Generating Real Numbers: Is It Possible?

[cragar]

Could it be possible to come up with a formula or infinite series or continued fraction to generate the real numbers? I might have to change something in my formula to generate another real. I couldn't just change one of my numbers in the formula because then i would be saying there is a one-to-one correspondence between the naturals and the reals and there isn't. But what if I used irrational numbers to change my formula or something like that.
And i don't think we could generate the reals in order because there is no next real on the number line, maybe its not possible, what do you guys think.

 

[HallsofIvy]

What do you mean by a "formula"? A formula in terms of what kind of "variable"? integers? rational numbers? Neither of those is possible because the set of all rational numbers (and so the set of all integers) are both countable while the set of all real numbers is uncountable. If you allow real numbers as variables, then the simplest such formula is "f(x)= x". That will give every real numbers as a function of some real number x!

 

[cragar]

okay what i mean is, and this probabaly won't work because their uncountable . But i can write e^x as a series and then evaulate e with this series, so why couldn't I write a series for every real number.

 

[SteveL27]

okay what i mean is, and this probabaly won't work because their uncountable . But i can write e^x as a series and then evaulate e with this series, so why couldn't I write a series for every real number.

Well you really can't write e^x as a series. What I mean is, you can't write down the entire series. What we do is write out an expression for the general term of the series; or we write down a finite number of terms of the series and end with dot dot dot. In either case we are writing a finite-length expression.

There are only countably many finite-length expressions from a finite or countable alphabet (you should prove this for yourself, it will give you a good feel for the countable nature of the set of finite expressions).

But there are uncountably many reals. So most of the reals can not be described by a finite expression.

If we allowed infinite expressions, then any real could be expressed by some infinite expression. For example we could write down the entire decimal expansion of each real. So if we allow for expressions of infinite length, there are indeed uncountably many of those.

 

[HallsofIvy]

Writing $e^x$ as a series still requires that you evaluate it for irrational x to get all real numbers as values.