Surds
A surd is a root that cannot be simplified to a whole number, such as the square root of 2, and is left in exact form.
▶ Interact
Play with the controls to see the idea in action.
📘 Learn
A surd is a root that cannot be simplified to a whole number, such as the square root of 2, and is left in exact form.
- Surds are irrational numbers
- Root 12 simplifies to 2 root 3
- Rationalise by removing surds from denominators
🧠 Quick quiz
Three questions. Score 2 or more to complete the topic.
What is Surds?
Within Mathematics, which covers numbers, algebra, geometry, trigonometry and statistics from first principles to exam level, this topic explains Surds. 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 Surds in your own words and give one example from everyday life. If you can do that, you have understood the core idea.
Understanding Surds gives you a reliable method you will reuse in later maths units, in science lessons and in data work.
Surds in detail
A surd is an irrational root that cannot be written exactly as a fraction or whole number, such as √2 or √7. Leaving an answer as a surd keeps it exact, whereas a decimal would be only an approximation.
To simplify, pull out square factors: √50 = √25 × √2 = 5√2. Useful rules are √a × √b = √(ab) and √a ÷ √b = √(a/b). To remove a surd from a denominator, called rationalising, multiply top and bottom by that surd, so 1/√3 becomes √3/3. Surds appear in Pythagoras, trigonometry and the quadratic formula.
How the interactive helps you understand Surds
Watch the right-angled triangle in the interactive. Use any sliders or buttons it offers to change the values and see how the right-angled triangle responds.
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 Surds.
Real-life uses and examples
- Spot Surds in exam questions, where it often sits inside a short word problem.
- Check a homework answer about Surds by estimating first, then working it out step by step.
- Use Surds when comparing prices, discounts or recipe quantities in everyday life.
Common mistakes to avoid
- Forgetting units, signs or brackets when you apply Surds.
- Memorising the steps for Surds 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.
Frequently asked questions about Surds
What is Surds?
A surd is a root that cannot be simplified to a whole number, such as the square root of 2, and is left in exact form.
What are the most important points about Surds?
The main points are: Surds are irrational numbers; Root 12 simplifies to 2 root 3; Rationalise by removing surds from denominators.
Why is Surds useful?
Understanding Surds gives you a reliable method you will reuse in later maths units, in science lessons and in data work.
How can I practise Surds?
Open the interactive at the top of this page and change the controls one at a time. Predict the result before each change, then take the three-question quiz. A score of two or more marks the topic as complete on this device.
Which topics are related to Surds?
Related topics in Mathematics include Bounds and Error, Arithmetic Sequences, Algebraic Expressions.