Complex Numbers

The imaginary unit i is defined by i² = −1. A complex number z = a + bi combines a real part a and an imaginary part b, and is plotted on the complex plane.

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No real number squares to a negative result: whatever you choose, the square is zero or positive. Rather than declare $\sqrt{-1}$ impossible, mathematics gives it a name — the imaginary unit $i$ — and everything that follows is built on that single definition.

Concept The imaginary unit

i = \sqrt{-1} \qquad \Longrightarrow \qquad i^2 = -1

The letter comes from imaginary. Raising it to successive powers produces a cycle of length four:

i^1 = i

i^2 = -1

i^3 = i^2 \cdot i = -i

i^4 = i^2 \cdot i^2 = 1

i^5 = i

, and the pattern repeats every four powers.

To evaluate any power of $i$ , divide the exponent by 4 and keep only the remainder.

Concept Definition of a complex number

z = a + bi

a is the real part.

b is the imaginary part.

If b = 0 the number is purely real.

If a = 0 the number is purely imaginary.

If neither is zero it is a genuine complex number.

Plotted this way, a complex number is a point of the plane: the horizontal axis carries the real part, the vertical axis the imaginary part. Multiplying by $i$ rotates that point 90° counter-clockwise — which is exactly why the powers of $i$ close up after four steps.

Example Adding and subtracting

Let $z_1 = 3 + 4i$ and $z_2 = 1 - 2i$ . Find $z_1 + z_2$ and $z_1 - z_2$ .

Combine real parts with real parts and imaginary with imaginary.

z_1 + z_2 = (3 + 1) + (4 - 2)i = 4 + 2i

z_1 - z_2 = (3 - 1) + (4 + 2)i = 2 + 6i

⟹ sum = 4 + 2i, difference = 2 + 6i

Example Multiplying

Multiply $(3 + 2i)(1 + 4i)$ .

Expand exactly as with binomials.

= 3 + 12i + 2i + 8i^2

Now replace

i^2

by

-1

.

= 3 + 14i - 8

⟹ −5 + 14i

The one substitution $i^2 = -1$ is what turns an ordinary expansion into complex arithmetic.

Summary

i is defined by i² = −1, and its powers cycle i, −1, −i, 1 every four steps.
A complex number is z = a + bi, with a the real part and b the imaginary part.
In the complex plane, the real part is the horizontal coordinate and the imaginary part the vertical.
Add and subtract part by part; multiply by expanding, then replace i² with −1.

231. Arithmetic Sequences and Series:Next

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