Imaginary Reality

Posted on Mar 11, 2022

Many years ago, Schrodinger figured out that imaginary numbers are the only way to make sense of reality. Professor F Dyson described it best in his recent lecture1:

…But then came the surprise. Schrodinger put the square root of minus one into the equation, and suddenly it made sense. Suddenly it became a wave equation instead of a heat conduction equation. …And that square root of minus one means that nature works with complex numbers and not with real numbers.

Imaginary numbers are very real. Consider trigonometry. The angles and ratios are natural, right? You can write them using exponents of irrational numbers (something that we cannot count) raised by imaginary numbers (something that doesn’t precisely exist).

\(\exp(ix)\) can be decomposed with \(\sin (x)\) and \(\cos(x)\) using the following formulas.

\[ \exp(ix) = \cos (x) + i \sin(x), \]

\[ \exp(-ix) = \cos(x) - i \sin(x). \]

Now, add up the two equations to get the value of \(\cos(x)\) and \(\sin(x)\).

\[ \cos(x) = \frac{\exp(ix) + \exp(-ix)}{2}, \]

\[ \sin(x) = \frac{\exp(ix) - \exp(-ix)}{2}. \]

I used this trick to solve my high school trigonometry problems. In a broader sense, this also means that we can represent a natural angle and ratio in terms of quantities we can’t define as naturally in real life. Isn’t this beautiful?

The only way to make sense of reality is to borrow from the imaginary world. We will create non-sense tools that speak well with what we do know. Some imaginary tools that bind with reality as we know it. Some day we will be able to reconcile all of that knowledge together. There is a lot we do not know.

  1. Freeman, D. (2009). Birds and Frogs. Chinese Journal of Nature. PDF.↩︎