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.
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