to know how to calculate percent decrease, we follow a very similar process as percent increase. Notice the slight modification of the formula:

Percent decrease = [(original value - new value)/original value] * 100

Suppose we have the same investment value after one year of \$1,445. A year later the value decreased to \$1,300. The percent decrease would be calculated as follows:

[(1,445 - 1,300)/1,445] * 100

(145/1,445) * 100

0.10 * 100 = 10 percent decrease

### 34. Pythagorean Theorem

This calculator solves the Pythagorean Theorem equation for sides a or b, or the hypotenuse c. The Pythagorean Theorem, also known as Pythagoras' theorem, is a fundamental relation between the three sides of a right triangle. Given a right triangle, which is a triangle in which one of the angles is 90°, the Pythagorean theorem states that the area of the square formed by the longest side of the right triangle (the hypotenuse) is equal to the sum of the area of the squares formed by the other two sides of the right triangle.

In other words, given that the longest side c = the hypotenuse, and a and b = the other sides of the triangle:

a2 + b2 = c2

The hypotenuse is the side of the triangle opposite the right angle.For right triangles only, enter any two values to find the third. See the solution with steps using the Pythagorean Theorem formula.

This calculator also finds the area A of the right triangle with sides a and b. The formula for area of a right triangle is:

A=ab/2

How to Use the Pythagorean Theorem Calculator?

The procedure to use the Pythagorean Theorem calculator is as follows:

1. Enter the values of two sides in the input fields and enter “x” for the unknown side value

2. Click the button “Solve” to get the unknown side measure

3. The value of the variable “x” will be displayed in the output “x” field

What is the Pythagorean theorem?

The Pythagorean theorem describes how the three sides of a right triangle are related in Euclidean geometry. It states that the sum of the squares of the sides of a right triangle equals the square of the hypotenuse. You can also think of this theorem as the hypotenuse formula. If the sides of a right triangle are a and b and the hypotenuse is c, the formula is

a² + b² = c²

The theorem was credited to the ancient Greek philosopher and mathematician Pythagoras, who lived in the sixth century BC. Although it was previously used by the Indians and Babylonians, Pythagoras (or his students) were credited to be the first to prove the theorem. It should be noted that there is no concrete evidence that Pythagoras himself worked on or proved this theorem.

Example: How to use the Pythagorean theorem

1. Input the two lengths that you have into the formula. For example, suppose you know a = 4, b = 8 and we want to find the length of the hypotenuse c.

2. After the values are put into the formula we have 4²+ 8² = c²

3. Square each term to get 16 + 64 = c²

4. Combine like terms to get 80 = c²

.5. Take the square root of both sides of the equation to get c = 8.94.

The Pythagorean theorem calculator will solve for the sides in the same manner that we displayed above. We have included the method to show you how you can solve your problem if you prefer to do it by hand.

What is the hypotenuse formula?

The hypotenuse formula is simply taking the Pythagorean theorem and solving for the hypotenuse, c. Solving for the hypotenuse, we simply take the square root of both sides of the equation a² + b² = c²and solve for c. When doing so, we get c = √(a² + b²). This is just an extension of the Pythagorean theorem and often is not associated with the name hypotenuse formula.

### 35. Generate Random Number

Generate one or more random numbers within a range that you define.Enter the minimum and maximum for the range you want and enter the number of numbers you want generated.You can also sort the generated numbers from lowest to highest, from highest to lowest, or choose to do not sort. Note that even if you choose not to sort that it is still possible for your numbers to be in some kind of order if they happen to be randomly chosen in order.

Generate a Random Number for a PIN.

To generate a 4 digit PIN without duplicate digits, choose Min = 0, Max = 9 and Generate 4 Numbers.

Example: 4 digit PIN without duplicates

A random number is a number chosen from a pool of limited or unlimited numbers that has no discernible pattern for prediction. The pool of numbers is almost always independent from each other. However, the pool of numbers may follow a specific distribution. For example, the height of the students in a school tends to follow a normal distribution around the median height. If the height of a student is picked at random, the picked number has higher chance to be closer to the median height than being classified as very tall or very short. The random number generators above assume that the numbers generated are independent of each other, and will be evenly spread across the whole range of possible values.

This calculator can create one or many random integers or decimals. It can deal with very large numbers with up to 999 digits of precision.A random number generator, like the ones above, is a device that can generate one or many random numbers within a defined scope. Random number generators can be hardware based or pseudo-random number generators. Hardware based random-number generators can involve the use of a dice, a coin for flipping, or many other devices.

A pseudo-random number generator is an algorithm for generating a sequence of numbers whose properties approximate the properties of sequences of random numbers. Computer based random number generators are almost always pseudo-random number generators. Yet, the numbers generated by pseudo-random number generators are not truly random. Likewise, our generators above are also pseudo-random number generators. The random numbers generated are sufficient for most applications yet they should not be used for cryptographic purposes. True random numbers are based on physical phenomenon such as atmospheric noise, thermal noise, and other quantum phenomena. Methods that generate true random numbers also involve compensating for potential biases caused by the measurement process.

### 36. Ratio Calculator

Ratio Calculator is a free online tool that displays the simplified ratio for the given numbers. Online ratio calculator tool makes the calculation faster, and it displays the simplified ratio in a fraction of seconds.

How to Use the Ratio Calculator?

The procedure to use the ratio calculator is as follows:

1. Enter the x and y value in the respective input field

2. Now click the button “Solve” to get the simplified form

3. Finally, the simplified ratio will be displayed in the output field

The ratio calculator performs three types of operations and shows the steps to solve:

1. Simplify ratios or create an equivalent ratio when one side of the ratio is empty.

2. Solve ratios for the one missing value when comparing ratios or proportions.

3. Compare ratios and evaluate as true or false to answer whether ratios or fractions are equivalent.

The ratio calculator will accept integers, decimals and scientific e notation with a limit of 15 characters.

What is Meant by Ratio?

In mathematics, the ratio is used to define the relative size of two objects. The ratio can be expressed in many ways. It can be expressed by using the symbols “:” or “/”. Even decimal and percentages are also used to define ratio. For example, in a class, there are 3 boys and 1 girl.

The different forms to express the above example are:

1: 3 (for every one girl, there are 3 boys)

¼ are girls, ¾ are boys

0.25 are girls (It means remaining 0.75 are boys)

25% are girls (It means remaining 75% are boys)

How to do ratios

A ratio is made up of two parts, the same as how a fraction is made up of two parts. There is the numerator (the top number of the fraction) and the denominator (the bottom number of the fraction). For example, suppose there is a pie cut into eight slices and three of the eight slices have been eaten. If we want to know the ratio of slices eaten compared to the entire pie, then we have to put the number eaten as the numerator and the total number of pieces as the denominator; 3/8. That is the most basic of ratios since no simplification is involved.

How to find a ratio

Suppose we have the same ratio of 3/8 but we want to scale it up to a larger, equivalent ratio with a denominator of 72. The way to do this is to set up a proportion, which is two ratios equal to each other and solve for the missing part. This is done as follows:

1. Write both ratios in terms of fractions, labeling the missing part with an x

2. Set the fractions equal to each other, forming a proportion.

3. Use the process of cross multiplication to isolate the variable.

4. Solve for the variable.

In the above example, the steps would look as follows:

3/8 = x/72

8 * x = 72 * 3

8x = 216

x = 27

For more complex ratios involving larger numbers or decimals, the ratio calculator is much more convenient to use. The proportion calculator, which does the same thing, may also be used to solve problems such as the one above.

### 37. Scientific Notation

Convert a number to and from scientific notation, e notation, engineering notation and real numbers. Enter a number or a decimal number or scientific notation and the calculator converts to scientific notation, e notation and engineering notation formats.

To enter a number in scientific notation use a carat ^ to indicate the powers of 10. You can also enter numbers in e notation. Examples: 3.45 x 10^5 or 3.45e5.

Scientific notation calculations

For 2 number x1 and x2:

x1 = a1 × 10b1

x2 = a2 × 10b2

x1 + x2 = a1 × 10b1 + a2 × 10b2

Scientific notations subtraction

x1 - x2 = a1 × 10b1 - a2 × 10b2

Scientific notations multiplication

x1 × x2 = a1a2 × 10b1+b2

Scientific notations division

x1 / x2 = (a1/a2) × 10b1-b2

What is Scientific Notation?

Scientific notation is generally used with very large or very small numbers in applications such as physics, engineering and chemistry. It condenses the numbers into the form into a number multiplied by 10 raised to an exponent, denoted as a * 10ⁿ.

How to Convert a Number to Scientific Notation

The proper format for scientific notation is a x 10^b where a is a number or decimal number such that the absolute value of a is greater than or equal to one and less than ten or, 1 ≤ |a| < 10. b is the power of 10 required so that the scientific notation is mathematically equivalent to the original number.

1. Move the decimal point in your number until there is only one non-zero digit to the left of the decimal point. The resulting decimal number is a.

2. Count how many places you moved the decimal point. This number is b.

3. If you moved the decimal to the left b is positive.

If you moved the decimal to the right b is negative.

If you did not need to move the decimal b = 0.

4. Write your scientific notation number as a x 10^b and read it as "a times 10 to the power of b."

5. Remove trailing 0's only if they were originally to the left of the decimal point.

Scientific Notation Rules

When converting a number into scientific notation, we must remember a few rules. First, the decimal must be between the first two non-zero numbers. The number prior to the multiplication symbol is known as the significant or mantissa. The numbers of digits in the significant depends on the application and are known as significant figures. The significant figures calculator can assist in this situation. The value of the exponent depends on whether or not the decimal place is moved to the right or left to return to the original number.

### 38. Fractions Simplifier

This calculator simplifies or reduces a fraction to its simplest or lowest term. In other words, the numerator and denominator of the fraction cannot both be divided by any number to further reduce the fraction.

There are two steps in reducing the fraction:

1. The Greatest Common Divisor (GCD) is determined. The Greatest Common Divisor is the largest integer value which will divide the numerator and denominator without a remainder. This calculator uses Euclid's Algorithm to determine the Greatest Common Divisor.

2. The numerator and denominator are simply divided by the GCD.

What is a fraction?

A fraction is a ratio of two numbers, the numerator and the denominator. With this definition in mind, we can see that 1/2 is a fraction, 10/12 is a fraction, 3.3/6.2 is a fraction, and so on. Typically, we use whole numbers within fractions as it's more intuitive and simpler to use, especially when adding or subtracting fractions.

Additionally, you can turn any decimal value into a fraction.

Now, it's high time to ask yourself "How do you simplify fractions?" e.g., 4/8. The solution is to find a common factor of both numbers and divide them by this value. 2 is one of the common factors, so:

4 divided by 2 is 2.

8 divided by 2 is 4.

As a result, we continue by reducing the fraction 4/8 to 2/4.

In other words, we can say that 4/8 and 2/4 are equivalent fractions.

Although the outcome is correct, we usually want to reduce the fraction to its simplest form in these kinds of computations.

Reducing fractions to lowest terms

Reducing a fraction to its simplest form (or lowest term) is almost the same as the procedure explained in the previous section. The most significant difference is that we need to divide both the numerator and denominator by the greatest common factor (GCF). Let's take a look at the same example, 4/8:

1) Find the GCF of 4 and 8, which is 4.

2) 4 divided by 4 is 1.

3) 8 divided by 4 is 2.

4) 4/8 in simplest form is 1/2.

5) You can always convert a fraction to decimal, in this case 0.5.

And what is the simplest form of a fraction with negative numbers, e.g., -4/6? That isn't difficult at all! The only thing you need to know is that the negative number factors are the same as the positive ones multiplied by -1.

As a result, we can reduce the fraction -4/6 to -2/3.

### 39. Standard Deviation

Standard deviation is a measure of spread of numbers in a set of data from its mean value. Use our online standard deviation calculator to find the mean, variance and arithmetic standard deviation of the given numbers.Standard deviation calculator calculates the sample standard deviation from a sample:

X: x1,x2,x3,..........xn,

using simple method. It’s an online Statistics and Probability tool requires a data set (set of real numbers or valuables). The result will describe the spread of dataset, i.e. how widely it is distributed about the sample mean.

It is necessary to follow the next steps:

Enter a sample (observed values) in the box. These values must be real numbers or variables and may be separated by commas. The values can be copied from a text document or a spreadsheet.

Press the "GENERATE WORK" button to make the computation.

Standard deviation calculator will give the sample standard deviation of the sample

X: x1,x2,x3,..........xn,

standard deviation calculator gives us the stepwise procedure and insight into every step of calculation. Before the final result of sample standard deviation is derived, it calculates the arithmetic mean of a sample. The sample standard deviation calculator also calculates square of the sample standard deviation and get magnitude known as variance. These values of the sample mean and the variance can be of benefit for further solving of problems and applications.

Standard Deviation Formula

Standard deviation of a data set is the square root of the calculated variance of a set of data.

The formula for variance is the sum of the squared differences between each data point and the mean, divided by the number of data points.

When working with data from a complete population the sum of the squared differences between each data point and the mean is divided by the size of the data set, n. When working with a sample, divide by the size of the data set minus 1, n - 1.

The formula for variance for a population is:

Variance = σ2=Σ(xi−μ)2/n

Standard deviation in statistics, typically denoted by σ, is a measure of variation or dispersion (refers to a distribution's extent of stretching or squeezing) between values in a set of data. The lower the standard deviation, the closer the data points tend to be to the mean (or expected value), μ. Conversely, a higher standard deviation indicates a wider range of values. Similarly to other mathematical and statistical concepts, there are many different situations in which standard deviation can be used, and thus many different equations. In addition to expressing population variability, the standard deviation is also often used to measure statistical results such as the margin of error. When used in this manner, standard deviation is often called the standard error of the mean, or standard error of the estimate with regard to a mean. The calculator above computes population standard deviation and sample standard deviation, as well as confidence interval approximations.

### 40. Subtracting Fractions

Subtracting Fractions Calculator is a free online tool that displays the difference value of two fractions. Online subtracting fractions calculator tool performs the calculation faster, and it displays the difference in a fraction of seconds.

How to Use the Subtracting Mixed Fractions Calculator?

The procedure to use the subtracting mixed fractions calculator is as follows:

1. Enter the two mixed fraction values in the respective input field

2. Now click the button “Solve” to get the difference value

3. Finally, the difference value will be displayed in the output field

What is Meant by the Subtracting Fractions?

In Mathematics, a mixed fraction is one of the types of fractions, and it is the combination of a whole number and a fractional number. For example, 2 ⅓ is a mixed fraction. Here, 2 is a whole number, and ⅓ is a fractional number. The arithmetic operations that can be performed on the mixed numbers are the addition, subtraction, multiplication and division. The subtraction of the mixed fraction is performed as follows:

Fractions with Same Denominator Values:

1. Convert the mixed fractions into an improper fraction.

2. Keep the denominator common and subtract the numerator.

3. Simplify the fraction if possible or convert the improper fraction into a mixed fraction.

Fractions with Different Denominator Values:

1. Convert the given mixed fraction to an improper fraction.

2. Find the LCM of the denominator values.

3. Multiply the numerator and denominator values by a number such that the fractions should have the LCM value as the new denominator.

4. Now, keep the denominator common and subtract the numerator values.

5. Simplify the fractions or convert into a mixed number.

### 41. Subtraction Calculator

Subtraction Calculator is a free online tool that displays the subtraction of two numbers. Online subtraction calculator tool makes the calculation faster and it displays the difference value in a fraction of seconds.

How to Use the Subtraction Calculator?

The procedure to use the subtraction calculator is as follows:

1. Enter the first and second number in the input field

2. Now click the button “Solve” to get the difference

3. Finally, the difference between the two numbers will be displayed in the output field

What is Meant by the Subtraction?

In mathematics, subtraction is one of the four basic arithmetic operations. The subtraction process gives the difference between the two numbers. It is the inverse of the addition operation. The notation used to describe the subtraction is “minus (-)”. The subtraction process starts from the unit digit, then ten’s digits, the hundreds digit, and it continues the process towards the highest digit places. If the minuend is greater than the subtrahend, the subtraction process is easy, and we can subtract directly. If the minuend is lesser than the subtrahend, then borrow the number from the next highest digit of the minuend to continue the subtraction process.

For example,

15 – 5 = 10

Here,

15 is minuend

5 is subtrahend

10 is difference

### 42. Trigonometry Calculator

Trigonometry Calculator is a free online tool that displays the values of six important trigonometric functions. Online trigonometry calculator tool makes the calculation faster, and it displays the trigonometric values in a fraction of seconds. Trigonometry calculator can support you in finding the trigonometric functions values or solving the right triangle.

Trigonometric functions

sin A = opposite / hypotenuse = a / c

cos A = adjacent / hypotenuse = b / c

tan A = opposite / adjacent = a / b

csc A = hypotenuse / opposite = c / a

sec A = hypotenuse / adjacent = c / b

cot A = adjacent / opposite = b / a

How to Use the Trigonometry Calculator?

The procedure to use the trigonometry calculator is as follows:

1. Enter the value of the base and perpendicular side in the respective input field

2. Now click the button “Calculate” to get the trigonometric value

3. Finally, the value of six trigonometric functions will be displayed in the output field

What is Meant by Trigonometry?

In Maths, trigonometry is a branch that deals with the sides and the measure of the angle for the right angle triangle. The trigonometric function is also known as the circular function. Based on the angles and the sides, the trigonometric functions are classified into six important types. They are Sine, Cosine, Tangent, Cosecant, Secant, and Cotangent. The ratio to define these function are given below:

Sin Function = Opposite Side/Hypotenuse

Tan Function = Opposite Side / Adjacent Side

Cosec Function = Hypotenuse / Opposite Side

Sec Function = Hypotenuse / Adjacent Side

Cot Function = Adjacent Side / Opposite Side

Trigonometry calculator finding sin, cos, tan, cot, sec, csc

To find the trigonometric functions of an angle, enter the chosen angle in degrees or radians. Underneath the calculator, six most popular trig functions will appear - three basic ones: sine, cosine and tangent, and their reciprocals: cosecant, secant and cotangent. Additionally, if the angle is acute, the right triangle will be displayed, which can help you in understanding how the functions may be interpreted.

Trigonometry calculator as a tool for solving right triangle:

To find the missing sides or angles of the right triangle, all you need to do is enter the known variables into the trigonometry calculator. You need only two given values in the case of:

1. one side and one angle

2. two sides

3. area and one side

Remember that if you know two angles, it's not enough to find the sides of the triangle. Two triangles having the same shape (which means they have equal angles) may be of different sizes (not the same side length) - that kind of relationship is called triangle similarity. If the sides have the same length, then the triangles are congruent.

### 43. Variance Calculator

The variance calculator finds variance, standard deviation, sample size n, mean and sum of squares. Variance is a measure of dispersion of data points from the mean. Low variance indicates that data points are generally similar and do not vary widely from the mean. High variance indicates that data values have greater variability and are more widely dispersed from the mean.

Variance Calculator is a free online tool that displays the variance for the given set of data. Online variance calculator tool makes the calculation faster, and it displays the variance in a fraction of seconds.

How to Use the Variance Calculator?

The procedure to use the variance calculator is as follows:

1. Enter the numbers separated by a comma in the respective input field

2. Now click the button “Calculate Variance” to get the result

3.Finally, the variance for the given set of data will be displayed in the output field

What is Meant by the Variance?

In Mathematics, the variance is a measure which can be calculated from the set of data. It can be calculated by taking the average of the squared difference from the mean. To find out the variance, follow the below procedure:

1. Find out the mean for the given set of data

2. For each number, subtract the mean and then square the obtained result

3. Finally, work out the average of those squared differences.

For example, 5, 6, 7 are the set of data

To find variance,

1. mean = (5 + 6 + 7)/3 = 18/3 = 6

2. (5 – 6) 2+ (6 – 6)2 + (7 – 6)2 = (-1)2 + 0 + (1)2

3. (1 + 0 + 1)/3 = ⅔ = 0.667

What is the definition of variance?

Variance is a measure of the variability of the values in a dataset.A high variance indicates that a dataset is more spread out.A low variance indicates that the data is more tightly clustered around the mean, or less spread out.

Variance formula

Variance (denoted as σ2) is defined as the average squared difference from the mean for all data points. We write it as:

σ2 = ∑(xi - μ)2 / N

where,

σ2 is the variance;

μ is the mean; and

xᵢ represents the ith data point out of N total data points.

You can calculate variance in three steps:

Find the difference from the mean for each point. Use the formula: xi - μ

Square the difference from the mean for each point: (xi - μ)2

Find the average of the squared differences from the mean which you found in step 2:

∑(xi - μ)2 / N.

This is the population variance formula.

### 44. Weighted Average

The Weighted average calculator helps you find the average when the values are not weighted equally.The weighted average (x) is equal to the sum of the product of the weight (wi) times the data number (xi) divided by the sum of the weights:

Example

Find the weighted average of class grades (with equal weight) 70,70,80,80,80,90:

Since the weight of all grades are equal, we can calculate these grades with simple average or we can cound how many times each grade apear and use weighted average.

2×70, 3×80, 1×90

x = (2×70+3×80+1×90) / (2+3+1) = 470 / 6 = 78.33333

what is a weighted average?

Weighted average (weighted arithmetic mean) is a concept similar to standard arithmetic mean (called simply the average), but in the weighted average not all elements are contributing equally to the final result. We can say that some values are more important than the others, so they are multiplied by a coefficient called the weight.

For example, during your studies you may encounter the situation where the grade from an exam is two times more important than the grade from a quiz - and that's exactly what we call the weighted average method. To define it in a more mathematical way, we can write the weighted average formula as:

weighted mean formula

(w1x1 + w2

x2 +........+wnxn) / (w1 + w2 +......+ wn)

where x1,x2...xn are our numbers, and w1,w2...wn are our weights - the importance of the numbers in averaging.

So, having A from an exam and C from a quiz, you'd get B as a standard average, but assuming that the exam is two times more important, you should get a B+.

How to calculate a weighted average

Let's find out how to calculate a weighted average - the easiest way is to look at the simple example:

Suppose a student has two four-credit classes, a three-credit class, and a two-credit class. Assume that the grades of the courses are as follows:

A for a four-credit class,

B for the other four credit class,

A for the three credit class,

C+ for the two credit class.

Then, we need to translate the letter grades into numerical values. Most schools in the US use a so-called 4.0 GPA scale, which is a 4 point grading scale.

So from the table we know that A = 4.0, B = 3.0 and C+ = 2.3. Now that we have all the information, we can have a look at how to calculate the GPA using a weighted average method:

1. Sum the number of credits. 4 + 4 + 3 + 2 = 13, that was a really easy step.

2. Take the value assigned to the grade and multiply by the number of credits. In our case, it will be:

A * 4 credits = 4.0 * 4 = 16

B * 4 credits = 3.0 * 4 = 12

A * 3 credits = 4.0 * 3 = 12

C * 2 credits = 2.3 * 2 = 4.6

3. Add all the values. 16 + 12 + 12 + 4.6 = 44.6

4. Divide the sum by the total number of credits. So for our example it's equal to 44.6/13 = 3.43

The whole weighted average formula may be written as:

(4 * 4 + 4 * 3 + 3 * 4 + 2.3 * 2) / (4 + 4 + 3 + 2) = 3.43

Let's compare this result to an average that is not weighted. Then we don't take the credits into account, and we divide the sum of grades by its total number.

(4 + 3 + 4 + 2.3) / 4 = 3.33

Notice how the weighted average changed. Sometimes it may be a really significant difference - like a grade difference or even whether you pass or fail your course.