Irrational Numbers

Irrational numbers are numbers that can’t be expressed as fractions of two integers. Simply put, irrational numbers are real numbers that are not rational numbers. Irrational numbers have always fascinated mathematicians as early as the times of Hipassus (credited for the discovery of \sqrt{2}). Even up to this point, the fascination towards famous irrational numbers such as \pi has not dwindled!

This is why we’ve allotted a special article covering all the fundamentals of irrational numbers. In this article, we’ll show you the definition and properties of irrational numbers. We’ll also show you some of the most known irrational numbers we know in math and highlight their importance in different fields in math and applied sciences!

What Are Irrational Numbers?

Irrational numbers are the set of real numbers that can’t be written as a simple ratio or fraction of the form, \dfrac{p}{q}. In short, irrational numbers are simply real numbers that are not rational.

You may have seen the left side of the diagram when learning about whole and rational numbers. This time, we’ll expand the set to account for irrational numbers. This means that all real numbers that are not rational numbers will belong inside the set of irrational numbers. 

Recall that rational numbers are numbers that can be written in the form of p/q, where p and q are integers but q can never be equal to zero. 

Rational NumberIrrational Number
\begin{aligned}\left.\begin{matrix}2.5  = \dfrac{5}{2}\end{matrix}\right\} \text{Ratio}\left(\dfrac{p}{q} \right )\end{aligned} \begin{aligned}\left.\begin{matrix}e =2.71828182845...= \dfrac{?}{?}\end{matrix}\right\} \text{Unknown ratio}\end{aligned}

Any real number that can’t be written in this form is automatically an irrational numbers. Here’s a fun fact: because of irrational number’s definition, we sometimes denote it as r \setminus q. The backlash symbol (also known as the set minus) highlights the idea that irrational numbers can’t be written as ratios of two integers. Why don’t we test our understanding of rational and irrational numbers by warming up on the sample problem below?

Problem 1

Determine whether the following numbers are rational or irrational.

a. 0.04

When checking out whether a given number is rational or irrational, we attempt to write the given number as a fraction or ratio of two integers.

\begin{aligned}0.04 &= \dfrac{4}{100}\\&= \dfrac{1}{25}\\&\Rightarrow \textbf{Rational} \end{aligned}

Since we can write 0.04 as a ratio of 1 and 25, so the decimal is indeed a rational number.

b.\sqrt{2}

By instinct, we can write \sqrt{2} as a fraction of \sqrt{2} and 1. 

\begin{aligned}\sqrt{2} &= \dfrac{\sqrt{2} }{1} \\&(\sqrt{2} \text{ is not an integer})\\&\Rightarrow \textbf{Irrational} \end{aligned}

However, \sqrt{2} is not an integer, so we can conclude that \sqrt{2} is irrational.

c.\sqrt{25}

Now, when given the square root of a perfect square, simplify the expression first: \sqrt{25} = 5. This simplified form of \sqrt{25} returns a whole number and as we know, all whole numbers are rational numbers.

d.\dfrac{-1 + \sqrt{5}}{2}

This radical expression is in fact one of the most famous ratio in sciences and art- \dfrac{-1 + \sqrt{5}}{2} is also known as the golden ratio ( \phi \approx 1.618034... ). By inspection, the numerator is not an integer, so the golden ratio is in fact, an irrational number.

Speaking of famous irrational numbers, let us show you a few more known irrational numbers in math. Some of them, you’ve already encountered in the past such as \pi and \sqrt{2}.

Famous Examples of Irrational Numbers

Here are some of the most common irrational numbers that you’ll encounter in your math classes and even advanced classes in science!

\boldsymbol{\pi}: Pi is probably one of the most known irrational numbers. 

\begin{aligned}\pi \approx 3.14159265… \end{aligned}

Mathematicians and programmers are motivated to determine as many digits as possible. As we write this article, researchers have successfully set up a new record of 62.8 trillion digits using a powerful supercomputer. This is also what makes irrational numbers different from repeating decimals – The decimal places are endless. 

\begin{aligned}\pi &\approx \dfrac{22}{7}\\&\approx 3.14 \end{aligned}

Since \pi is often used in circles and areas involving circular regions, we’ve established a close approximation for \pi – the rational number: \dfrac{22}{7} or 3.14.

\boldsymbol{\sqrt{2}}: The square root of 2 and in fact, roots of many numbers that can’t be simplified.

When the nth root of a given number can’t be simplified to a whole number of fraction, the root is an irrational number. So why did we single out \sqrt{2}? Because it’s a famous hypotenuse of an isosceles right triangle with legs measuring one unit each. This is a special right triangle – and you’ll learn more about this in geometry and trigonometry. 

\boldsymbol{e}: The Euler number, e is another famous irrational number.

\begin{aligned}e \approx 2.71828182845… \end{aligned}

The Euler number, e, is an irrational number and is the base used in natural logarithms. Together with the following constants, \{0, 1, i, \pi\}, these five numbers will form one of the most famous and beautiful equations in mathematics: 

\begin{aligned}e^{i\pi} + 1 = 0\end{aligned}

The Euler number is also crucial in modeling rapid changes – both growth and decay.

There are a lot of irrational numbers you’ll encounter in math. These three irrational numbers are also a great reminder that even when these numbers look different from the rest of the real numbers out there, they continue to play crucial roles in math. A lot of researchers and programmers are fascinated by the nature of irrational numbers.

Problem 2

Wrapping up what we’ve just learned about common and famous irrational numbers, which of the following are irrational numbers?

\begin{aligned} \boldsymbol{\left\{e, \sqrt{4+ 5}, \pi, \dfrac{3 + \sqrt{5}}{2}, 0, \dfrac{22}{7}, \sqrt[4]{36}\right\}} \end{aligned}

We’re given seven rational numbers, so let’s inspect each number and check whether the number is either rational or irrational.

  • We’ve discussed that e is a famous irrational number called the Euler number.
  • Simplifying \sqrt{4 + 5}, we have \sqrt{9} = 3, so the number is rational.
  • As we have established, pi (or \pi) is irrational.
  • Since the numerator of \dfrac{3 +\sqrt{5}}{2} is irrational, the entire fraction is also irrational.
  • The number, 0, will always be rational.
  • Although \dfrac{22}{7} is an approximation of pi, since numerator and denominator are integers, so it is rational.
  • We can’t simplify \sqrt[4]{36} into a whole or rational number, so the fourth root of 36 is irrational.

Hence, of the seven numbers, the following numbers are irrational: \left\{e,\pi, \dfrac{3 + \sqrt{5}}{2}, \sqrt[4]{36}\right\} .

What Are Properties of Irrational Numbers?

As we have established earlier, irrational numbers are still part of subsets of real numbers. What does this mean for their properties? Irrational numbers will still follow fundamental rules and properties established for all real numbers. 

Here are some interesting properties to learn about irrational numbers:

  • Irrational numbers will always contain decimals that never end and never repeat with a pattern.
  • All irrational numbers are real numbers.
  • The sum or difference of two irrational numbers may or may not be irrational. 

For example, the difference between \pi and itself is equal to zero, which is a rational number.

  • When an irrational number is multiplied by a nonzero rational number, the product will always be an irrational number.
  • The operations done exclusively between two irrational numbers may or may not be rational or irrational.

A faster way to know these properties by heart is by thinking of actual examples on your own that satisfy these statements. We’ve also prepared a few more examples for you to work on to master this topic!

Problem 3

From the last bullet, we’ve mentioned that the product of two irrational numbers may or may not be irrational. Show examples that satisfy the last statement.

To easily find examples, try to multiply square roots of numbers that share different bases. Now, duplicate one of the numbers and try to find their product instead. See what happens – for our solution, we’ll use \sqrt{5} and \sqrt{6}.

\boldsymbol{\sqrt{5}\times \sqrt{6}}\boldsymbol{\sqrt{5}\times \sqrt{5}}
\begin{aligned}\sqrt{5}\times \sqrt{6}&= \sqrt{30}\\&\Rightarrow \textbf{Irrational} \end{aligned} \begin{aligned}\sqrt{5}\times \sqrt{5}&= \sqrt{25}\\&= 5\\&\Rightarrow \textbf{Rational} \end{aligned}

These two examples alone show that it is indeed possible for the product of two irrational numbers may or may not be irrational.

Problem 4

The figure below is a right triangle with irrational side lengths. Determine whether the right triangle’s perimeter and area are irrational or rational.

We can find the perimeter of the right triangle by adding all the legs’ side lengths. Keep in mind that when adding radical numbers, we can only combine them when the base inside and root are the same. 

\begin{aligned}\text{Perimeter} &= \sqrt{3} + 2\sqrt{3}+ \sqrt{15}\\&= 3\sqrt{3} + \sqrt{15}\end{aligned}

This means that for our right triangle, the perimeter is equal to 3\sqrt{3} + \sqrt{15} units. Since we can no longer simplify the irrational expression further, we can conclude that the right triangle’s perimeter is irrational. 

Now, let’s take a look at its area. Recall that the area of a triangle is simply one half its base times its height: \text{Area} = \dfrac{1}{2}bh. For the right triangle, we have b = 2\sqrt{3} and h = \sqrt{3}

\begin{aligned}\text{Area} &= \dfrac{1}{2}bh\\&= \dfrac{1}{2}(2\sqrt{3})(\sqrt{3})\\&= \dfrac{1}{2}(2)(\sqrt{3} \times \sqrt{3})\\&= 1(\sqrt{3})^2\\&= 3\end{aligned}

Hence, the area of our right triangle is equal to 3 squared units. Since the area is a whole number, we can conclude that the area of our right triangle is a rational number.