Numbers and algebra

1748

Euler’s formula

Leonhard Euler

What could spinning in a circle have to do with interest at the bank?

Euler linked the number e (the natural base of logarithms), the imaginary number i, and the sine and cosine of trigonometry in one line. Spinning in a circle and growing exponentially turned out to be the same idea.

First, make a guess

Where does eiπe^{i\pi} land on the circle?

How they figured it out

The number e came from money. Jacob Bernoulli asked in 1683: if a bank pays 100% interest, what happens if it compounds twice a year, monthly, daily, every second? The answer creeps up towards 2.71828…, never beyond.

Leonhard Euler named it e and, in 1748, showed that raising it to an imaginary power walks you around a circle. Euler was the most prolific mathematician in history. He lost the sight of one eye in his thirties and became almost completely blind in his sixties, and kept producing work at the same pace, dictating it.

The equation

eiθ=cos⁡θ+isin⁡θe^{i\theta} = \cos\theta + i\sin\theta

Raising e to an imaginary power walks you around a circle.

Everyday echoEngineers describe alternating current and radio signals this way.

See it in real life

₹100 and a very generous bank

Put ₹100 in an account paying 100% a year. Compounded once, you get ₹200. Compounded every second, you get ₹271.83: that is e. And the 50 Hz mains in your wall is a point spinning 50 times a second, written V0ei 2π50tV_0 e^{i\,2\pi 50 t}.

Where engineers use it

  • AC circuits

    Electrical engineers write voltages as VeiωtV e^{i\omega t} so that capacitors and inductors become simple multiplication.

  • Signal processing

    Every Fourier transform, filter and radio modulator uses Euler’s formula.

Worked example: Half a turn

  1. eiπ=cos⁡π+isin⁡π=−1+0ie^{i\pi} = \cos\pi + i\sin\pi = -1 + 0i
  2. So eiπ+1=0e^{i\pi} + 1 = 0.

Where it came from

What it led to