You know that $0.9$ means not quite $1$, and $0.99$ means a bit closer but still not $1$, and that $0.999$ means even closer still, but again not quite there. You following so far?
Good!Ok, what about $$0.9...$$? What do you think the three extra dots mean? They're technically called ellipsis!
Well they generally mean 'and so forth', in essence I'm implying that those 9's go on ad infinitum.
Keeping up ok? Well this is the mindblowing bit.
Consider a number $$s = 0.9... $$(remember $$0.9...$$ means the 9's go on for ever).
I'm going to prove to you that $$s = 1$$, not nearly $$1$$, but exactly $$1$$. Does that seem counter-intuitive to you? How can a number that appears to be not quite $$1$$ be exactly $$1$$ at the same time. Here goes (don't worry the maths is fairly straightforward).
$$s = 0.9...$$
Multiply both sides of the equation by $$10$$
$$10s = 9.9...$$
Subtract one '$$s$$' from both sides
$$9s = 9$$
Divide both sides by $$9$$
$$\frac{9s}{9} =\frac{9}{9}$$
Simpify
$$s = 1$$ QED
Does that blow your mind? It should, it certainly blew mine!
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