Let ε < 0.

In honor of tax day…

The Salary Theorem. The less you know, the more money you make.

Proof. It is known in universities that knowledge is power, hence

Similarly, it is known in business that time is money, whence

From physics, we have by definition that power is the ratio of work to time, so that

Making the substitutions above, we have

Solving for money, we get:

Thus, as knowledge approaches zero, money approaches infinity, regardless of the amount of work done. Q.E.D.

Theorem. A peanut-butter and jelly sandwich is better than life itself.

Proof. A peanut-butter and jelly sandwich is better than nothing.

Nothing is better than life itself.

By transitivity, a peanut-butter and jelly sandwich is better than life itself. Q.E.D.

Theorem. A sheet of paper is a lazy dog.

Proof. A sheet of paper is an ink-lined plane.

An inclined plane is a slope up.

A slow pup is a lazy dog.

By transitivity, a sheet of paper is a lazy dog. Q.E.D.

Theorem. A cat has nine tails.

Proof. No cat has eight tails.

A cat has one tail more than no cat.

Therefore, a cat has nine tails. Q.E.D.

Theorem. A dollar is equal to one penny.

Proof. We shall use the conventional physics notation and write units behind the numbers. That is, 1 $ means 1 dollar, and 5 c means five cents. Proceeding:

1 $
= 100 c
= (10 c)²

= (0.1 $)²
= 0.01 $
= 1 c. Q.E.D.

The following generalizes yesterday’s theorem.

Theorem. Any integer equals its successor, i.e. n = n+1 any integer n.

Proof.

Q.E.D.

Theorem. Zero equals unity, i.e. 0 = 1.

Proof.

Q.E.D.

Theorem. The integral of cosine over its first period is zero.

Proof.

Q.E.D.

This was an honest-to-God answer that a calculus student wrote on a final exam! I kid you not — I shared an office with the lucky guy who graded it at the time. As an interesting post script to this story, the student actually demanded he be awarded partial credit, because he got the right answer.

Here is a slightly more rigorous version of yesterday’s theorem.

Theorem. Every natural number can be unambiguously described in 14 words or less.

Proof. Suppose there is some natural number which cannot be unambiguously described in fourteen words or less. Then there must be a smallest such number. Let’s call it n.

But now n is “the smallest natural number that cannot be unambiguously described in fourteen words or less.” This is a complete and unambiguous description of n in fourteen words, contradicting the fact that n was supposed not to have such a description!

Therefore, all natural numbers can be unambiguously described in fourteen words or less. Q.E.D.