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 )