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# Cool proofs and origins

I am always interested in simple/elegant/cool proofs of mathematical facts. I’m also interested to know their origins. Does anyone know who came up with the following rather elegant proof of ? It’s likely that the origins are lost in time but I thought I would ask anyway.

Proof:

I like this because of the use of 2-1=1. Another situation where the use of 1 gives a cool proof is the following from complex analysis. Does anyone know the origin of the proof?

Theorem: If is complex Riemann integrable and is Riemann integrable, then .

Proof: If the left-hand side is zero, then the statement is trivial. Hence, assume that the left-hand side is non-zero. Define the complex number by

We have,

Do you have any cool proofs? (With or without clever use of 1!)

The second proof “trick” is very common when dealing with complex vector spaces (it must be used 20 times in a typical course on Banach spaces). Integration is a linear functional from $C[a,b]$ to the field– you normally define it for real scalars, and then extend to complex scalars by “taking real and imaginary parts”. Once you’re in this situation, this idea of “rotating the problem onto the real line” is very common.

Indeed, rather than define $c$ by division, I think I’d say “Let $c=e^{i\theta}$ for some real $\theta$ chosen so that $c \int_a^b g(t) \ dt$ is non-negative.” Whether that’s easier for students to understand is open to debate I guess…

Indeed the underlying trick is a common one but I think that this is the first point at which most students would meet it. I prefer dressing it up so that the c introduced earlier magically disappears at the end. I enjoy the proof more that way!

The $c=e^{i\theta }$ proof requires more initial understanding from students and likely leads to a deeper understanding when grasped so I would leave it to a more advanced course.