That's the point! Using the algebra from the article:
(1 - 1 + 1 - 1 ...) = 0 + (1 - 1 + 1 - 1 ...)
and therefore:
(1 - 1 + 1 - 1 ...) = (0 + 1 - 1 + 1 - 1 ...)
And as they're mathematically the same (again, in the article's algebra), why not replace one with the other?
The problem is that when you do, and you sum the two infinite series using the "zipping" method (as in the numberphile video) the two equations equate to different results.
Can someone explain to me why I am allowed to add a padding zero at the start without taking it out at the end?
By adding it, I am giving the sum an offset, and I can kind of understand why adding zero is not a big deal, but they reasoned:
Then S1 + S1 = (1 - 1 + 1 - 1 ...) + (1 - 1 + 1 - 1 ...)
= (1 - 1 + 1 - 1 ...) + (0 + 1 - 1 + 1 - 1 ...)
= (1 + 0) + (1 - 1) + (1 - 1) ....
= 1 + 0 + 0 ... = 1
But I might as well leave the leading Zero out and argue that:
Then S1 + S1 = (1 - 1 + 1 - 1 ...) + (1 - 1 + 1 - 1 ...)
= (1 - 1 + 1 - 1 ...) + (1 - 1 + 1 - 1 ...)
= (1 + 1) + ( -1 - 1) + (1 + 1) ....
= (2 - 2) + (2 - 2) + (2 - 2)... = 0