Thursday, September 27, 2007

1 = 1

Two forums I read regularly have recently had threads on the subject of 0.999999.... = 1. Amazingly, lots of people still seem to believe it's not true, or not quite true or that maybe it only approaches one. Sadly for them, as soon as you admit that an infinite series can sum to a finite number (which it can) 0.9999... = 1 follows immediately.

Proof



Let x = 0.9999....

Then 10x = 9.9999....

10x - x = 9.9999.... - 0.9999....

9x = 9

x = 1.

Not a proof but perhaps more psychologically convincing



Nobody seems to have a problem with 1/3 = 0.3333....

In fact

1/9 = 0.1111111....
2/9 = 0.2222222....
3/9 = 0.3333333....
4/9 = 0.4444444....
5/9 = 0.5555555....
6/9 = 0.6666666....
7/9 = 0.7777777....
8/9 = 0.8888888....
9/9 = 0.9999999....

QED

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