Real Number System

Introduction to Surds

**Concepts Related to Surds**

Look at the following numbers.

All these are rational numbers as .

Now, observe the numbers . These numbers are irrational.

**Roots of rational numbers:**

Suppose 5 is the square of a rational number, then

*x*^{2} = 5

⇒ *x *=

Here, 5 is a rational number, but is not a rational number. Thus, *x* can not be a rational number.

Now, let us assume that 10 is the cube of a rational number, therefore* **y*^{3} = 10.

⇒ *y*=

Here, 10 is a rational number. Since cube root of 10 is not a rational number, *y *cannot be a rational number.

Similarly, there are many rational numbers that are not square, cube, etc. of any rational number. In other words, we can say that there are many rational numbers whose roots are irrational.

**Irrational root of a positive rational number is called surd.**

For example:

, , , , etc.

It can be generally defined in the following way:

If is an irrational number such that *x *is a positive rational number and *a* (*a *≠ 1) is a natural number, then is known as a surd. Here, is the **radical sign**, *a *is the **order **of the surd and *x *is the **radicand**.

When *a* = 2, the surd is called a quadratic surd.

Now, consider the number .

Is it a surd?

No, it is not.

Since is the root of the negative rational number −4, it cannot be called as surd. Similarly, is the root of an irrational number π, so it is not a surd.

Now, observe the number .

Is it a surd?

Yes, it is.

By just looking at the number, it seems that is not a surd, but it can be reduced to the surd form in the following way:

is a surd.

**Rules for surds:**

Let Q and N be the sets of rational numbers and natural numbers respectively.

If *x*, *y *∈ Q, *x*, *y *> 0 and *a*, *b*, *c *∈ N, then

These rules are very useful to solve the problems related to surds.

Let us go through a few examples to understand the concept better.

**Example 1:**

Identify the surds among the given numbers and observe their orders.

**Solution:**

_{Example 2: Simplify the following using the rules of surds.}

_{Solution:}

**Various Forms of Surds and Their Conversions**

Surds can be represented in two main forms, which are pure and mixed form.

**Pure form: **A surd of the form is said to be in **pure form when** *k *∈ Q, such that *k *= ±1.

For example, are pure surds.

**Mixed form:**

A surd of the form is said to be in **mixed form** when *k *∈ Q, such that *k *≠ 0 and *k *≠ ±1.

For example, are mixed surds.

We can easily convert mixed surds to pure surds and vice versa. Let us study a few examples to understand the conversion.

Therefore, mixed surd can be written in pure surd form as .

Let us take another example.

Therefore, pure s…

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