I came across the following while browsing a collection of maths “jokes”. I’m not sure how they got into the list, but they’re an excellent way to test how well you can spot the holes in an apparently plausible mathematical argument…

**Theorem 1**: .

**Proof**. We start with

Subtracting from each side and factorising on the LHS, we have

Adding to each side now yields

This may be written as

Finally, we take the square roots of both sides, obtaining

and cancelling the terms we are left with as required. QED.

**Remark**. This argument can be generalised to show that for any numbers .

**Theorem 2**: £1.00 = 1p. (A very useful theorem in modern macroeconomics!)

**Proof**. This is purely arithmetical. We have . QED.

**Theorem 3**: .

**Proof**: By one of the standard log laws, we have

On the other hand, we also know that

Combining these two results gives and hence as required. QED.

**Theorem 4**. All positive integers are equal.

**Proof**. It is sufficient to show that for any two positive integers, and , . Further, it is sufficient to show that for all , if and (both positive integers) satisfy then .

We proceed by induction.

(i) If , then and , being positive integers, must each be 1. So .

(ii) Assume that the result holds for some value . Now take and with . Then , and hence . Consequently, .

(iii) We have shown that the result holds for and that if it holds for then it must also hold for . Hence, by induction, the theorem is true for all . QED.

Please use the comments thread to point out the flaws!

(DP)

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