Complex Numbers

What number multiplied by itself gives negative one? No real number works, so mathematicians simply invented one. Every real number lives on a line. Squares of reals are never negative, so the equation x squared equals minus one has no real solution. We define a new number called i, with the single rule: i squared equals minus one. Everything else follows from that rule.

A complex number has a real part a and an imaginary part b. We write it as a plus b i.

Since two independent numbers are needed, z lives on a plane, not a line. Real part goes across, imaginary part goes up. Take z equals four plus three i. Its real part is the green horizontal leg, four. Its imaginary part is the blue vertical leg, three. The point z sits at the far corner of this triangle.

The distance from the origin to z is the modulus. By Pythagoras, it is the hypotenuse of this triangle.

Let us compute it. Square each leg and add them, exactly like the Pythagorean theorem.

Sixteen plus nine is twenty five.

And the square root of twenty five is five. So the modulus of four plus three i is exactly five.

Now multiplication. Expand like ordinary algebra, then replace i squared by minus one. Try one plus two i, times three plus i.

Multiply every term: one times three, one times i, two i times three, and two i times i.

Now use the rule i squared equals minus one, so two i squared becomes minus two.

Collect real parts and imaginary parts separately. The product is one plus seven i.

Here is the deep reason multiplication is special. Write z with its length r and its angle theta from the real axis.

Multiply two such numbers. The cross terms combine using the angle addition formulas for cosine and sine.

So multiplying complex numbers multiplies lengths and adds angles. Multiplication is a rotation plus a stretch. Check with i itself. It has length one and angle ninety degrees, so multiplying by i twice rotates by one hundred eighty, giving minus one. A common misconception is that imaginary numbers are fake or useless. They are just as consistent as negatives, and engineers rely on them daily.

• Imaginary does not mean useless

Real use: alternating current and signals. A wave's amplitude and phase fit in one complex number, and phase shifts become simple rotations. One limit to remember: complex numbers have no natural ordering, so we cannot say one is bigger than another, only compare moduli. Bigger picture: every polynomial has a complex root, and Euler's formula links e, angles and i. Now test your understanding.

• No order, only modulus compares

Summary: i squared is minus one, numbers become points in a plane, and multiplying means rotate by the angle and scale by the length.