Quantum Diaries

I was quite surprised to see an article on the main NY Times website this morning on complex numbers.

I thought the article was good both for the mathematical content and for the prose:

Complex numbers are magnificent, the pinnacle of number systems. They enjoy all the same properties as real numbers — you can add and subtract them, multiply and divide them — but they are better than real numbers because they always have roots. You can take the square root or cube root or any root of a complex number and the result will still be a complex number.

One of the things the article doesn’t mention (which is ok, I didn’t expect a textbook) is the polar representation of complex numbers. Real positive numbers correspond to an angle of 0 and real negative numbers correspond to an angle of 180 degrees or π (pi) radians. The name ‘imaginary’ numbers lead one to imagine that this part of math is just a game with rules – not anything based in nature. But quantum mechanics is full of them – any quantum mechanical phenomena we witness could not be explained without complex numbers. The ‘imaginary’ part is the phase – something that is not observable but is carried along. For instance, a wave has a phase and an amplitude. What we measure is the amplitude. But if two waves interact, their phases interact and the waves could be ‘in phase’ (adding amplitude) or ‘out of phase’ (subtracting amplitude). It is relatively straightforward in the world of complex numbers.

Here is the finale of complex numbers: Euler’s Identity. It brings together the 5 most important numbers in math

• Start with e, the base of the exponential.
• When raised to an imaginary power (i), it is a rotation in the complex plane.
• What angle to rotate it? How about π (pi), the magic number of circles.
• e^iπ (e to the i pi) rotates from 1 halfway around the complex plane, resulting in -1.
• This gives us 3 numbers – the final numbers are 1 and 0.

So Euler’s Identity becomes:

I considered this for a long time for a tattoo. I hope to find something this elegant in physics for my next tattoo, or else someone might mistake me for a mathematician.