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# Proof that 1 + 1 = 1. Not 2!?!

Here is a quick demonstration of what can go wrong when you violate the rules of mathematics.

1. Let  $a = 1$  and  $b = 1$ .

2. Now this means that  $a = b$ .

3. If we multiply both sides by  $a$  we get  $a^{2} = ab$ .

4. If we then subtract  $b^{2}$  from both sides we would have  $a^{2} - b^{2} = ab - b^{2}$ .

5. We can then factorise both sides to get  $(a + b)(a - b) = b(a - b)$ .

6. Dividing both sides by  $(a - b)$  would give us  $a + b = b$ .

7. Substituting back the values of  $a = 1$  and  $b = 1$  would give us that  $1 + 1 = 1$ .

8. So this "proves" that  $1 + 1 = 1$  not  $2$ .

Except that in step 6, when we are dividing by  $a - b$ , we are in fact dividing by zero. This is a violation of the rules of mathematics and hence the reason why our conclusion is invalid.

Why not try this demonstration with someone you know and see if they can spot the problem?

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CITE THIS AS:
Ovens, Matthew. "Proof that 1 + 1 = 1. Not 2!?!" Retrieved from YourStatsGuru.

First published Dec, 2011 | Last updated: 20 January 2018