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?