Although the name may sound strange, a surd is just a type of number with certain characteristics. In this lesson you’ll learn what surds are and when to use them. You’ll also be introduced to simplifying surds as well as adding, subtracting, multiplying and dividing them.

Introduction to Surds

What is a Surd?

√4 = 2, √64 =8, √81 = 9, √225 = 15, √169 = 13

What is √65 equal to?

Using a calculator, √65 = 8.062257748…

The numbers after the decimal point continue forever and have no recurring patterns.

Numbers like this are called irrational numbers. For example, π = 3.1415… is also an irrational number. Irrational numbers cannot be written as a fraction.

A surd is a root whose value is irrational. A surd can be a square root, a cube root, or higher root whose value is not equal to a whole number. When writing a surd in decimal form, the numbers after the decimal point continue infinitely and have no recurring pattern.

Surd Examples

The following numbers are all examples of surds. Their values are all irrational numbers.

• √3 = 1.732050808…
• √7 = 2.645751311…
• √8 = 2.828427125…
• √13 = 3.605551275…
• ∛10 = 2.15443469…
• ∜50 = 2.659147948…

These numbers are not surds as their values are not irrational numbers.

• √25 = 5
• √0 = 0
• ∛27 = 3
• ∛1 = 1
• ∜16 = 2

When Do We Use Surds?

Consider these two examples using the Pythagorean theorem.

Sometimes in mathematics you may calculate an answer those value is a surd. It is especially common when using the Pythagorean theorem to calculate a missing side length of a right angle triangle.

Instead of rounding the answer to a chosen number of decimal places or significant figures, you can write the answer in surd form. This gives the exact value of the surd and is more accurate than rounding.

More About Surds…

Just like any other type of number, surds can be added, subtracted, multiplied, and divided.

Surds can also be simplified. In a similar way as to how some fractions can be written in a simpler way, some surds can also be written in a similar way.

The rest of this lesson will introduce all of these ideas.

Simplifying Surds

Learn how to simplify surds here.

Adding and Subtracting Surds

Learn how to add surds here.

Multiplying Surds

Learn how to multiply and divide surds here.

Expanding Brackets with Surds

Surds can appear in questions involving expanding single and double brackets. If you’ve mastered adding, subtracting, multiplying and dividing surds then this is the next step towards mastery!

Learn how to expand brackets with surds here.

Rationalising the Denominator

Learn how to rationalise the denominator here.