Rational and Irrational Numbers
Rational numbers are fractions; irrational numbers cannot be written as fractions.
About this topic
Rational numbers can be written as a fraction of two integers, and their decimals either end or repeat. Examples are 3/4, −2 and 0.333…. Irrational numbers cannot be written that way, and their decimals go on forever without repeating, as with √2 and π.
Together they make up the real numbers. The ancient Greeks were shocked to find that the diagonal of a unit square is irrational. Between any two rational numbers there are infinitely many more of both kinds. To change a repeating decimal into a fraction, let x = 0.333…, so 10x = 3.333…, and subtracting gives 9x = 3, so x = 1/3.
Square roots of non-perfect squares, such as √2 and √5, are irrational, as are π and e. The circle's circumference to diameter ratio is π, about 3.14159, but 22/7 is only a close rational approximation, not the true value. Engineers use rational approximations for practical work, while mathematicians treat irrational numbers exactly.
A common misconception is that a decimal that looks long, like 0.123456789, must be irrational. If it ends, it is rational. Another is that the sum of two irrationals is irrational: √2 + (−√2) = 0, which is rational. Also, a rational plus an irrational is always irrational, a useful fact when classifying numbers.
Key ideas
- Rationals have terminating or repeating decimals
- π and √2 are irrational
- Together they form the real numbers
Quick check
Use cases: where Rational and Irrational Numbers helps
- Homework: review the key idea behind Rational and Irrational Numbers before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Rational and Irrational Numbers without notes.
- Visual learning: play with the interactive lab to see how Rational and Irrational Numbers changes when you adjust its inputs.
- Teaching a friend: explain Rational and Irrational Numbers in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Rational and Irrational Numbers?
Rational numbers are fractions; irrational numbers cannot be written as fractions.
What will I learn in Rational and Irrational Numbers?
Rationals have terminating or repeating decimals. π and √2 are irrational. Together they form the real numbers.
Is Rational and Irrational Numbers part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Rational and Irrational Numbers lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Rational and Irrational Numbers?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.