Difference between Absolute Value and Modulus with Examples

Difference between absolute value and modulus is a foundational concept in mathematics that helps students understand how numbers are measured in relation to zero.Absolute value and modulus are two names for the same concept and the two terms are often used interchangeably at the school level. It is the non-negative distance of a number from the origin on the number line, regardless of whether the number is positive or negative. The term ‘modulus’ carries extra weight in more advanced mathematics, particularly when complex numbers are involved. In this guide, students will learn how absolute value and modulus are represented, how they behave in different situations, and why the result is always non-negative.

Table of Contents

• What Is Absolute Value?
• What Is Modulus?
• Modulus of a Complex Number
• Absolute Value vs Modulus

What Is Absolute Value?

The absolute value of a number is simply its distance from zero on the number line. 

Notation: the absolute value of a number x is written |x|, read as ‘mod x’ or ‘absolute value of x.’

The formal definition:

|x| = x, if x ≥ 0

|x| = −x, if x < 0

Examples: |7| = 7, |-7| = 7, |0| = 0

What Is Modulus?

The word modulus comes from Latin, meaning ‘a small measure.’ It was introduced into mathematics in 1806 by the French mathematician Jean-Robert Argand.

For real numbers (all ordinary numbers: integers, decimals, fractions), modulus and absolute value are completely interchangeable. |x| is both the absolute value and the modulus of x.

Modulus of a Complex Number 

For a complex number z = a + bi, its modulus (distance from the origin in the complex plane) is:

|z| = √(a² + b²)

The complex number forms a right-angled triangle with legs a and b, and the modulus is the hypotenuse. The modulus gives you how far the number is from the origin.

Notice that for a real number x (where b = 0):

|x + 0i| = √(x² + 0²) = √(x²) = |x|

So the modulus of a complex number reduces to the ordinary absolute value when the imaginary part is zero. This is why both terms co-exist and overlap Absolute value is a special case of modulus.

Example: Modulus of 3 + 4i = |3 + 4i| = √(3² + 4²) = √(9 + 16) = √25 = 5.

Absolute Value vs Modulus