We are back for another Solve it Sundays, and this one I didn’t find too tricky. Good luck! and as always, the answers are in the comments.

It is tempting, soothing even, to think of mathematics as a perfect edifice of logic and order. The truth however is that it is an art as well as a science, and it has places where absolutism breaks down.

For this example, we will show that 0 = 1. Firstly, however, I should point out that when adding a series of numbers, the associative law says that you may bracket the sums as you like without any effect.

1+2+3 = 1+ (2+3)= (1+2) + 3.So, with that established, consider adding an infinite number of zeroes. No matter how much nothing you gather, you will still always have nothing.

0 = 0+0+0+0+0+…Since 1-1 = 0, you can replace each zero in your sum, like so:

0 = (1-1)+(1-1)+(1-1)+(1-1)+(1-1)+…From the associative law, you may arrange the brackets in your sum as you see fit. Which means:

0 = 1+(-1+1)+(-1+1)+(-1+1)+(-1+1)+(-1+1)+…However, as established, (-1+1) = 0, so this sequence can also be stated as:

0 = 1+0+0+0+0+0+…Or, for simplicities sake:

## 0 = 1

Something is clearly incorrect. But what?

ANSWER BELOW

The error is assuming that the associative law applies to an infinite calculate. That isn’t necessarily true. Infinity is uncountable, and therefore indefinite – it goes on for ever, after all – and if your chain of sums isn’t fixed you can’t freely rearrange things. It is the chain of infinite (+1-1) and (-1+1) expressions that is equal, not the whole equation.

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