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.
Let x = 0.9999....
Then 10x = 9.9999....
10x - x = 9.9999.... - 0.9999....
9x = 9
x = 1.
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
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
# posted by JeremyP : 6:43 pm
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I remember at school years ago when Mr Thingymathsteacher asked (in a test) "What is 3.3333.... plus 6.6666.....?". A few clever people wrote "10" and some other people wrote "9.9999....". He wanted to just give the 'clever' people the mark for getting it right, (being clever himself) until one of the others pointed out that 9.9999... actually equals 10 so his trick question misfired on him and he had to give everyone the point.
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