Real numbers

Commutative and Associative Properties of Rational Numbers

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

x2 = 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 y3 = 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:

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

x2 = 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 y3 = 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 surd can be written in mixed surd form as .

It should be noted that it is not possible to express every pure surd as mixed surd.

For example:

, etc.

In such cases, the radicand is a prime number or it has the factors whose roots are irrational.

Let us go through a few examples to get more familiar with the concept.

Example 1: Convert the following mixed surds to pure surds.

Solution:

Example 2: Convert the following pure surds to mixed 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 surd can be written in mixed surd form as .

It should be noted that it is not possible to expr…

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