# Absolute Values

## Meaning of Absolute Value in Maths

The absolute value in maths is defined as the value which describes the distance of a number on the number line from 0 without taking into consideration which direction from zero the number lies. The absolute value of a number can never have negative values.

• The absolute value of digit  5 is 5.

The distance from 0 to 5 units is represented below in number line

• The absolute value of -5 is 5

The distance from 0 to -5 unit is represented below in number line

The absolute value of 0 is  0.

In this article, we will study the meaning of absolute values in Maths, the absolute value of a number meaning, the absolute value of 17, the absolute value of real numbers and the absolute values of complex numbers.

### What is the Absolute Value of a Number?

The absolute value of a number is always positive excluding zero, as zero is neither a positive number nor a negative number. Absolute value refers to the distance of a number from zero, irrespective of direction. The distance is always positive, as the absolute value of a number can never be negative. We use the term absolute values to refer to the distance of a point or number from the origin (zero) of a number line. The symbol used to represent the absolute value of a number is two vertical lines ‘| |’

For example: ‘ |x| ‘, where x is the number whose absolute value has to be calculated.

What is the absolute value of 17?

The absolute value of  17  is “17” because the absolute value of 17 is seventeen units away from zero on the number line. In another way,

• |17| indicates that the absolute value of 17 is 17.

• |-17| indicates that the absolute value of -17 is 17 because absolute value can never be negative.

### Absolute Value of a Real Number

Let us consider θ a real number. Then the absolute value of θ is represented as follows:

• For θ = 0 or θ > 0, | θ | = θ

• For θ < 0, | θ | = – θ

Alternatively, we can say that the absolute value of a real number θ is equal to the positive square root of θ²:

| θ | = ( θ²) ½.

### Absolute Value of a Complex Number

As we know, complex numbers include both real numbers and imaginary numbers. Hence, it is difficult to find the absolute value of a complex number. Suppose, x+yi is the given complex number, where x and y are real numbers. The absolute value of a complex number, x+yi (also known as the modulus) is defined as the distance between the origin (0,0) and the point (x,y) in the complex plane.

x + yi = $\sqrt{x^{2} + y^{2}}$

### Absolute Value of a Complex Number Example

|-2+ 3i| = $\sqrt{-2^{2} + 3^{2}}$

= $\sqrt{4 + 9}$

= $\sqrt{13}$

### Properties of the Absolute Value of a Number

If β and θ are real numbers and then the absolute values satisfying the following properties of the absolute value of a number

1. Non – Negativity

|  β | ≥ 0, it implies that | β| can never be zero

1. Positive – Definiteness:

| β | = 0 ↔ k = 0

1. Multiplicatively:

|β × θ| = | β| × |θ |, it implies that the absolute value of a times b and the ( absolute value of a ( and the absolute value of b ) is same

|  β + θ | is less than and equals to  | β| + |θ|

1. Symmetry :

| – β| = |  β |

1. Identity of Indiscernible (Equivalent to Positive – Definiteness) :

|  β – θ | = 0 ↔ x = y

1. Triangle Inequality (Equivalent to Subadditivity ) :

|  β – θ |   ≤ |  β – θ  | + | α –  β |

1. Preservation of Division (Equivalent to Multiplicatively) :

|β | / | θ |  is equals to | β / θ|

| β – θ |  is greater than and equals to || β | – | θ||

### Absolute Value of a Number Example

1. Solve the absolute value equation -7| 9- 2x| + 9=  -12

Solution: -7| 9- 2x| + 9=  -12

The absolute value equation is not yet isolated. First, eliminate the +9 and then the -7 which is presently multiplying the absolute value expression.

-7| 9-2x| + 9=  -12

-7| 9-2x| + 9 – 9 =  -12-9

-7| 9-2x| = – 21

| 9-2x| = -21/-7

|9-2x| = 3

Now let’s split |9-2x| = 3  in two parts and solve each equation.

First Part

9-2x = 3

9 – 9- 2x = 3 – 9

-2x = -6

-2x/-2 = -6/-2

x= 3

Second Part

9-2x = -3

9- 2x = -3

9 – 9- 2x = -3 – 9

-2x = -12

-2x/-2 = -12/-2

x = 6

Hence, the value of x = 3 or x = 6.

### Solved Examples

1. Find the value of 2|5x-1| if x = -2

Solution: 2 |5x-1| = 2| 5(-2)-1|

2 |5x-1| = 2|-10-1|

2 |5x-1| = 2|-11|

2 |5x-1| = 2|11|

2 |5x-1| = 22

2. Find the absolute  value of 2| p-2| = 16

Solution: 2| p- 2| = 16

| p-2| = 16/2

| p- 2| = 8

p – 2 = 8 or p – 2 = – 8

p = 10 + 2 or p = – 8 + 2

p = 12 or -6

### Quiz Time

1. The integer between -3 and 2 comprises.

1. -1,-2,0,1

2. -2,-1,0,1

3. 0,1,-1,-2

4. 0,-2,-1,1

2.   Find the absolute of -6

1. 6

2. -6

3. 0

4. 1

5. -1

1. When do we use Absolute Values When Solving Radicals?

Radicals are real numbers that can be represented as powers with rational exponents.

x√n = n1/x.

√x or 4√x then the value under the radical has to be positive i.e. x > 0

Hence √x² = |x|, because an even-root radical is a positive quantity, as the absolute value.

Hence, we use absolute values while solving even-root radicals.

The absolute value symbol is not needed if the radical is an odd-root radical as the value under an odd-root radical can be positive or negative.

3√-8 = -2 and 3√8 = 2

2. What are the Real Numbers and Complex Numbers?

Complex Numbers- These are the numbers that can be represented in the form of a+ib where i is an imaginary number known as iota and has the value of (√-1). For example, 5+6i is a complex number, where 5 is considered as a real number and 6i is considered as an imaginary number. Hence, the complex numbers are the combination of both the real number and imaginary numbers,

Real Numbers- Any number which is included in a number system such as positive, negative, zero, integer, rational, irrational, fractions, etc. are known as real numbers. It is denoted as Re(). For example: 10, -25, 0, 3/5, 8.9, √4 are all real numbers.