# Operations on Real Numbers

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You have learnt, in earlier classes, that rational numbers satisfy the commutative, associative and distributive laws for addition and multiplication.

Moreover, if we add, subtract, multiply or divide (except by zero) two rational numbers, we still get a rational number (that is, rational numbers are ‘closed’ with respect to addition, subtraction, multiplication and division).

It turns out that irrational numbers also satisfy the commutative, associative and distributive laws for addition and multiplication. However, the sum, difference, quotients and products of irrational numbers are not always irrational. For example, (

Let us look at what happens when we add and multiply a rational number with an irrational number. For example,

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**Example 11 : Check whether 75,7**

**Now, let us see what generally happens if we add, subtract, multiply, divide, take square roots and even nth roots of these irrational numbers, where n is any natural number. Let us look at some examples.**

**Example 12 : Add 2 2 +53 and 2−**

**Example 13 : Multiply 6 5 by25**

**Example 14 : Divide 8 15 by23.**

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These examples may lead you to expect the following facts, which are true:

**(i) The sum or difference of a rational number and an irrational number is .**

**(ii) The product or quotient of a non-zero rational number with an irrational number is .**

**(iii) If we add, subtract, multiply or divide two irrationals, the result may be or irrational.**

We now turn our attention to the operation of taking square roots of real numbers. Recall that, if a is a natural number, then a = b means

Let a > 0 be a

we saw how to represent n for any positive integer n on the number line. We now show how to find

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**For example, let us find it for x = 3.5, i.e., we find **

Mark the distance 3.5 units from a fixed point A on a given line to obtain a point B such that AB =

From B, mark a distance of

More generally, to find

Notice that, in Fig is shown above. ∆ OBD is a right-angled triangle. Also, the radius of the circle is

Therefore, OC = OD = OA =

Now, OB =

So, by the Pythagoras Theorem, we have

This shows that BD =

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We would like to now extend the idea of square roots to cube roots, fourth roots, and in general nth roots, where n is a positive integer. Recall your understanding of square roots and cube roots from earlier classes.

What is

From these examples, can you define

Let a > 0 be a real number and n be a

We now list some identities relating to square roots, which are useful in various ways. You are already familiar with some of these from your earlier classes. The remaining ones follow from the distributive law of multiplication over addition of real numbers, and from the identity

Let a and b be positive real numbers. Then

(i)

(ii)

(iii)

(iv)

(v)

(vi)

**Let us look at some particular cases of these identities.**

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**Example 15 : Simplify the following expressions:**

**Remark** : Note that ‘simplify’ in the example above has been used to mean that the expression should be written as the sum of a rational and an irrational number. We end this section by considering the following problem. Look at

**Example 16 : Rationalise the denominator of **

**Solution :** We want to write

**Example 17 : Rationalise the denominator of **

**Solution :** We use the Identity (iv) given earlier. Multiply and divide

**Example 18 : Rationalise the denominator of **

**Solution :** Here we use the Identity (iii) given earlier.So,

**Example 19 : Rationalise the denominator of **

**Solution :**

So, when the denominator of an expression contains a term with a square root (or a number under a radical sign), the process of converting it to an equivalent expression whose denominator is a rational number is called rationalising the denominator.