Rational and Irrational Numbers

What is Rational and Irrational Numbers?

Within Mathematics, which covers numbers, algebra, geometry, trigonometry and statistics from first principles to exam level, this topic explains Rational and Irrational Numbers. Maths builds step by step, so each topic here gives you a rule, a picture and a quick check you can use on homework and in exams.

Check that you can explain Rational and Irrational Numbers in your own words and give one example from everyday life. If you can do that, you have understood the core idea.

Understanding Rational and Irrational Numbers gives you a reliable method you will reuse in later maths units, in science lessons and in data work.

Rational and Irrational Numbers in detail

A rational number can be written as a fraction of two integers, so it is a terminating or recurring decimal. Examples are 3/4, −5, 0.2 and 0.333…. An irrational number cannot be written this way, and its decimal expansion goes on forever without repeating, as with π, √2 and e.

Together the rational and irrational numbers form the real numbers. A useful fact is that the sum or product of two rational numbers is rational, but combining an irrational number with a non-zero rational one gives an irrational result. Recognising which type a number is helps in exact working and proof.

How the interactive helps you understand Rational and Irrational Numbers

Move the slope and y-intercept sliders. The line and its equation update, so you can see how the slope sets the steepness and the intercept sets where the line crosses the y-axis.

Before you change anything, predict the result. Then change one value at a time and check whether your prediction was right. This is a quick way to build a correct mental model of Rational and Irrational Numbers.

Real-life uses and examples

• Use Rational and Irrational Numbers when comparing prices, discounts or recipe quantities in everyday life.
• Explain Rational and Irrational Numbers to a friend with a simple picture, such as a row of tiles or a bag of sweets.
• Spot Rational and Irrational Numbers in exam questions, where it often sits inside a short word problem.

Common mistakes to avoid

• Forgetting units, signs or brackets when you apply Rational and Irrational Numbers.
• Memorising the steps for Rational and Irrational Numbers without checking why they work.
• Learning the definition without testing it. Use the quiz and the interactive to check that you can apply the idea.