Surds and Irrational Numbers

Rational and irrational numbers

Every number you have met so far is either rational or irrational.

A rational number is any number that can be written as a fraction a/b, where a and b are whole numbers and b is not zero. This includes whole numbers (5 = 5/1), terminating decimals (0.75 = 3/4) and repeating decimals (0.333… = 1/3).

An irrational number cannot be written as such a fraction. Its decimal expansion goes on forever without ever repeating a pattern. Famous examples include:

• √2 = 1.41421356… (the diagonal of a unit square)
• π = 3.14159265… (the ratio of a circle's circumference to its diameter)
• √3, √5, √7 and most other roots of whole numbers

The number line is completely filled by these two families together — between any two fractions there are infinitely many irrational numbers, and vice versa.

What is a surd?

A surd is a root — usually a square root — that is irrational, meaning it does not simplify to a rational number. So √2, √3 and √10 are surds, but √4 (= 2) and √9 (= 3) are not, because they give whole-number answers.

We keep surds in root form because that form is exact. Writing √2 ≈ 1.41 throws away accuracy; writing √2 keeps the value perfect. This matters in trigonometry, in the Pythagorean theorem, and whenever an exam asks for an "exact answer".

Why this exists: The ancient Greeks discovered that the diagonal of a square with side 1 has length √2, and proved this length can never be an exact fraction. It was a famous shock — it meant not every length is a tidy ratio. Surds are how we write those lengths honestly.

Simplifying surds

To simplify a surd, pull out the largest perfect-square factor hiding inside it. Perfect squares are 4, 9, 16, 25, 36, 49, 64, … The rule used is:

√(a × b) = √a × √b

Worked example 1: Simplify √12.

1. Find a perfect-square factor of 12. Since 12 = 4 × 3 and 4 is a perfect square, use that.
2. Split the root: √12 = √4 × √3.
3. Simplify the perfect square: √4 = 2.
4. So √12 = 2√3. (Check: 2√3 ≈ 2 × 1.732 = 3.464, and √12 ≈ 3.464. Correct.)

Worked example 2: Simplify √72.

1. Look for the largest perfect square dividing 72. Options: 4, 9, and 36. Use 36 (the biggest) because 72 = 36 × 2.
2. √72 = √36 × √2 = 6√2.
3. If you had only spotted 4 first, you would get √72 = 2√18, then √18 = √9 × √2 = 3√2, giving 2 × 3√2 = 6√2 — the same answer, just more steps.

Worked example 3: Simplify √200.

1. 200 = 100 × 2, and √100 = 10.
2. √200 = 10√2.

Multiplying and dividing surds

Multiplying is straightforward — multiply the numbers under the roots:

√a × √b = √(ab)

Worked example 4: Work out √3 × √12.

1. Multiply inside: √3 × √12 = √(3 × 12) = √36.
2. √36 = 6. The surds disappeared completely because 36 is a perfect square.

Division works the same way:

√a ÷ √b = √(a/b)

Worked example 5: Work out √20 ÷ √5.

1. √20 ÷ √5 = √(20 ÷ 5) = √4 = 2.

Adding and subtracting surds

You can only add or subtract surds when they have the same number under the root — these are called like surds. Treat the surd like an algebraic letter, exactly as you would with algebra basics:

• 2√5 + 3√5 = 5√5 (just as 2x + 3x = 5x)
• 7√2 − 4√2 = 3√2

Unlike surds (such as √2 and √3) cannot be combined — √2 + √3 stays as it is.

Worked example 6: Simplify √18 + √8.

1. Simplify each first: √18 = √9 × √2 = 3√2, and √8 = √4 × √2 = 2√2.
2. Now they are like surds: 3√2 + 2√2 = 5√2.

This is the key insight — surds that look different often become like surds once simplified.

Worked example 7: Simplify √75 − √12 + √48.

1. Break each into a perfect-square factor × something: √75 = √25 × √3 = 5√3; √12 = √4 × √3 = 2√3; √48 = √16 × √3 = 4√3.
2. All three are now multiples of √3, so they are like surds.
3. Combine the numbers in front: 5√3 − 2√3 + 4√3 = 7√3.

Whenever a question mixes several roots, simplify them all first — they very often collapse onto the same surd.

Expanding brackets with surds

Surds follow the same expansion rules as algebra. When you expand brackets, multiply every term by every term, then collect like surds.

Worked example 8: Expand and simplify √2(√2 + √6).

1. Multiply √2 by each term: √2 × √2 = √4 = 2, and √2 × √6 = √12 = 2√3.
2. So the answer is 2 + 2√3.

Worked example 9: Expand (1 + √3)(2 − √3).

1. Use FOIL: 1 × 2 = 2; 1 × (−√3) = −√3; √3 × 2 = 2√3; √3 × (−√3) = −√9 = −3.
2. Collect: 2 − 3 = −1 for the whole numbers, and −√3 + 2√3 = √3 for the surds.
3. Result: −1 + √3.

Rationalising the denominator

By convention we do not leave a surd on the bottom of a fraction. Rationalising removes it by multiplying the top and bottom by a chosen surd, which is the same as multiplying by 1 (so the value is unchanged).

Worked example 10: Rationalise 1/√2.

1. Multiply numerator and denominator by √2:

1/√2 × √2/√2 = √2 / (√2 × √2).

1. The denominator becomes √2 × √2 = √4 = 2.
2. Result: √2 / 2. The root is now on top, where it is allowed.

Worked example 11: Rationalise 6/√3.

1. Multiply by √3/√3: 6√3 / (√3 × √3) = 6√3 / 3.
2. Simplify the fraction: 6/3 = 2, giving 2√3.

A reference table

Surd	Simplified form	Approximate value
√8	2√2	2.828
√12	2√3	3.464
√18	3√2	4.243
√50	5√2	7.071
√75	5√3	8.660
1/√2	√2 / 2	0.707

Where surds are used

Surds appear the moment you measure a slanted or curved length exactly. Using the Pythagorean theorem, a right triangle with both short sides equal to 1 has a hypotenuse of exactly √2. In trigonometry, exact values like sin 60° = √3/2 are surds. Engineers, physicists and architects keep surds during a calculation to avoid rounding errors building up, only converting to a decimal at the very end.

Practice activity

Simplify or evaluate each. Keep answers exact.

1. √45
2. √98
3. √6 × √24
4. 5√3 − 2√3
5. √12 + √27
6. Rationalise 10/√5

Answers:

1. 45 = 9 × 5, so √45 = 3√5.
2. 98 = 49 × 2, so √98 = 7√2.
3. √6 × √24 = √144 = 12.
4. Like surds: 5√3 − 2√3 = 3√3.
5. √12 = 2√3 and √27 = 3√3, so the sum is 5√3.
6. 10/√5 × √5/√5 = 10√5/5 = 2√5.

Summary

A rational number can be written as a fraction; an irrational number cannot, and its decimal never ends or repeats. A surd is an irrational root that we keep in exact form. Simplify a surd by extracting its largest perfect-square factor, multiply and divide by combining what is under the roots, add or subtract only like surds, and rationalise a denominator by multiplying top and bottom by a surd. Surds give exact answers — the reason they are prized throughout geometry, trigonometry and physics.