Solve it Sundays: Absolutely Nothing

Solve it Sundays: Absolutely Nothing

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?

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One response

  1. 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.

    Like

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