Thread: Rationalizing An Irrational Number

Excuse me ... .

A set of the rational numbers is denoted as , where is the set of integers ... .

A set of the irrational numbers consists of the real numbers which cannot be denoted as such that and ... .

The problem is in following ... .

For example, in present time, here is which be agreed as an irrational number ... .

Can the irrational number be rationalized to be a rational number if a person is able to denote the as , where and , in the next time ... ? [Here, the absolute values of and may be the very very huge integers ... .]

Thank you for the answer ... .

Originally Posted by trfrm

Here, the absolute values of and may be the very very huge integers ... .

It seems to me you've just modify your definition of just in the middle of your question.

Is it all integers or only small ones ? What is small ? What is huge ?

I don't actually undrstand what you are trying to do. You define irrational numbers in a way that mean they are ... irrational. What do you mean by"rationalize"? (I can't follow from your notation)

Here is a simple proof that is irrational: Proof that the square root of 2 is irrational number

You can do this with most (all?) irrational numbers, for example: Proof that is irrational

Originally Posted by Boing3000

Is it all integers or only small ones ? What is small ? What is huge ?

Oh, I'm sorry ... . The term "small" and "huge" are relative ... .

Originally Posted by Strange

I don't actually undrstand what you are trying to do. You define irrational numbers in a way that mean they are ... irrational. What do you mean by"rationalize"? (I can't follow from your notation)

I just tried to regard an irrational number as a rational number ... .

Originally Posted by Strange

Here is a simple proof that is irrational: Proof that the square root of 2 is irrational number

You can do this with most (all?) irrational numbers, for example: Proof that is irrational

Thank you very much for the explanation ... . Now, I understand why the square root of 2 and the pi are regarded as the irrational numbers ... .

All rational numbers and all irrational number can form a set of real numbers ... .

There are significant differences between character of and character of ... .

If in example we define the set and the set , then we can see the significant difference ... .

We can find at least one bijective map , in example and , where is a constant ... .

But, however, we cannot find at least one bijective map ... .

How can it ... ? Until now I have not understood ... .

"But, however, we cannot find at least one bijective map ... .

How can it ... ? Until now I have not understood ... ."
We can:
Since all rationals are countable, and all rationals between -1 and 1 are countable, we can make lists of both sets, and then match them up.

pi * infinity = infinity

1 / pi = 1

Food for thought.

Originally Posted by Zesterer

1 / pi = 1

Er... no.

1/pi = 0.31831 (approx.)

That is not true. The function maps the set of all rational numbers one-to-one onto the set of rational numbers between -1 and 1.
1) if x is a rational number then x/(x- 1) is also rational.

2) For any x, x+1> x so x/x+1 is between -1 and 1.

3) For any y between -1 and 1, x= y/(y-1) maps to y.

Originally Posted by HallsofIvy

That is not true. The function maps the set of all rational numbers one-to-one onto the set of rational numbers between -1 and 1.

Excuse me ... . The plot of f(x) = x/(x + 1) is in the following site ... .

x/(x+1) - Wolfram|Alpha

The domain of the function is ... , and the range (or image) of the is ... . Thus, the function is not bijection from to ... .

I'm sorry ... .

Originally Posted by Zesterer

pi * infinity = infinity

1 / pi = 1

Food for thought.

... is this some kind of childish joke where you say or window?

As strange pointed out:...

That's not food for thought, that's just trash, no offense.