The absolute value of a complex number a + bi is √(a² + b²), the straight-line distance from the origin to the point (a, b) in the complex plane. This quantity is also called the modulus of the complex number, and it is written |a + bi| or |z|. The result is always a non-negative real number, because the formula squares both the real part a and the imaginary part b before adding them and taking a square root; squaring removes any sign, and the principal square root returns the non-negative root. So for z = 3 + 4i, you get |3 + 4i| = √(3² + 4²) = √(9 + 16) = √25 = 5. For z = -3 - 4i, the answer is identical: |-3 - 4i| = √((-3)² + (-4)²) = √25 = 5. The two points are mirror images of each other through the origin, so they sit the same distance from it. The same rule also covers purely real inputs: when b = 0, |a| = √(a² + 0²) = |a|, which is the absolute value you already know from the real number line.
The Formula for |a + bi|
Every complex number z can be written in the form a + bi, where a is the real part and b is the imaginary part. The absolute value, also called the modulus, is defined as the distance from z to the origin (0, 0) in the complex plane, which by the Pythagorean theorem is:
|z| = |a + bi| = √(a² + b²)
Three things to notice about this formula:
- It uses a + bi, not (a, b) — the imaginary unit i is just a label, and the calculation works on the real numbers a and b.
- It always returns a real number, even though the input z is complex.
- It collapses to the real absolute value when b = 0, since √(a² + 0²) = √(a²) = |a|.
The Geometric Meaning: Distance from the Origin
Plot any complex number z = a + bi as a point in the plane, with the real part a along the horizontal axis and the imaginary part b along the vertical axis. The absolute value of z is the length of the straight line from that point back to the origin. The horizontal leg of the right triangle has length |a|, the vertical leg has length |b|, and the hypotenuse has length √(|a|² + |b|²) = √(a² + b²). This is exactly the Pythagorean theorem applied to the triangle formed by z, its projection on the real axis, and the origin.
The geometric picture is the easiest way to remember why the formula never produces a negative result. A length cannot be negative, and a square root of a sum of non-negative numbers cannot be negative, so |z| ≥ 0 for every complex z. The only complex number whose absolute value is zero is 0 + 0i itself, the point that sits exactly at the origin.
How to Calculate the Absolute Value of a Complex Number
Follow these steps to find |a + bi| for any complex number:
- Write the number in standard form a + bi. For z = 3 + 4i, a = 3 and b = 4. For 5i, write it as 0 + 5i, so a = 0 and b = 5. For -7, write it as -7 + 0i, so a = -7 and b = 0.
- Square the real part to get a². Use the sign of a as written, because the square will erase it. For a = 3, a² = 9.
- Square the imaginary part to get b². For b = 4, b² = 16.
- Add the two squares. a² + b² = 9 + 16 = 25.
- Take the principal (non-negative) square root of the sum. √25 = 5, so |3 + 4i| = 5.
Each of the arithmetic operations above is a real-number calculation, and the Absolute Value Calculator can be used to double-check the magnitudes. Type any real intermediate result into the input and read |x| straight from the result panel — for instance, enter 25 to confirm that the absolute value of 25 is 25, which lines up with the final answer of 5 squared.
Worked Example: |3 + 4i| = 5
Take z = 3 + 4i. The real part is a = 3 and the imaginary part is b = 4. Plugging into the modulus formula:
|3 + 4i| = √(3² + 4²) = √(9 + 16) = √25 = 5
Geometrically, the point (3, 4) sits on a 3-4-5 right triangle with the origin, so its straight-line distance from the origin is 5 units. If you flip the sign of both parts to get -3 - 4i, the formula gives the same answer because the signs disappear inside the squaring step: |-3 - 4i| = √(9 + 16) = 5. If only one part changes sign — for example 3 - 4i or -3 + 4i — the result is again 5, because a² and b² are unchanged.
To sanity-check the arithmetic with the tool, type 9 into the Absolute Value Calculator and confirm the answer is 9, which is the contribution of the real part. Repeat for 16 (the contribution of the imaginary part) and for 25 (the sum before taking the square root). Each value comes back as itself because all three are non-negative reals, and the final principal square root √25 is 5.
Common Uses of the Complex Modulus
The complex absolute value is the standard way to report magnitude in fields that work with complex numbers. Here is how a few areas use it:
| Field | What |z| measures |
|---|---|
| Electrical engineering (AC circuits) | Impedance magnitude |Z| of a load, the effective resistance to alternating current |
| Signal processing | Amplitude of a complex-valued signal at a single sample, useful in Fourier analysis |
| Quantum mechanics | Magnitude of a probability amplitude in a state vector |
| Computer graphics | Length of a 2D vector, mathematically identical to a complex modulus |
| Control theory | Magnitude of a transfer function, plotted as the Bode magnitude plot |
In every one of these, the formula is the same: square the real part, square the imaginary part, add, then take the non-negative square root. The Absolute Value Calculator handles the real-number side of the calculation — the squaring and absolute-value checks — so you can keep the imaginary-unit bookkeeping separate from the arithmetic.
Useful Properties to Remember
Three properties of the complex absolute value come up often in practice:
- |z| is never negative. Because it is a distance, |z| ≥ 0 for every complex z, and |z| = 0 only when z = 0.
- |z| = |-z|. Negating both parts leaves the squares untouched, so the modulus is unchanged.
- |z| = |z̄|. The complex conjugate has the same real-part magnitude and imaginary-part magnitude, so its modulus equals the original. For 3 + 4i, the conjugate is 3 - 4i and both have modulus 5.
For the real-number case — including the rules |ab| = |a|·|b| and |a + b| ≤ |a| + |b| — see our guide to calculating the absolute value of any number. The complex modulus extends the same idea to two dimensions, swapping the one-dimensional distance on the real line for a two-dimensional distance in the complex plane.