# A Surprising Result. Yes, another one.

I realised that I haven’t made any maths posts in a while, so here you are. This one’s really surprising, and I didn’t know about it until this morning. Take a progression that is just eternally adding and subtracting ones:

1 – 1 + 1 – 1 + 1 – 1… to infinity terms

Say s = 1 – 1 + 1 – 1…

1 – s = 1 – (1- 1 + 1 – 1…)

1 – s = 1 -1 + 1 – 1… (as all signs inside the bracket are reversed)

1 – s = s

1 = 2s, so s = 1/2

So, if you keep adding and subtracting ones, you’ll end up with 1/2.

In mathematical notation, s can be represented by Σ (for k = 0 to infinity) (1 – 1). Sigma notation (this is a capital sigma: Σ) is used to add something multiple times. (For example, Σ (for k = 1 to n) (2k + 1) = n2)

So, basically, the expression Σ (1 – 1) represents (1 – 1 + 1 – 1… to infinity, depending on the upper and lower bounds you set on k; here, it’s to infinity.)

But Σ (1 – 1) = Σ 0, and however many times you add 0 to itself, it’ll never quite reach 0.000000000…(insert infinity zeroes here)….00001. It’ll always stay at 0, but here, from the same expression, we get two values, 0 and 1/2.

Therefore, 2*0 = 2*(1/2) —> 0 = 1.

Infinity works in mysterious ways, doesn’t it?