Complex modulus is one of 3 complex numbers formulas in the precalculus section of this library, and it is used at high school · ap level.
Why complex modulus works
Plot a + bi as the point (a, b) on the complex plane. The real part runs along the horizontal axis and the imaginary part runs up the vertical one, and the two axes meet at a right angle. The modulus is just the hypotenuse of that right triangle, which is why the Pythagorean pattern appears.
What each symbol means
$a$ is real part and $b$ is imaginary coefficient.
Complex modulus: when it holds
$a,b$ are real; modulus is always nonnegative.
When it stops applying
The modulus records distance and throws direction away, so it cannot tell two numbers apart. Both 3 + 4i and −4 + 3i have modulus 5 while pointing at completely different parts of the plane, which is why the argument is needed alongside it.
Complex modulus: a worked example
$|3-4i|=\sqrt{9+16}=5$.
The mistake to avoid
What people do: Students subtract the parts, reporting the modulus of 3 − 4i as 3 − 4 = −1.
Why it goes wrong: A modulus is a distance, so it can never be negative. The minus sign in the number tells you the point sits below the axis, not that the distance shrinks.
Do this instead: Square both parts before adding, which kills the sign: (−4)² is 16, so the modulus is √(9 + 16) = 5. Both 3 + 4i and 3 − 4i sit 5 units from the origin.
Complex modulus: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $a$ is real part and $b$ is imaginary coefficient.
- Check the conditions before substituting. $a,b$ are real; modulus is always nonnegative.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most precalculus slips.
Where this formula fits
- Subject
- Precalculus formulas — 10 entries in this library
- Topic
- Complex numbers
- Level
- High school · AP
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where complex modulus comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- The Quadratic Formula Explained — the lesson behind this formula: use the formula and interpret its discriminant.
- Precalculus Calculator — check your substitution and the value it produces.
- Study precalculus — the subject guide that explains the ideas these formulas compress.
- Precalculus Practice — questions that make you retrieve the formula instead of recognising it.
- All 10 precalculus formulas — the full grouped reference, or the complete formula library.
Questions about complex modulus
What is the modulus of a plain real number?
It is the ordinary absolute value. With b = 0 the formula reduces to √(a²), so the modulus of −7 is 7, and the complex idea contains the real one as a special case.
Is |z|² the same as z²?
No. Squaring 3 − 4i gives −7 − 24i, while |z|² is the plain number 25. The correct identity is |z|² = z times its conjugate, which is what makes the imaginary parts cancel.
How is the modulus different from the argument?
The modulus says how far the point is from the origin, and the argument says which direction it lies in. Together they fix the number exactly, the way distance and bearing fix a location.
Does the modulus of a product equal the product of the moduli?
Yes, and that makes multiplication easy to picture. Multiplying two complex numbers multiplies their distances from the origin while adding their angles.