What Is an Irrational Number?
Irrational numbers are real numbers that cannot be written as a fraction of two integers. That means you can’t write it as p/q, where p and q are whole numbers and q isn’t zero.
Some numbers never quite fit into a neat fraction. They go on forever without repeating and can’t be written as a ratio like ½ or ¾. Mathematicians call them irrational numbers.
In contrast, a rational number is any number that you can write as a fraction, like 3/4 or -5/1, where both the top and bottom are whole numbers and the bottom isn’t zero.
In this post, you will learn what an irrational number is, how to spot one, and see some classic examples.
What are Irrational Numbers?
Irrational numbers have decimal expansions that go on forever without repeating. For example, the number √2 is approximately 1.414213…, but the digits never stop or follow a pattern. That’s what makes it irrational.
In short: Irrational numbers cannot be expressed as a simple ratio.
Irrational Numbers Definition
If someone asks, “What is an irrational number?” you can say:
It’s a number that you can’t write as a fraction, and its decimal goes on forever without repeating.
In mathematics, the word irrational means “not a ratio.” So irrational numbers are simply numbers that don’t come from dividing one whole number by another.
Examples of Irrational Numbers
Here are some famous examples of irrational numbers:
- √2 ≈ 1.414213…
- √3 ≈ 1.732050…
- √5 ≈ 2.236067…
- π (pi) ≈ 3.141592…
- e (Euler’s number) ≈ 2.718281…
- φ (the golden ratio) ≈ 1.618033…
There are infinitely many of them between any two real numbers. Even between 1 and 2, you’ll find an endless supply of them!
Additionally, the square root of any prime number is always irrational because you cannot write it as a simple fraction.
Note: Not all square roots are irrational. For example, √4 = 2, which is a rational number because you can write it as 2/1. But the square root of any number that isn’t a perfect square will be irrational. Learn more about Square Roots Explained.
How to Know if a Number Is Irrational
To check if a number is irrational, ask yourself:
- Can you write it as a simple fraction like p/q?
- Does its decimal go on forever without repeating?
If the answer is no to the first and yes to the second, it’s an irrational number.
For example:
- √2 is irrational because you can’t write it as a fraction.
- π is irrational because it never ends or repeats.
Is Pi an Irrational Number?
Yes, pi (π) is irrational. Its digits go on forever without a pattern, and you cannot be write it exactly as a fraction. While 22/7 is a common approximation, it’s only that—an estimate. The true value of π begins 3.14159… and never ends. Learn more about it in What is Pi? Understanding the Number & Symbol.
Properties of Irrational Numbers
Irrational numbers have some interesting properties:
- Adding a rational number to an irrational number gives an irrational number. Example: 2 + √3 is irrational.
- Multiplying an irrational number by a nonzero rational number is still irrational. Example: 5 × √2 is irrational.
- Two irrational numbers added or multiplied might give a rational result—or might not. Example: √2 × √2 = 2 (which is rational). But √2 + √3 is irrational.
- The set of irrational numbers is not closed under addition or multiplication. This means the result of combining two of them is not guaranteed to stay irrational. Example: (3 + √2) + (5 − √2) = 3 + 5 + √2 − √2 = 8, a rational number.
Are Irrational Numbers Real Numbers?
Yes! They are part of the real number system. That means you can plot them on a number line, even if you can’t write them as fractions.
All irrational numbers are real, but not all real numbers are irrational. Real numbers include:
- Rational numbers (like ½ or -4)
- Irrational numbers (like π and √2)
Learn about Natural Numbers: Definition & Examples.
How Were Irrational Numbers Discovered?
The first known discovery of an irrational number happened over 2,500 years ago in ancient Greece, around 500 BCE. It’s often credited to Hippasus of Metapontum, a Greek philosopher and early follower of Pythagoras.
The Pythagoreans believed that all numbers could be written as whole number ratios—what we now call rational numbers. But Hippasus showed that this belief wasn’t always true.
According to legend, he proved that the diagonal of a square with side length 1 could not be written as a ratio of two integers. That diagonal is √2, and it goes on forever without repeating. In other words, √2 is an irrational number.
This idea was shocking at the time. It challenged the Pythagoreans’ view that numbers were neat and orderly. Some versions of the story say that Hippasus was punished—or even drowned at sea—for revealing this “dangerous” truth. While that part is probably myth, it shows how unsettling the discovery was to early thinkers.
Today, Hippasus is remembered for uncovering something big: that irrational numbers exist and are a natural part of the real number system.
FAQs About Irrational Numbers
What is an irrational number?
An irrational number is a real number that you can’t write as a simple fraction. It has a decimal that goes on forever without repeating. Examples include √2 and π.
Are irrational numbers real numbers?
Yes, all irrational numbers are real numbers.
What are five examples of irrational numbers?
Some examples are: √2, π, e, φ, and √11.
What numbers are not rational?
Any number that you cannot write as p/q is not rational. These are irrational numbers, such as √5 and π.
What is a irrational number vs a rational number?
A rational number can be written as a fraction (like ¾), while an irrational number cannot (like √3).
Are integers irrational numbers?
No, integers are not irrational. They are all rational numbers. You can write any integer, whether positive, negative, or zero, as a fraction with a denominator of 1. For example: -7, 0, and 9 are rational numbers because you can write them as -7/1, 0/1, and 9/1.
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