Prove that 3.999... = 4

The two solutions I have seen so far:

The one given:

Since multiplying by 10 just moves the decimal point, since we are in base 10, 39.99... has the same number of digits as 3.999... But .999... = .999 and so must have the same number or digits. (reflexive property) But 39 has 2 digits and 3 has one. So 3.999... x 10 must not equal 39.999...

And I haven't yet found any PROOF that 1/9 = .1111... You can prove that .25 is 1/4 because .25*4 = 1 (exactly) but .111... x 9 = .9999... and that is circular reasoning to try to prove it is equal to 1. (I am not saying it can't be done, but we were trying to prove this in the first place!)

Posted by Gamer on 2003-04-11 15:14:20