As an undergrad, I took a couple “math for physicists” courses that I found to be quite helpful. One of the more humorous concepts a professor of mine conveyed to me was the idea of a “physicist’s proof”. These would be sort of intuitive proofs that a mathematician may scoff at, but physicists tend to appreciate. Below is an example of a physicist’s proof showing the following mathematical relation:
This relation can more easily be stated in words. It says that if you add up consecutive odd numbers starting at 1, you get a perfect square. For instance, or .
Here is the idea:
For n=0, you get a 1×1 square as is seen in the following image:
For n=1, you add the 3 next squares and you get the following 2×2 square:
For n=2 and n=3, one adds 5 and 7 squares respectively to get the following 3×3 and 4×4 squares:
By now, you can probably see the pattern and why this mathematical relation holds (hopefully where I have put the dashes and dots helps you see this!). To me, these kinds of proofs, while lacking in mathematical rigor, do much for the intuition. In fact, they even suggest an algebraic manner by which to prove the sum rigorously.
Long live the physicist’s proof!