## Irrational Numbers

Corresponding to every rational
number, there is a point on the number line. We can say that corresponding to
every point on the number line, there is a rational number. Draw a number line
and mark a point A so that OA = 1 unit. Draw a square OABC on side OA. Now,
using Pythagoras theorem which states that the square of the hypotenuse of a right-angled triangle is equal to the sum of the squares of
the other two sides. We have

Now, with O as center and OB radius, draw an arc which cuts the number line at D. Thus,

OD = √2 . But √2 is not a
rational number as it cannot be expressed in the form

*p/q*. From this we infer that there are points on the number line which are not rational. If we express √2 in the decimal form, we observe that it does not terminate and is non-recurring. Such numbers are called**irrational numbers**.
The numbers √2 , √3 , √6 , √7 ,
2 + √3 , 2 – √3 , √2 + √3 etc., are examples of irrational numbers.

All non-terminating and non-recurring decimal number are
irrational numbers.

##
Real Numbers

The combination of rational and
irrational numbers are called

**real numbers**. It is denoted by**R**. Thus, R = {*x*:*x*is a rational or irrational number}.
Note that

**N**⊂**W**⊂**Z**⊂**Q**⊂**R.**
All decimal numbers
(terminating, recurring, non-terminating and non-recurring) are

real numbers. Let us observe the
following diagram which shows the relation among the

different kinds of numbers.

##
The Wheel of Theodorus

Around 425 B.C., Theodorus of
Cyrene, a philosophy of ancient Greece discovered the

construction below. It is called

**the wheel of the Theodorus**.
To construct the wheel of Theodorus, we proceed as:

1. Mark a point O and draw OA =
1 unit.

2. At A, draw _OAB
= 90° and cut off AB = 1 unit. Join OB.

3. At B, draw _OBC
= 90º and cut off BC = 1 unit. Join OC.

4. At C, draw _OCD
= 90º and cut off CD = 1 unit. Join OD.

Go on repeating the step 2 at E,
F, G and so on and, we get the value of OE = √5, OF = √6,

OG = √7 and so on.

All these lengths can be plotted
on the number line. Thus, corresponding to any real number, there is a unique
point on the number line and corresponding to every point on the number line,
there is a unique real number.

###
Important Facts
about Irrational Numbers

##
Rationalization

If the product of two irrational
numbers is a rational number then each number is said to be the rationalizing
factor of the other number.

For example:

a. √6 × √6 = 6
,
therefore, √6
is
a

**rationalizing factor**of √6.
b. (2 – √3) (2 + √3) = (2)

^{2}– (√3)^{2}= 4 – 3 = 1, Therefore 2 + √3 and 2 – √3 are rationalizing factor of each of other.
This process of multiplying an
irrational number by its rationalizing factor is called

**rationalization.**

###
Rule for Rationalization

In order to rationalize,
multiply and divide the denominator of the irrational number by the rationalizing
factor of the denominator and then simplify.

**Example 1:**Rationalize the denominator of √2/√7.

**Solution:**The rationalizing factor of √7 is √7.

So, multiply by √7 in the
numerator and denominator of √2/√7.

(√2 × √7)/(√7
× √7) = √14/7

**Example 2:**Write in ascending order: 2√3, 3√2, 2√5, 5√6, 4

**Solution:**

*Let us write all the numbers as square roots under one radical.*

2√3 = √4 × √3 = √12; 3√2 = √9 × √2 = √18; 2√5 = √4 × √5 = √20 ; 5√6 = √25 × √6 = √150,
4 = √16

Now, 12 < 16 < 18 < 20 < 150 therefore, √12 < √16 < √18 < √20 < √150

⇒ 2√3 < 4 < 3√2 < 2√5 < 5√6

Thus, the given numbers in
ascending order are 2√3, 4,
3√2, 2√5 and 5√6.

**Example 3:**Insert four irrational numbers between √2 and √11.

**Solution:**

*Consider the square of √2 and √11, we have (√2)*

^{2}= 2 and (√11)

^{2}= 11.

Now, 2 < 3 < 5 < 6 <
7 < 8 < 10 < 11 ⇒ √2 < √3 < √5 < √6 < √7 < √8 < √10 < √11

Hence, four irrational numbers
between √2
and
√11
are
√3,
√5, √6 and √7.

**Related Topics:**