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Fuzzy Math... Almost Is Enough

A surprisingly simple manipulation shows how infinity can trick our intuition.

By Alex Stone
Oct 12, 2006 5:00 AMNov 12, 2019 4:06 AM


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Infinity seems like a wildly remote concept, but sometimes you can tame it with some very simple math. Consider a zero followed by a decimal and an infinitely long string of nines, written as 0.99999.... What is the value of that unending series? The answer, strangely enough, is that it just equals 1. Here's a way to prove it.

1) First, define x to be 0.99999.... 2) Next, multiply x by 10 to get 10x. We know that 10 times 0.99999.... equals 9.99999...., so this means 10x equals 9.99999....3) Now subtract x from 10x to get 9x. This is just 9.9999.... minus 0.99999...., which equals 9 (the parts to the right of the decimal disappear because they are the same). 4) So 9x equals 9, which means x equals 1. 5) But by definition, x equals 0.99999...., so we conclude that 0.99999.... equals 1!

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