The Number That Turned Sideways

Summary

The Number That Turned Sideways A complex number comes in pairs: $a + bi$, where $a$ and $b$ are real numbers and $i$ is the square root of negative one. The magnitude is $$|z| = \sqrt{a^2 + b^2}$$ Given the magnitude and an angle $\theta$, you can reconstruct the number: $$z = |z|(\cos\theta + i\sin\theta)$$ These numbers are two-dimensional, like fractions — defined by two pieces of information. They live on a plane: the horizontal axis for the real part, the vertical axis for the imaginary. Every complex number is a point; every point is a complex number. Multiply a complex number by a real scalar — say, $7 \times (2 + 3i) = 14 + 21i$ — and the magnitude changes. The number stretches or shrinks along its direction. But put $i$ in the exponent, and something else happens: Raise a number to the power of $i$, and it rotates. $i$ itself rotates $\frac{\pi}{2}$. Multiply by $i$ twice — that's $i^2 = -1$ — and you've rotated a full $\pi$. Three times, $\frac{3\pi}{2}$. Four times, back to where you started: a complete circle. It doesn't have to be $i \times i$; it can be $e^{i\pi}$ or $3^{2i}$. For some mysterious reason, when $i$ appears in the exponent, the number rotates. This is what the textbooks teach. This is what earns students high marks. And yet: why does the exponent rotate? Who decided $\sqrt{-1}$ belongs on a vertical axis? What does the exponential function — the function of growth and decay — have to do with circles? This wasn't one genius with a complete vision. It was a relay race across three centuries. The race begins in 1572, when an Italian mathematician named Rafael Bombelli defined the arithmetic rules that make $i$ behave the way it does. He built the engine. In 1748, Leonhard Euler fed that engine into his infinite series — what we now call the Taylor expansion of $e^x$ — and discovered that growth and oscillation were the same thing in disguise. And between 1799 and 1831, three men — a Norwegian surveyor, a Parisian bookkeeper, and the prince ...