# Surds and Irrational Numbers

> Rational and irrational numbers, simplifying surds, adding, multiplying and dividing them, and rationalising denominators.

From HSC Maths Advanced on WhizzNotes, free study notes for NSW students: https://whizznotes.com/mathematics-advanced/surds/. Updated 3 November 2026.
Year 11 and Year 12 boxes go deeper than the Year 10 text around them.

### Story time: The square tile
Ava is tiling a bathroom floor with square tiles that are exactly 1 m wide. She wants to cut one tile corner to corner. How long is the cut?

Pythagoras says the diagonal is $\sqrt{1^2 + 1^2} = \sqrt{2}$ m. Her calculator shows 1.414 213 562... and the digits never stop and never repeat. No fraction is exactly equal to it. Builders round it to 1.41 m, but mathematicians keep it exact as $\sqrt{2}$, a **surd**. Working with surds lets you keep answers exact all the way through a problem.

**In short:** A surd is a root that is an irrational number, like $\sqrt{2}$ or $\sqrt{5}$, so its decimal goes on forever without repeating. Simplify a surd with its largest square factor, as in $\sqrt{50} = 5\sqrt{2}$, and rationalise to clear a surd from the bottom of a fraction.

## Starting point: Years 9 and 10

### Rational and irrational numbers

A **rational** number can be written as a fraction $\dfrac{a}{b}$ of two whole numbers (with $b \neq 0$). Its decimal either stops or repeats: $\dfrac{3}{4} = 0.75$ and $\dfrac{1}{3} = 0.333\ldots$

An **irrational** number cannot be written as a fraction. Its decimal goes on forever with no repeating pattern. Examples are $\pi$, $\sqrt{2}$ and $\sqrt{5}$.

Together, rational and irrational numbers make the **real numbers**.

### What is a surd?

A **surd** is a root of a whole number that is irrational. $\sqrt{7}$ and $\sqrt[3]{10}$ are surds. $\sqrt{9} = 3$ is not a surd, because it is a whole number.

### Two rules for roots

For $a, b \geq 0$:

$$\sqrt{ab} = \sqrt{a} \times \sqrt{b} \qquad \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$$

Also $(\sqrt{a})^2 = a$, so $\sqrt{a} \times \sqrt{a} = a$.

### Simplifying surds

Find the **largest square factor**, then take its root out the front.

### Worked example: Simplify
$\sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3}$

$3\sqrt{20} = 3\sqrt{4 \times 5} = 3 \times 2\sqrt{5} = 6\sqrt{5}$

### Exam tip: Know your squares
4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144. Spot one of these inside the number and the surd simplifies.

### Calculator tip: Exact or decimal
Most scientific calculators show roots in exact form: $\sqrt{12}$ comes out as $2\sqrt{3}$. Press `S⇔D` to switch between the exact form and the decimal.

To check a simplified or rationalised answer, find both your answer and the original as decimals. If they match, your working is right. In the exam still show every step: a question that asks for an exact answer needs the surd, not a decimal.

### Adding and subtracting

Only **like surds** (same number under the root) can be added, just like like terms in algebra.

$$5\sqrt{3} + 2\sqrt{3} = 7\sqrt{3} \qquad 4\sqrt{2} - \sqrt{2} = 3\sqrt{2}$$

Sometimes you must simplify first: $\sqrt{12} + \sqrt{27} = 2\sqrt{3} + 3\sqrt{3} = 5\sqrt{3}$.

### Multiplying and dividing

Multiply the numbers outside together and the numbers inside together.

$$2\sqrt{3} \times 5\sqrt{2} = 10\sqrt{6} \qquad \frac{12\sqrt{10}}{4\sqrt{5}} = 3\sqrt{2}$$

Expand brackets as normal: $\sqrt{2}(3 + \sqrt{2}) = 3\sqrt{2} + 2$.

### Rationalising the denominator

Mathematicians prefer no surd on the bottom of a fraction. Multiply top and bottom by the surd:

$$\frac{5}{\sqrt{2}} = \frac{5}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{5\sqrt{2}}{2}$$

### Key idea: Surd rules
- $\sqrt{ab} = \sqrt{a}\sqrt{b}$ and $\sqrt{a} \times \sqrt{a} = a$.
- Add and subtract only like surds.
- Multiply outside with outside, inside with inside.
- Rationalise by multiplying top and bottom by the surd.

### Watch out: Roots do not split over plus
$\sqrt{9 + 16} = \sqrt{25} = 5$, but $\sqrt{9} + \sqrt{16} = 7$. So $\sqrt{a + b} \neq \sqrt{a} + \sqrt{b}$.

**Mind map**

```text
Surds
  Number types
    Rational: fraction
    Irrational: endless, no repeat
    Real numbers: both
  Simplify
    Largest square factor
  Add and subtract
    Like surds only
  Multiply and divide
    Outside with outside
    Inside with inside
  Rationalise
    Times surd over surd
    Conjugate for binomials
```

### Year 11: Binomial denominators and expanding
**Expanding pairs of brackets** with surds uses the same patterns as algebra:

$$(2 + \sqrt{3})^2 = 4 + 4\sqrt{3} + 3 = 7 + 4\sqrt{3}$$

$$(\sqrt{5} + \sqrt{2})(\sqrt{5} - \sqrt{2}) = 5 - 2 = 3$$

The second line shows why the **conjugate** is useful. To rationalise $\dfrac{4}{3 - \sqrt{5}}$, multiply by $\dfrac{3 + \sqrt{5}}{3 + \sqrt{5}}$:

$$\frac{4(3 + \sqrt{5})}{9 - 5} = 3 + \sqrt{5}$$

Exact answers in trigonometry, such as $\sin 45^\circ = \dfrac{1}{\sqrt{2}} = \dfrac{\sqrt{2}}{2}$, use these skills.

When the top also has a surd, multiply out the top after using the conjugate, and write the answer as $a + b\sqrt{n}$:
$$\frac{\sqrt{3} + 2}{\sqrt{3} - 1} \times \frac{\sqrt{3} + 1}{\sqrt{3} + 1} = \frac{3 + \sqrt{3} + 2\sqrt{3} + 2}{3 - 1} = \frac{5 + 3\sqrt{3}}{2}$$

**Solving equations with surds.** Simplify each surd first, then solve as usual. For $x\sqrt{8} = \sqrt{50} - \sqrt{2}$: $2\sqrt{2}\,x = 5\sqrt{2} - \sqrt{2} = 4\sqrt{2}$, so $x = 2$.

A neat pattern: $\dfrac{1}{\sqrt{n + 1} + \sqrt{n}} = \sqrt{n + 1} - \sqrt{n}$ (multiply by the conjugate). So a long sum of these fractions collapses, because the middle terms cancel.

### Year 12: Surds as powers in calculus
In calculus, write surds as fractional powers: $\sqrt{x} = x^{\frac{1}{2}}$ and $\sqrt[3]{x^2} = x^{\frac{2}{3}}$. Then the power rule works:

$$\frac{d}{dx}\sqrt{x} = \frac{1}{2}x^{-\frac{1}{2}} = \frac{1}{2\sqrt{x}}$$

Exam answers are often expected in simplest surd form, for example a stationary point at $x = \sqrt{12} = 2\sqrt{3}$.

Conjugates also help with **limits**. When putting in the value gives $\frac{0}{0}$, multiply by the conjugate to cancel the problem factor:
$$\lim_{x \to 9} \frac{x - 9}{\sqrt{x} - 3} = \lim_{x \to 9} \frac{(x - 9)(\sqrt{x} + 3)}{x - 9} = \lim_{x \to 9} (\sqrt{x} + 3) = 6$$

## Quick summary

- Rational numbers are fractions. Irrational numbers have endless, non-repeating decimals.
- A surd is an irrational root, like $\sqrt{2}$.
- Simplify using the largest square factor: $\sqrt{50} = 5\sqrt{2}$.
- Add only like surds. Multiply outside with outside and inside with inside.
- Rationalise: $\dfrac{a}{\sqrt{b}} = \dfrac{a\sqrt{b}}{b}$. For $a + \sqrt{b}$ on the bottom, use the conjugate.

## Quiz

Try these questions from the chapter quiz.

1. Which number is irrational?
   - A. $\dfrac{2}{3}$
   - B. 0.25
   - C. $\sqrt{11}$
   - D. $\sqrt{49}$
   - Answer: C. $\sqrt{11}$. $\sqrt{11}$ is not a whole number, so its decimal never ends or repeats.

2. Which of these is a surd?
   - A. $\sqrt{16}$
   - B. $\sqrt{6}$
   - C. $\sqrt[3]{8}$
   - D. $\sqrt{100}$
   - Answer: B. $\sqrt{6}$. The others are whole numbers: 4, 2 and 10.

3. Simplify $\sqrt{45}$.
   - A. $5\sqrt{3}$
   - B. $3\sqrt{5}$
   - C. $9\sqrt{5}$
   - D. $15$
   - Answer: B. $3\sqrt{5}$. $45 = 9 \times 5$.

The whole chapter quiz is on the page, for signed-in students: https://whizznotes.com/mathematics-advanced/surds/#quiz