Irrational Numbers

In this article, you will learn everything you need to know about irrational numbers and how to identify them in various numerical sets.
Shall we begin?

Irrational numbers are those that cannot be represented as the quotient of integer numbers - numerator and denominator.
In other words, an irrational number is one that, when trying to represent it in fraction form, will not give us an integer numerator and an integer denominator.

When representing an irrational number as a decimal, it will be infinite and its decimal digits will not be periodic.
That is, after the decimal point, an infinite number of digits will appear that do not repeat in a periodic manner.

Example of an Irrational Number - $\sqrt2$

The number $\sqrt2$ cannot be represented as the quotient of integers, therefore, it is irrational.

Determine which numbers are irrational within the following set:

$15, 0, 8.56845623......, 5$

Solution:

Only the number $8.56845623$ is irrational, all the others are rational.
$15$ $~$ - > rational, can be represented as a fraction of two integers $22\over1$
$0$ $~$ - > rational, represented as a fraction $0\over1$
$8.56845623$ $~$ - > irrational, represented as a decimal number carries an infinite number of digits that do not repeat periodically.
$5$ $~$ - > rational, can be represented as a fraction of two integers: $5\over1$

Determine which numbers are irrational within the following set:
$5.369369369,1...  , 6.53248 ,0.020202...$

Solution:
$0.020202$ $~$- > rational, represented as a decimal number carries digits that repeat periodically to the right of the decimal point, therefore, it is rational.
$6.53248...$ $~$- > irrational, represented as a decimal number carries an infinite amount of digits that do not repeat periodically, therefore, it is irrational.
$1$: rational, can be represented as a fraction of two integers -> $1\over1$
$5.369369369$ $~$- > rational, represented as a decimal number carries digits that repeat periodically to the right of the decimal point, therefore, it is rational.

Determine which numbers are irrational within the following set:
$\frac{6}{3},\sqrt2   , .3.98765... ,0.100100010000$

Solution:
$0.100100010000$ $~$- > irrational, represented as a decimal number has digits that do not repeat periodically to the right of the decimal point, therefore, it is irrational.
(You might have thought at first glance that there is a periodic repetition here, but, it is very important that you pay attention and check if the figures really repeat periodically).
$3.98765...$ $~$- > irrational, represented as a decimal number carries an infinite amount of digits that do not repeat periodically, therefore, it is irrational.
(Even though the digits after the decimal point decrease successively, it does not mean they are periodic, therefore, the number is irrational).
$\sqrt2$ $~$- > irrational, if we tried to represent it as a fraction of two integers - integer numerator and integer denominator - we would not succeed, therefore, it is an irrational number.
$6 \over 3$ $~$- > rational, fraction with integer numerator and integer denominator.