Absolute Value

The absolute value of a number represents its distance from zero on the number line, regardless of direction. It is always non-negative and is denoted by the symbol ∣ ∣.

The absolute value of x tells how far it is from 0. Whether x is positive or negative, the distance remains the same.

Examples:

• ∣5∣ = 5 (since 5 is 5 units away from 0)
• ∣−5∣ = 5 (since -5 is also 5 units away from 0)
• ∣0∣ = 0                         

Absolute Value of Zero

Zero is neither positive nor negative, and its absolute value is ∣0∣ = 0

Since the distance of 0 from itself is 0, its absolute value remains 0.

Absolute Value of Real Number

For a real number the absolute value is the value of the number without any sign and it satisfies the condition.

• |x| = +x for x ≥ 0
• |x| = -x for x < 0

For example:

• |4| = 4
• |-4| = 4

Absolute Value of Complex Number

A complex number consists of a real part and an imaginary part. The absolute value (or modulus) of a complex number z = a + ib is given by:

|z| =√(a²+b²)

Where a and b are  real numbers

Example: Find the absolute value of z = 3 + 4i

|3 + 4i| = √(3² + 4²) = √(9 + 16)  = √(25) = 5

Thus, |3 + 4i| = 5

Related Articles

• Absolute value function
• Solving absolute value equations
• How to Find the Absolute Value of a Complex Number?

Solved Examples

Example 1: Solve 3 | x – 2 | = 15

Solution:

Given, 3| x – 2 | = 15
| x – 2 | = 5 
x - 2 = 5 or x - 2 = -5
x = 7 or x = -5 + 2 = -3

Example 2: Solve | 2x² - 1 | = | x² + 2 |

Solution:

Given, | 2x² - 1 | = | x² + 2 |

Using Property, | x | = | y | ⇒  x = ± y

2x² - 1 = x² + 2 and 2x² - 1 = - ( x² + 2 )
⇒ x² = 3 and  2x² -1 = -x² -2
⇒ x =  ±√ 3 and 3x² = -1 
⇒  x =  ±√ 3 and x = ±√( -1 / 3 ) = ± i / √ 3 =  ± √( 3 ) i/3
⇒  x =  ±√ 3 and x = ± (√3) i/3

Example 3: What is the value of 5 | 7x – 1 | if x = – 2?

Solution:

Given, find the value of, 5| 7x – 1 | if, x = – 2

= 5 | 7(-2)  – 1 |
= 5 | -14 -1 |
= 5 | -15 |
= 5 ( 15 )
= 75

Value of 5 | 7x – 1 | = 75, when x = -2

Practice Problems

Question 1: Arrange in ascending order: |-1|, |2|, -|7|, |-9|, -|5|, |-18|, |-5|, |16|.

Question 2: Find the absolute value of a number, -3/4.

Question 3: Determine the absolute value of complex number 4 + 9i.

Question 4: Find the absolute value of the complex number 3 - 2i.

Question 5: Evaluate |7-16|.

Question 6: Evaluate |-(8-12)|.

Question 7: Evaluate |-(-4+9)|.

Question 8: Given that z = 5 + 6i, find |z|.

Question 9: Write the answer in standard form: (2 – 7i)(3 + 7i) and find the absolute value of the complex number formed.

Question 10: Solve the following equation: |4p - 7| = 3 for p.

Hint for Q9 : i² = -1 and Q10 : Take 4p - 7 = -3 OR 4p - 7 = 3 , Now using the two Equations, Find value of p