A complex number $a + bi$ (from complex arithmetic) has two independent parts, just as a point has two coordinates. So we can picture each complex number as a point, and the distance formula then measures its size.
Key idea
In the complex plane, $a + bi$ is the point $(a, b)$: the real part across, the imaginary part up. Its modulus is its distance from $0$: $$|a + bi| = \sqrt{a^2 + b^2}.$$ Adding complex numbers adds them like vectors, the conjugate $a - bi$ is the reflection across the real axis, and $|z - w|$ is the distance between $z$ and $w$.
Why it works
Two complex numbers are equal only when both parts match, so a complex number carries exactly the information of a point. Adding real parts and imaginary parts separately is the same as adding vector components, which is why sums follow the tip-to-tail picture.
Worked examples
Worked example: Plot and measure
Plot $-2 + 3i$ and find its modulus.
- It is the point $(-2, 3)$: two left, three up, in Quadrant II.
- $|-2 + 3i| = \sqrt{4 + 9} = \sqrt{13}$.
Answer: $\sqrt{13} \approx 3.61$
Worked example: Adding as vectors
Add $(3 + i) + (-1 + 2i)$ and describe the picture.
- Real parts: $3 - 1 = 2$. Imaginary parts: $1 + 2 = 3$.
- The arrows to $3 + i$ and $-1 + 2i$ form two sides of a parallelogram; the sum is the far corner.
Answer: $2 + 3i$
Worked example: A distance
How far apart are $1 + 2i$ and $4 - 2i$?
- Subtract: $(1 + 2i) - (4 - 2i) = -3 + 4i$.
- $|-3 + 4i| = \sqrt{9 + 16} = 5$.
Answer: $5$
Common mistake
Computing $|3 + 4i|$ as $3 + 4 = 7$, or as $\sqrt{3^2 + (4i)^2} = \sqrt{9 - 16}$. The modulus is a distance, found from the coordinates $3$ and $4$ (no $i$ inside the root): $\sqrt{9 + 16} = 5$. A modulus is never negative and never imaginary.