1.4
Irrational numbers:
Every rational number can be expressed as a decimal
number.
For example 1/2 = 0.5, 1/4= 0.25, 1/8 = 0.125, 1/5 = 0.2 and
so on.
These rational numbers have fixed number of digits after the
decimal point.
These are rational numbers whose reminder becomes zero
after few successive divisions (division is exact).
However, there are rational numbers like 1/3 and 1/7 where
the reminder does not become zero even after several successive divisions.
Moreover, we notice that 1/3 = 0.33333 .. And 1/7 = 0.142857142857142857….
The recurring part of non-terminating recurring decimal is
called the ‘period’ and the number of
digits in the recurring part is called ‘periodicity’
1/3 is also represented as 0.
(Implying
(meaning) that the digit 3 repeats itself. In this case 3 is the period
and periodicity is 1)
1/7 is also represented as 0.
(Implying that
the group 142857 repeats itself. In this case 142857 is the period and
periodicity is 6)
In the case of 1/4, the decimal has only 2 digits after
the decimal point and they are called terminating
decimals. Where as in the case of 1/3 and 1/7 there is no fixed number
of digits after the decimal point and the group of digits repeat themselves. Such
decimals are called non terminating and recurring decimals.
Terminating decimals and recurring decimals can be
expressed as rational numbers which is of the form p/q with q 
0.
But, non terminating and non
recurring decimals cannot be expressed in the form p/q with q 0.
Definition: Non terminating
and non recurring decimals/numbers
which cannot be written in the form p/q
with q 0 are called irrational numbers.
Examples are 
=1.41421356237310 and
= 2.23606797749979
In Sulabhasutras
which dates back to Vedic period, value of is given as = 1 +1/+{(1/4)*(1/3)} – {(1/34)*(1/4)*(1/3)}
= 1.41421356
Another irrational number is 
whose approximate
value = 3.14159265358979
|
Aryabhatta the Indian mathematician of 5th century AD
was the first one to give approximate value of His
formula is : The approximate circumference of a circle of diameter 20000 units is got by adding 62000 to the result of 8
times the sum of 100 and 4. Circumference
= 62000+ 8(100+4) = 62832; diameter
= 200
|
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Note: Since is an irrational number, 4+ is also an irrational number and hence 
is also an irrational
number.
The square roots and cube roots of natural numbers whose
exact value cannot be obtained are irrational numbers (Ex:
,
and also 5,8)
1.4 Summary of learning
|
No |
Points studied |
|
1 |
Irrational
numbers |
Additional
Points:
1.4 Problem
1: Show that
0.477777 is a rational number
We need to show that the given number is of the form p/q
Solution:
Let x = 0.4
. Note that only one digit, 7 (periodicity = 1) repeats, so
we multiply both sides by 10.
(As a general rule we multiply both sides by 10n,
where n is the ‘periodicity’ (number of digits that repeat): If 3 digits repeat,
multiply both sides by 103)
10x = 4.77777…. = 4.3+0.47777… = 4.3+x
9x = 4.3 = 43/10
x = 43/90
Verify that 43/90 = 0.47777
Thus, a number whose decimal expansion is terminating or non-terminating
and recurring is rational.
Observations:
1. The sum or
difference of a rational number and an irrational number is irrational (4 + )
2. The product or quotient
of a non-zero rational number with an irrational number is irrational ( 2, (1/2)*)
3.
Addition/subtraction/multiplication/division of irrational numbers may
be rational or irrational
( -= 0 is rational,+ =2 is irrational, * = 5 is rational, 
* = 
is irrational, ÷ = 1 is rational, ÷ = 
is irrational )