Transcript Decimal Expansion of Rational Numbers

Decimal Expansion of
Rational Numbers
By
Arogya Singh
Seema KC
Prashant Rajbhandari

Rational Number
 A rational number is any number that can be
expressed as the ratio of two integers p/q, where q≠0.
Some examples of rational numbers are:
2 1  50
1
25
0
1 4
3
2
, ,
,
,
,
 Rational numbers that can be expressed in a decimal
form either terminates or repeats. For Example:
3/5 = 0.6 (Terminate)
2/3 = 0.6666 (Repeat)
Conditions (Terminate)
The conditions for a rational numbers to terminate
is:

A fraction (in simplest form/lowest terms) terminates in its decimal
form, if the prime factors of the denominator are only 2’s and 5’s or
a product of prime factors of 2’s and 5’s. The product of prime
factors can also be expressed as
(Q = P1n1*P2n2………..……………Pknk).

The decimal expansion of an irrational number never terminates.
Conditions (Cont…..)
Terminate
1 , Where 2 in the denominator is the prime factor of
2 (2*1).
Repeat
1
3
4 , Where 5 in the denominator is the prime factor of 1
5 (5*1).
15
Where 3 in the denominator is not the prime factor of
2’s & 5’s.
Where 15 in the denominator is not the prime factor
of 2’s & 5’s.
3 , Where 5 in the denominator is the prime factor of 7 Where 21 in the denominator is not the prime factor
of 2’s & 5’s.
10 (2*5).
21
7 , Where 5 in the denominator is the prime factor of 7 Where 30 in the denominator is not the prime factor
of 2’s & 5’s.
20 (2*2*5).
30
4 , Where 5 in the denominator is the prime factor of 1 Where 66 in the denominator is not the prime factor
66
of 2’s & 5’s.
25 (5*5).
Condition (Repeat)
The conditions for a rational numbers to repeat is:
 A fraction (in simplest form/lowest terms) repeats in its decimal
form, if the prime factors of the denominator are not 2’s and 5’s or
a product of prime factors, 2’s and 5’s.
 The decimal expansion of an irrational number does not repeat.
Some real numbers cannot be expressed by fractions. These
numbers are called irrational numbers. For example:
2 = 1.414213562

= 3.141592653
r = 0.10110111011110
Conjecture
Since p/q = p(1/q), it is sufficient to investigate the decimal
expansions of 1/q. Our conjecture or assumption is that the
decimal expansion of 1/q for enough positive integers can either
be terminates or repeats. As it has already been describe above
that a fraction (in simplest form/lowest terms) terminates in its
decimal form if the prime factors of the denominator are only 2’s
and 5’s or a product of primes factors, 2’s and 5’s. Otherwise it
repeats. The product of prime factors can also be expressed in
the equation of;
Q = P1n1*P2n2………..……………Pknk
Conjecture (Cont….)
1/2 = 0.5
Terminate
1/3 = 0. 3
Repeat
1/4 = 0.25
Terminate
1/5 = 0.20
Terminate
1/6 = 0.1 6
Repeat
1/7 = 0. 142857
1/8 = 0.125
Repeat
Terminate
1/9 = 0. 1
Repeat
1/10 = 0.1
Terminate
Conjecture ( Cont…..)
If we take 100 as a denominator it terminates
because 100 is the product of prime factors of
2*2*5*5. Similarly, if we take 66 it repeats because 66
is the product of prime factors of 2*3*11. Therefore
any number that is in the denominator and is the
combination of 2’s and 5’s is terminated. The
denominators of the first few unit fractions having
repeating decimals are 3, 6, 7, 9, 11, 12, 13, 14, 15,
17, 18, 19, 21, 22, 23, 24, 26, 27, 28, and 29.
That’s all folks