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
nonterminating 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 5^{th} century AD was the first one to
give approximate value of _{} to 4 correct decimal
places (3.1416). 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 _{}= circumference ÷ diameter = 62832 ÷ 20000= 3.1416 

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 10^{n}, where n is the ‘periodicity’ (number of digits that
repeat): If 3 digits repeat, multiply both sides by 10^{3})
_{}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 nonterminating 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 nonzero 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 )