What surds are

Some roots come out exactly: $\sqrt{16} = 4$ and $\sqrt[3]{8} = 2$. Oth­ers never do: $\sqrt{2} = 1.41421\ldots$ goes on for ever with­out repeat­ing. Roots of the sec­ond kind are called surds, and they turn up every­where in geom­e­try and alge­bra, from the diag­o­nal of a square to the solu­tions of qua­dratic equa­tions. This les­son says exactly what a surd is, how it dif­fers from other irra­tional num­bers, and names the two types you will use most: mixed surds and con­ju­gate surds.

Roots that come out exactly#

The $n$th root of a num­ber $a$, writ­ten $\sqrt[n]{a}$, is the num­ber which, mul­ti­plied by itself $n$ times, gives $a$. The small num­ber $n$ is the order of the root, and $a$ is the rad­i­cand. For a square root the order $2$ is not writ­ten: $\sqrt{a}$ means $\sqrt[2]{a}$.

Some roots are ratio­nal num­bers:

$$\begin{aligned} \sqrt[3]{8} &= \sqrt[3]{2 \times 2 \times 2} = 2 \\ \sqrt{16} &= \sqrt{4 \times 4} = 4 \\ \sqrt[5]{32} &= \sqrt[5]{2^{5}} = \sqrt[5]{2 \times 2 \times 2 \times 2 \times 2} = 2 \end{aligned}$$

In each case the rad­i­cand splits into $n$ equal fac­tors, so the root is a whole num­ber.

But $\sqrt{2}$ is not a ratio­nal num­ber: no frac­tion mul­ti­plied by itself gives exactly $2$. Nor is $\sqrt[3]{5}$: $5$ does not split into three equal ratio­nal fac­tors.

What a surd is#

A surd (also called a pure surd) is a root $\sqrt[n]{a}$ that is not a ratio­nal num­ber, where $a$ is a pos­i­tive ratio­nal num­ber and $n$ is a nat­ural num­ber greater than $1$.

Exam­ples of surds:

$$\sqrt{2}, \quad \sqrt[4]{5}, \quad \sqrt[6]{3}, \quad \sqrt[7]{8}, \quad \sqrt[9]{10}$$

Each of these is a root of a ratio­nal num­ber, and none of them comes out exactly. Notice $\sqrt[7]{8}$: even though $8 = 2^{3}$, the sev­enth root does not come out exactly, because $8$ does not split into seven equal ratio­nal fac­tors.

Not surds: $\sqrt{16}$, $\sqrt[3]{27}$ and $\displaystyle \sqrt{\frac{9}{4}}$, because they equal $4$, $3$ and $\displaystyle \frac{3}{2}$, all ratio­nal.

Exam­ple 1#

Which of these are surds? $\sqrt{25}$, $\sqrt{7}$, $\sqrt[3]{27}$, $\sqrt[3]{10}$, $\sqrt[4]{16}$, $\sqrt{8}$

• $\sqrt{25} = 5$: ratio­nal, not a surd.
• $\sqrt{7}$: $7$ is not a per­fect square, so this is a surd.
• $\sqrt[3]{27} = 3$: ratio­nal, not a surd.
• $\sqrt[3]{10}$: $10$ is not a per­fect cube, so this is a surd.
• $\sqrt[4]{16} = 2$, since $2 \times 2 \times 2 \times 2 = 16$: not a surd.
• $\sqrt{8}$: $8$ is not a per­fect square, so this is a surd.

Surds and irra­tional num­bers#

An irra­tional num­ber is a num­ber that can­not be writ­ten as a frac­tion. Its dec­i­mal form never ends and never repeats (it is non-ter­mi­nat­ing and non-recur­ring).

Every surd is an irra­tional num­ber. But not every irra­tional num­ber is a surd. The famous num­bers

$$\pi = 3.14159\ldots \qquad e = 2.71828\ldots$$

are irra­tional, with non-ter­mi­nat­ing, non-recur­ring dec­i­mals, but they are not roots of any ratio­nal num­ber, so they are not surds.

Remem­ber. Every surd is irra­tional, but an irra­tional num­ber need not be a surd.

Types of surds#

Mixed surds#

A surd mul­ti­plied by a ratio­nal num­ber, in the form $a\sqrt[n]{b}$, is called a mixed surd. Exam­ples:

$$2\sqrt{3}, \qquad 5\sqrt[6]{3}, \qquad 4\sqrt{2}$$

In $2\sqrt{3}$ the ratio­nal part is $2$ and the surd part is $\sqrt{3}$. You will see in a later les­son that a mixed surd can always be rewrit­ten as a pure surd, and that a pure surd like $\sqrt{12}$ can often be rewrit­ten as a mixed surd, $2\sqrt{3}$.

Con­ju­gate surds#

Two surds of the form

$$a + \sqrt{b} \quad \text{and} \quad a - \sqrt{b}$$

are called con­ju­gate surds of each other. They are the same except for the sign in front of the square-root part. Exam­ples of con­ju­gate pairs:

Con­ju­gates mat­ter because their prod­uct has no surd in it. For exam­ple, $(2 + \sqrt{3})(2 - \sqrt{3}) = 4 - 3 = 1$. You will work this out prop­erly in the next les­son, and use it again and again later in the chap­ter.

Exam­ple 2#

Write the con­ju­gate of (a) $6 + \sqrt{5}$ (b) $3 - \sqrt{2}$ (c) $\sqrt{11} + 4$.

(a) $6 - \sqrt{5}$. (b) $3 + \sqrt{2}$. (c) $-\sqrt{11} + 4$, which is the same as $4 - \sqrt{11}$.

Exam­ple 3#

State the order and the rad­i­cand of $\sqrt[5]{12}$, and say whether it is a surd.

The order is $5$ and the rad­i­cand is $12$. Since $12$ is not the fifth power of a ratio­nal num­ber, $\sqrt[5]{12}$ is a surd.

Prac­tice#

1. Which of these are surds? $\sqrt{36}$, $\sqrt{11}$, $\sqrt[3]{64}$, $\sqrt[3]{9}$, $\displaystyle \sqrt{\frac{4}{9}}$, $\sqrt[4]{81}$
2. State the order and the rad­i­cand of $\sqrt[4]{7}$.
3. Is $\pi$ a surd? Give a rea­son.
4. Is every irra­tional num­ber a surd?
5. Which of these are mixed surds? $3\sqrt{5}$, $\sqrt{20}$, $7\sqrt[3]{2}$, $\sqrt[3]{9}$
6. Write the con­ju­gate of $5 + \sqrt{7}$.
7. Write the con­ju­gate of $8 - \sqrt{3}$.
8. Show that $\sqrt[3]{125}$ is not a surd.
9. Is $\sqrt{18}$ a surd? Is $\displaystyle \sqrt{\frac{18}{2}}$?

Answers#