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 impossible, mathematics gives it a name — the imaginary unit — and everything that follows is built on that single definition.

Concept The imaginary unit

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

, and the pattern repeats every four powers.

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

Concept Definition of a complex number

3 + 2i a = 3 b = 2 Re Im
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 rotates that point 90° counter-clockwise — which is exactly why the powers of close up after four steps.

Example Adding and subtracting

Let and . Find and .

Combine real parts with real parts and imaginary with imaginary.
⟹ sum = 4 + 2i, difference = 2 + 6i
Example Multiplying

Multiply .

Expand exactly as with binomials.
Now replace by .
⟹ −5 + 14i

The one substitution is what turns an ordinary expansion into complex arithmetic.

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