Irrational Numbers, Real Numbers, Rationalization

# Irrational Numbers, Real Numbers, Rationalization

## 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.

## 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: 23, 32, 25, 56, 4

Solution: Let us write all the numbers as square roots under one radical.
23 = 4 × 3 = 12; 32 = 9 × 2 = 18; 25 = 4 × 5 = 20 ; 56 = 25 × 6 = 150, 4 = 16
Now, 12 < 16 < 18 < 20 < 150 therefore, √12 < 16 < 18 < 20 < 150
23 < 4 < 32 < 25 < 56
Thus, the given numbers in ascending order are 23, 4, 32, 25 and 56.

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.

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