2010-03-11, 06:59 PM
Hi there guys, Shouri here, math major. Just took Real Analysis in which this topic came up on about the first week. The complicated version for why .9999.... = 1 deals with something called cauchy sequences and equivalence classes.
When we only deal with the rational numbers (numbers of the form a/b where a and b are integers and b=/=0 ), things like .999=/=1. But in order to build the real numbers, we needed numbers that could possibly have an infinite number of decimal places. We defined number such as pi in terms of a cauchy sequence of rational numbers. For instance, the cauchy sequence of pi is {3, 3.1, 3.14,....} We then said that the number pi is the limit of that sequence. But the thing is that there is more than one sequence which can get you to pi. For instance, another cauchy sequence for pi is {3.2, 3.15, 3.142, 3.1416, 3.14160, 3.141593,....}. We can call these two sequences equivalent since the differences in the nth terms for these sequences approaches zero.
Now we look at two cauchy sequences, one for 0.9999..... and one for 1:
{0.9, 0.99, 0.999, 0.9999, 0.99999,.....} Call this sequence S and call the nth term sn (term 3 would be s3 = 0.999)
and
{1.0, 1.0, 1.0,...} Call this sequence R, and the nth term rn.
Notedly cauchy sequences are allowed to be just one number repeated in which case, they represent that very number.
The differences in the sequence are:
sn-rn ={.1, .01, .001,....} which is
nth term = 1/(10^n)
You then take the limit of this sequence as n approaches infinity which is zero. Thus the two sequences R and S are part of the same equivalence class.
If two numbers x and y have cauchy sequences in the same equivalence class then x=y.
Thus .9999......=1
When we only deal with the rational numbers (numbers of the form a/b where a and b are integers and b=/=0 ), things like .999=/=1. But in order to build the real numbers, we needed numbers that could possibly have an infinite number of decimal places. We defined number such as pi in terms of a cauchy sequence of rational numbers. For instance, the cauchy sequence of pi is {3, 3.1, 3.14,....} We then said that the number pi is the limit of that sequence. But the thing is that there is more than one sequence which can get you to pi. For instance, another cauchy sequence for pi is {3.2, 3.15, 3.142, 3.1416, 3.14160, 3.141593,....}. We can call these two sequences equivalent since the differences in the nth terms for these sequences approaches zero.
Now we look at two cauchy sequences, one for 0.9999..... and one for 1:
{0.9, 0.99, 0.999, 0.9999, 0.99999,.....} Call this sequence S and call the nth term sn (term 3 would be s3 = 0.999)
and
{1.0, 1.0, 1.0,...} Call this sequence R, and the nth term rn.
Notedly cauchy sequences are allowed to be just one number repeated in which case, they represent that very number.
The differences in the sequence are:
sn-rn ={.1, .01, .001,....} which is
nth term = 1/(10^n)
You then take the limit of this sequence as n approaches infinity which is zero. Thus the two sequences R and S are part of the same equivalence class.
If two numbers x and y have cauchy sequences in the same equivalence class then x=y.
Thus .9999......=1