Operations on Real Numbers
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.
In other words, 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:
What about 2 +
Therefore, both
Examples 11: Check whether 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.
Examples 12: Add
Examples 13: Multiply
Examples 14: Divide
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
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
Let x be a positive rational number. Let AB be of a length 'x' units on the numberline.
Now, extend AB upto a point C on the numberline such that BC is 1 unit.
Taking AC of length
Using D as the center, draw a semi-circle/ circle and also extend AC further, to see where this circle intersects the numberline.
From point B, draw a perpendicular intersecting the circle. Join BE.
The length of the segment BE is equal to
But how do we get that? Let's find out!
We can see that DE =
From the figure, we can see: DB = DC - BC =
Looking at triangle DEB, using pythagoras theorem, we get:
=
=
Thus, BE =
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
Well, we know it has to be some positive number whose cube is 8, and you must have guessed
Let us try . Do you know some number b such that
The answer is
From these examples, can you define for a real number a > 0 and a positive integer n?
Let a > 0 be a real number and n be a
Note that the symbol ‘
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.
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
Using this, we get:
Example 17: Rationalise the denominator of
Solution : We use the Identity (iv) given earlier.
Multiply and divide
Thus, the answer is 2 -
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.