What is a Surd?

Definition

A quantity that expresses an unresolved root of a number is called a surd.

Introduction

The irrational numbers are one type of real numbers, whereas there are some special forms of irrational numbers and they express the unresolved roots of positive integers. So, they are called the surds.

Let’s start the concept of a surd in mathematics to know what a surd really is.

Now, let’s see some more examples to clearly know what a surd is in mathematics.

Examples

$\sqrt{3}$, $\sqrt{4}$ and $\sqrt{5}$

The values of square roots of $3$, $4$ and $5$ are given here.

$(1).\,\,$ $\sqrt{3}$ $\,=\,$ $1.7320508075\cdots$

$(2).\,\,$ $\sqrt{4}$ $\,=\,$ $2$

$(3).\,\,$ $\sqrt{5}$ $\,=\,$ $2.2360679774\cdots$

Now, let’s discuss about above irrational form real numbers.

1. The number of decimal places is infinite, in the case of both the square root of $3$ and the square root of $5$, because the square roots of $3$ and $5$ are unsolvable. So, $\sqrt{3}$ is a surd and $\sqrt{5}$ is also a surd.
2. The square root of $4$ is $2$ because it is solvable. So, $\sqrt{4}$ is not a surd.

Let’s see some more examples of a surd.

$(1).\,\,$ $\sqrt[\Large 3]{2}, \, \sqrt[\Large 3]{7}, \, \sqrt[\Large 3]{10}$ and so on.

$(2).\,\,$ $\sqrt[\Large 4]{3}, \, \sqrt[\Large 4]{5}, \, \sqrt[\Large 4]{6}$ and so on.

$(3).\,\,$ $\sqrt[\Large 5]{2}, \, \sqrt[\Large 5]{4}, \, \sqrt[\Large 5]{9}$ and so on.

The degree of a root can be any positive integer and every unresolved root of a positive real number is a surd. A surd is also a radical but remember, every radical is not a surd.

Notation

When $a$ denotes a positive rational number and $n$ denotes a positive integer, the $n$th root of a expresses an unresolved root of a number and it is called a surd.

In general, a surd is written as $\sqrt[\Large n]{a}$ algebraically in mathematics.