Irrational Numbers
Real numbers that cannot be written as a fraction, such as √2 and π.
Key idea
Real numbers that cannot be written as a fraction, such as √2 and π.
In depth
An irrational number cannot be written as a ratio of integers; its decimal never ends or repeats. Examples are √2 and π. √2 is irrational since if √2 = p/q in lowest terms, p² = 2q², forcing both even, a contradiction.
Worked example
√2 ≈ 1.41421356..., with no repeating pattern.
Real-world uses
- Calculating the diagonal of a square in carpentry
- Using π in circle formulas in design and physics
Common mistake
Thinking 22/7 equals π. It is only an approximation.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Irrational Numbers?
Real numbers that cannot be written as a fraction, such as √2 and π. It is a topic at the advanced level in Arithmetic & Number Sense.
Where is Irrational Numbers used in real life?
Real-world examples: Calculating the diagonal of a square in carpentry; Using π in circle formulas in design and physics.
What should I learn before Irrational Numbers?
Make sure you are comfortable with Rational Numbers, the previous topic in Arithmetic & Number Sense.
What is a common mistake with Irrational Numbers?
Thinking 22/7 equals π. It is only an approximation.
How can I practise Irrational Numbers?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Irrational Numbers?
Move on to Real Numbers next.
Related advanced topics in Arithmetic & Number Sense
- Repeating Decimals – Decimals with a repeating block are rational and convertible to fractions.
- Compound Interest – Interest earned on principal and past interest.
- Inverse Proportion – When one quantity doubles, the other halves.
- Rational Numbers – Numbers expressible as p/q with integers p, q, q≠0.
- Real Numbers – All rational and irrational numbers; the continuous number line.
- Square Roots of Non-squares – Approximate or simplify surds like √50 = 5√2.