What are irrational real numbers?

Question

I was given a question saying:

"One can show that the union of two countable sets is countable. Is the set of irrational real numbers countable?"

I don't know what irrational real numbers are. Can someone please give me an example and a definition please?

Answer 1

Irrational numbers are real numbers that cannot be expressed a fraction of two integers. Examples include $\pi$ and $\sqrt{2}$ etc.

See https://en.wikipedia.org/wiki/Irrational_number

Answer 2

Here is the definition (in terms of sets) of an Irrational number: The set of Real numbers $\mathbb{R}$ minus the set of Rational numbers $\mathbb{Q}$ is the set of Irrational numbers which is written $\mathbb{R} \setminus\mathbb{Q}$.

Less formally, a definition of an Irrational number is a number that cannot be written in the form $\cfrac{p}{q}$ where $p\in \mathbb{Z}$ and $q\in \mathbb{N^{+}}$.

In simple English a Rational number is any fraction such as $\cfrac{3}{4}$, $\cfrac{6}{1}=6$ etc.

The other way to tell if a number is rational is to see if it's decimal digits recur (repeat) such as $\cfrac{1}{9}=0.111111$ and $\cfrac{2}{15}=0.13333333$, also $\cfrac{1}{7}=0.$$\color{blue}{142857}$$142857$$\color{blue}{142857}$$142857$ $\implies$ ($\color{red}{142857}$ recur in this case)

An Irrational number is $\sqrt{3}$ for example, from which it can be seen that its decimal digits do not recur (although no-one of course has "thoroughly" checked this throughout the infinite decimal): $\sqrt{3} = 1.73205080756887729352744634150587236 ....$

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A word of caution:

If you are simply told that a number is not rational and nothing else, that does not mean that it is irrational, and vice versa.

However, since $\left(\mathbb{R} \setminus\mathbb{Q}\right) \subset \mathbb{R}$

It is okay to say that if a real number is not irrational then it must be rational, and vice versa.

This is because we are referring to a subset of the real numbers.

Answer 3

By the way, with respect to the OP question: Since the union of the rational reals $\mathbb{Q}$ and the irrational reals $\mathbb{P} \equiv \mathbb{R} \setminus \mathbb{Q}$ is simply the reals $\mathbb{R}$, and $\mathbb{Q}$ is countable, then if $\mathbb{P}$ were also countable, what would we be able to say about $\mathbb{R}$? And what is actually true about $\mathbb{R}$, and what can we therefore conclude about $\mathbb{P}$?

Answer 4

• A real number number is rational if it can be expressed as the ratio of two integers.

Thus $x$ is rational if it can be expressed as $x = \frac pq$ where $p$ and $q$ are integers.

The set of rational numbers is countable. see this

• A real number is irrational if it is not rational.

The famous, and probably the first, example is that $x = \sqrt 2$ is irrational see this.

The set of irrational numbers is uncountable.