Proof that irrational numbers do not exist

In summary, the conversation discusses the form of any real number in terms of its decimal expansion and the concept of a countable infinity. The argument is made that any real number can be represented as a ratio of infinitely large integers, but this idea is not universally accepted. Additionally, the conversation touches on the existence of irrational numbers and their relationship to rational numbers. The concept of "larginess" or cardinality is also mentioned in relation to infinite sets.
  • #36
Robert1986 said:
I understand exactly what you are saying. But you haven't added anything new the mathematics. It is, in fact, well known that every rational number is the limit of a sequence of irrational numbers. This is pretty much what you have shown. It is not a surprise to anyone who has taken a semester of Real Analysis.


The fact that there are reals that cannot be written in terms of integer ratios is, in fact, an important distinction if only because of the fact that they are countable whereas the entire real line isn't. There is nothing trivial about that.


Putting this aside, I grant that what you are saying is, in some sense, correct (at least what you wrote in the OP is "correct", even if it is trivial). So, what is the result you have proven with respect to prime numbers?
I dare say that what the OP purports to prove is false because all rational numbers are either finite decimal numbers or infinite repeating decimals. That is only some infinite decimal representations, e.g. 12.66123123123..., where the ending part, i.e. 123, repeats forever are rational numbers while those infinite decimal numbers which do not have an repeating ending are irrational numbers.
 
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  • #37
ramsey2879 said:
I dare say that what the OP purports to prove is false because all rational numbers are either finite decimal numbers or infinite repeating decimals. That is only some infinite decimal representations, e.g. 12.66123123123..., where the ending part, i.e. 123, repeats forever are rational numbers while those infinite decimal numbers which do not have an repeating ending are irrational numbers.

When I said that his OP was "correct" I only meant that he was correct in asserting that irrational numbers are the limits of sequences of rational numbers. But, as I said, that is not new to anyone who was taken a course in Real Analysis.


I am just interested in what he has allegedly proven about prime numbers.
 
<h2>1. What is an irrational number?</h2><p>An irrational number is a real number that cannot be expressed as a simple fraction or ratio of two integers. It is a number that goes on infinitely without repeating in its decimal representation.</p><h2>2. How can we prove that irrational numbers do not exist?</h2><p>We can prove that irrational numbers do not exist by using a proof by contradiction. This means assuming that irrational numbers do exist and then showing that this leads to a contradiction or an impossibility.</p><h2>3. Can you give an example of a proof that irrational numbers do not exist?</h2><p>One example of a proof that irrational numbers do not exist is the proof that the square root of 2 is irrational. This proof involves showing that if we assume the square root of 2 is rational, then it leads to a contradiction, thus proving that it is irrational.</p><h2>4. Why is it important to prove that irrational numbers do not exist?</h2><p>Proving that irrational numbers do not exist is important because it helps us understand and define the concept of a real number. It also allows us to make precise and accurate calculations and measurements in fields such as mathematics, physics, and engineering.</p><h2>5. Are there any other proofs that irrational numbers do not exist?</h2><p>Yes, there are many other proofs that irrational numbers do not exist. Some examples include the proofs that the square root of any prime number is irrational and that the decimal expansion of an irrational number is non-repeating and non-terminating.</p>

1. What is an irrational number?

An irrational number is a real number that cannot be expressed as a simple fraction or ratio of two integers. It is a number that goes on infinitely without repeating in its decimal representation.

2. How can we prove that irrational numbers do not exist?

We can prove that irrational numbers do not exist by using a proof by contradiction. This means assuming that irrational numbers do exist and then showing that this leads to a contradiction or an impossibility.

3. Can you give an example of a proof that irrational numbers do not exist?

One example of a proof that irrational numbers do not exist is the proof that the square root of 2 is irrational. This proof involves showing that if we assume the square root of 2 is rational, then it leads to a contradiction, thus proving that it is irrational.

4. Why is it important to prove that irrational numbers do not exist?

Proving that irrational numbers do not exist is important because it helps us understand and define the concept of a real number. It also allows us to make precise and accurate calculations and measurements in fields such as mathematics, physics, and engineering.

5. Are there any other proofs that irrational numbers do not exist?

Yes, there are many other proofs that irrational numbers do not exist. Some examples include the proofs that the square root of any prime number is irrational and that the decimal expansion of an irrational number is non-repeating and non-terminating.

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