I and others, have PROVEN .9999... = 1/1You are right, but what fraction is .999999.... ?

Here link to full, old proof, I gave at http://www.sciforums.com/showthread...ophy-of-Math&p=3136195&viewfull=1#post3136195

but essence of it is:

The objectors tend to fall into two classes: idiots and those not fully understanding the meaning of the "Bases, places and decimal point" notational system, so I will explain it in another posts, soon.... I first illustrate, several of the infinite numbers of examples, of true statements concerning terminating or infinitely Repeating Decimals. (And then a procedure for finding the Rational Fraction that equals to ANY given RD):

1/3 =0.333333.... and 1/1 = 0.99999.... are rational fraction numbers with a "repeat length" of 1 in their equivalent decimal versions.

12/99 = 0.12121212... and 19/99 = 0.1919191919... and 34/99 = 0.343434... are rational numbers with a "repeat length" of 2 in their equivalent decimal versions.

In general, any integer less than 99 divided by (and not a factor of) 99 will produce a decimal repeating with length 2. Some of the factors will too. For example 3/99 = 03/99 = 0.03030303... does; but not 11 or 33. I.e. 11/99 =0.111... and 33/99 = 0.3333333333... In ALL cases with 99 as the denominator, the decimal repeats two numbers in blocks of two) is still true, as well as in "blocks" of one for some cases as bold type helps you see.

Likewise any integer less than 999 divided by 999 will be a decimal fraction with repeat length not more than 3 and always will repeat in blocks of 3, but for somecases, like 333 /999 = 1/3 the least long repeat block is less than 3. Check with your calculator if you like. Etc. For example, 678 /999 = 0.678,678,678, .... and that is slightly larger than 678 /1000, which equals 0.678 and should given you a hint of the proof to come.

However, any integer divided by a factor of the number base (1, 2 & 5 for base 10) or any product of these factors (like 4, 16, 2^n, 5 or 5^m, {2^n x 5^m} ) will terminate, not repeat. For example 17 /(1x4x5) = 0.85

The proof I and others have given that 1 = 9/9 = 0.99999.... is just particular case of the fact ALL rational fractions like a/b or a/9 (both a & b being integers and a < b) are equal to an infinitely repeating decimal (if they are not a finite decimal when b is a factor or product of factors of the base).

For example, the general proof of this goes like:

Rational Decimal, RD = 0.abcdefg abcdefg abcdefg .... Where each letter is one from the set (0,1,2...8,9) and the spaces are just to make it easier to see the repeat length in this case is 7.

Now for this repeat length 7 case, moving the decimal point 7 spaces to the right is not a multiply operation, but a notional change with the same effect on meaning as multiplying RD by 10,000,000. I. e. 10,000,000 RD = a,bcd,efg . abcdefg abcdefg ... Is a 2nd equation with comas for easy reading the integer part.

Now, after noting (10,000,000 - 1) = 9,999,999 and subtracting the first equation from the second, we have:

a,bcd,efg = 9,999,999 x RD. Note 9,999,999 certainly is not zero so we can divide by it to get:The Rational Fraction, RF = RD = a,bcd,efg / 9,999,999I. e. the rational fraction of two integers exactly equal to the infinitely long repeating (with repeat length =7 in this case) decimal, RD.

Now lets become less general and consider just one of the repeat length = 7 cases. I. e. have a=b=c=d=e=f=g = 9 and recall RD was DEFINED as 0.abcdefg...so is now in this less general RD = 0.9,999,999,... and from green part of line above, The Rational Fraction which equals RD is 9,999,999 / 9,999,999, which reduces to the fraction 1/1 which is unity as the numerator is identical with the non-zero denominator. I.e. the least numerator rational fraction equal to 0.999,999... is 1/1.

By exactly the same procedure the RD = 0.123,123,123,.... is a case with repeat length of 3, can be shown be equal to the RF = 41/333 (as your calculator will show, as best as it can, if used to divide 41 by 333.

To prove this one moves the decimal point of the RD 3 places to the right get: RD' which is 1000 times larger than RD. I.e. RD' = 123.123123123... as the 2nd equation and then subtract the first from it to get after division by (1000 -1): RD = 123 /999, which reduces to 41/333 as the RF = 0.123123123...

The objectors to 1 =0.99999 need not only to give extraordinary proof for their objection but also need to explain why ONLY ONE of the other infinite number of successes of this procedure fails to produce the rational fraction that is exactly equal to the infinite Repeating Decimal , RD.