Studying Mathematics Thread

[polymath257]

Not bad. One point: in the definition of rational numbers, we do not allow 0 in the denominator (no division by 0).

One issue that comes up in this is the meaning of infinite decimal expansions. So, for example, when we write
pi=3.141592653589793....
what, precisely, is being said?

One way to answer this is through the concept of a limit. In essence, a limit describes what a process is getting closer and closer to. In this case, what we have is a sequence of *rational approximations*
3
3.1
3.14
3.141
3.1415
3.14159
...

And the idea is that these rational approximations are getting closer and closer to some fixed irrational number, which happens to be pi. There are many *other* sequences of rational approximations, by the way. And to tell when two different rational approximations represent the same real number, we just ask if the approximations themselves are getting closer and closer to the other.

So, for example, and a source of many internet discussions, consider the representation
1.99999999.....
where the sequence of 9's is infinite. As was noted above, since the sequence repeats in a cycle, this number is rational. But we can do better. Consider the successive approximations
1
1.9
1.99
1.999
1.9999
1.99999
...

and ask "Is there is some *fixed* number that these are better and better approximations to?" A bit of thought will readily give
the answer as 2. And, in fact,
2=1.99999.....

This confuses many people because they say the right hand side 'never gets there'. And that is even correct, in a sense: the sequence of approximations never has a term equal to 2. But that wasn't the question. The question is what fixed number these are approximating better and better. And there is precisely one such number and that number is 2.

Now, we can do algebraic manipulations to reach the same answer:
If x=1.9999...., then
10x=19.999999....
We can subtract and get
9x=10x-x=19.9999... -1.9999..... =18,
so x=2.

The main reason I don't like this is that it ignores the underlying meaning of the notation. The notation *means* the value of the limit. That is one, fixed number. It just has two different *representations* in terms of decimals. But that is fine, it also has more than one representation in terms of fractions: 2=2/1=4/2=6/3... As long as we recognize this ambiguity in our notation, there is no problem with the numbers themselves.