Surds and Irrational Numbers

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 $\frac{a}{b}$ of two whole numbers (with $b\ne 0$). Its decimal either stops or repeats: $\frac{3}{4}=0.75$ and $\frac{1}{3}=0.333\dots$

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 $a,b\ge 0$:

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

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.

Adding and subtracting

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

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

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.

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

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:

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

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: $\frac{a}{\sqrt{b}}=\frac{a\sqrt{b}}{b}$. For $a+\sqrt{b}$ on the bottom, use the conjugate.

Quiz