# Are all non-irrational numbers rational? [duplicate]

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I am working on a simple proof involving rational and irrational numbers. Is it safe to assume that if a number is not rational, it is irrational, and that if a number is not irrational, it is rational?

Example: Let $P(x)=\text{x is rational}$ and $Q(x)=\text{x is irrational}$.

Then is it true that $\forall x\ P(x)\text{ xor }Q(x)$?

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• A double negation is an affirmation. – Pedro Tamaroff Sep 16 '16 at 1:47
• As long as by "number" you mean real number. – Joey Zou Sep 16 '16 at 4:56
• I feel that it's important to point out that double negation isn't always an affirmation. See constructive mathematics. – James Wood Sep 17 '16 at 20:45

## 3 Answers

The real numbers are composed of the rational numbers and the irrational numbers so yes if a number is not one, it is the other.

See Dominik's answer:

Are there real numbers that are neither rational nor irrational?

There he provided this excellent picture:

A rational number is defined as a number that can be expressed as the ratio of two integers, i.e. $\frac{p}{q}$, where $q\ne0$. An irrational number is a real number that cannot be expressed as a ratio. So, yes a real number is either rational or irrational, but not both.

An irrational number is a number that is not rational.

Thus a non-irrational number is not (a number that is not rational), thus it is rational.

In set notation irrational numbers are $\mathbb{R}\setminus\mathbb{Q}$, so non-irrational numbers are $\mathbb{R}\setminus(\mathbb{R}\setminus\mathbb{Q})=\mathbb{Q}$.