Absolute Value Calculator (|x| of real numbers and |a + bi| of complex numbers)

Understanding the absolute value calculator — concept and principle

This calculator finds the absolute value of a real number, |x|, or the modulus of a complex number, |a + bi|. It also shows the exact form, a² + b², the angle (argument) and which other numbers share the same absolute value. Use it for homework, error sizes and distances.

How it is calculated

For a real number: |x| = x if x ≥ 0, and |x| = −x if x < 0

For a complex number z = a + bi: |z| = √(a² + b²)

Argument = angle of the point (a, b) from the positive real axis

• With whole-number a and b the root is simplified, for example |2 + 4i| = √20 = 2√5.
• For a real number, the numbers with the same absolute value are x and −x.

The principle

Absolute value is distance from zero, and distance is never negative. On the number line, −7.5 and 7.5 are both 7.5 units from 0. A complex number a + bi is a point in a plane with a real axis and an imaginary axis, so its distance from the origin comes from the Pythagorean theorem: √(a² + b²). For example |3 + 4i| = √(9 + 16) = 5. The absolute value together with the angle gives the polar form of a complex number.

Worked example

Default inputs: a = −7.5, b = 0.

1. The number is the real number −7.5, measured on the number line.
2. Since −7.5 < 0, |−7.5| = −(−7.5) = 7.5; exact form 7.5.
3. a² + b² = 56.25 + 0 = 56.25, and √56.25 = 7.5 confirms the result.
4. The point lies on the negative real axis, so the argument is 180°.
5. Numbers with the same absolute value: 7.5, −7.5.

How to read the result

The absolute value is always 0 or positive, and it is 0 only for 0 itself. In applications it measures size without direction: an error of −0.3 and +0.3 has the same size, and the distance between two numbers p and q is |p − q|. For complex numbers, all points with the same modulus lie on a circle around the origin.

Common mistakes

• Removing minus signs term by term inside an expression: |a − b| is not |a| − |b|.
• Writing |a + bi| = |a| + |b| instead of √(a² + b²).
• Forgetting that |x| = 5 has two solutions, 5 and −5.
• Giving the argument in radians when degrees are expected, or the reverse.

Frequently asked questions

What is the absolute value of −7.5?

7.5.

What is the absolute value of 3 + 4i?

5, because √(3² + 4²) = 5.

Can absolute value be negative?

No. It is a distance, so it is always 0 or positive.

Sources: OpenStax, Prealgebra 2e (2020), §3.1 Introduction to Integers · OpenStax, Precalculus 2e (2021), §8.5 Polar Form of Complex Numbers