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Absolute Value of a Complex Number — Definition

Absolute Value of a Complex Number
Modulus of a Complex Number

The distance between a complex number and the origin on the complex plane. The absolute value of a + bi is written |a + bi|, and the formula for |a + bi| is \(\sqrt {{a^2} + {b^2}}\). For a complex number in polar form  r(cos θ + isin θ)  the modulus is r.

 

Complex plane showing vector from origin to point a+bi, with |a+bi| as its length and argument as its angle from real axis.

 

 

See also

Real number, imaginary number, argument of a complex number

Key Formula

∣a+bi∣=a2+b2|a + bi| = \sqrt{a^2 + b^2}
Where:
  • aa = The real part of the complex number
  • bb = The imaginary part of the complex number (the coefficient of i)
  • ∣a+bi∣|a + bi| = The absolute value (modulus) of the complex number, representing its distance from the origin

Worked Example

Problem: Find the absolute value of the complex number 3+4i3 + 4i.
Step 1: Identify the real part aa and the imaginary part bb.
a=3,b=4a = 3, \quad b = 4
Step 2: Substitute into the formula ∣a+bi∣=a2+b2|a + bi| = \sqrt{a^2 + b^2}.
∣3+4i∣=32+42|3 + 4i| = \sqrt{3^2 + 4^2}
Step 3: Square each part and add.
=9+16=25= \sqrt{9 + 16} = \sqrt{25}
Step 4: Take the square root to get the final answer.
=5= 5
Answer: ∣3+4i∣=5|3 + 4i| = 5

Another Example

This edge case shows that when the imaginary part is zero, the absolute value of a complex number reduces to the ordinary absolute value of a real number. Notice that ∣−5∣=5|-5| = 5 whether you use the real-number definition or the complex-number formula.

Problem: Find the absolute value of −5+0i-5 + 0i (a purely real number treated as a complex number).
Step 1: Identify the real and imaginary parts. Since there is no imaginary component, b=0b = 0.
a=−5,b=0a = -5, \quad b = 0
Step 2: Apply the formula.
∣−5+0i∣=(−5)2+02|-5 + 0i| = \sqrt{(-5)^2 + 0^2}
Step 3: Simplify.
=25+0=25=5= \sqrt{25 + 0} = \sqrt{25} = 5
Answer: ∣−5+0i∣=5|-5 + 0i| = 5

Frequently Asked Questions

What is the difference between absolute value of a real number and absolute value of a complex number?
For a real number aa, the absolute value ∣a∣|a| gives its distance from 0 on the number line. For a complex number a+bia + bi, the absolute value a2+b2\sqrt{a^2 + b^2} gives its distance from the origin on the two-dimensional complex plane. The complex formula generalizes the real one: if b=0b = 0, then a2+02=∣a∣\sqrt{a^2 + 0^2} = |a|, so the two definitions are fully consistent.
Why is the absolute value of a complex number always positive or zero?
Because a2a^2 and b2b^2 are each non-negative, their sum a2+b2a^2 + b^2 is non-negative, and the principal square root of a non-negative number is always non-negative. The only way the result equals zero is when both a=0a = 0 and b=0b = 0, meaning the complex number itself is 00.
How do you find the absolute value of a complex number in polar form?
If a complex number is written in polar form as r(cos⁡θ+isin⁡θ)r(\cos\theta + i\sin\theta), the absolute value is simply rr, the radial distance from the origin. You do not need to convert back to rectangular form — the modulus rr is already built into the polar representation.

Absolute Value (Modulus) vs. Argument of a Complex Number

Absolute Value (Modulus)Argument of a Complex Number
What it measuresDistance from the origin (how far)Angle from the positive real axis (which direction)
Formula (rectangular form)a2+b2\sqrt{a^2 + b^2}θ=arctan⁡(ba)\theta = \arctan\left(\frac{b}{a}\right) (adjusted for quadrant)
Result typeA non-negative real numberAn angle, typically in radians or degrees
In polar form r(cos⁡θ+isin⁡θ)r(\cos\theta + i\sin\theta)Equals rrEquals θ\theta
Geometric meaningRadius of a circle centered at the originDirection measured counterclockwise from the positive real axis

Why It Matters

You encounter the absolute value of a complex number frequently in precalculus and physics when converting between rectangular and polar forms. It is essential for multiplying and dividing complex numbers in polar form, since ∣z1⋅z2∣=∣z1∣⋅∣z2∣|z_1 \cdot z_2| = |z_1| \cdot |z_2|. In electrical engineering, the modulus represents the amplitude of alternating current signals modeled by complex numbers.

Common Mistakes

Mistake: Adding aa and bb before squaring, computing (a+b)2\sqrt{(a + b)^2} instead of a2+b2\sqrt{a^2 + b^2}.
Correction: You must square the real and imaginary parts separately, then add the squares. For example, ∣3+4i∣=9+16=5|3 + 4i| = \sqrt{9 + 16} = 5, not (3+4)2=7\sqrt{(3+4)^2} = 7.
Mistake: Forgetting to include the sign of aa or bb when squaring, especially with negative values like −3+4i-3 + 4i.
Correction: Squaring always makes the result positive: (−3)2=9(-3)^2 = 9. The formula a2+b2\sqrt{a^2 + b^2} handles negative components automatically, so ∣−3+4i∣=9+16=5|-3 + 4i| = \sqrt{9 + 16} = 5, the same as ∣3+4i∣|3 + 4i|.

Related Terms