1. Decimal Expansion of Rational Numbers: An Analysis
Project 1.2
Decimal Expansion of Rational Numbers:
An Analysis

2. Rational to Decimal
When a rational number is converted to decimal, it is represented in the form p/q.
In this form p/q = p(1/q).
To determine which decimals will repeat or terminate, we need only investigate 1/q.

3. Determining those that Repeat
When q is in the form of 2x5y and x and y are positive integers then 1/q will terminate .
For all other values of q, 1/q will repeat.
Values for which 1/q will terminate or repeat
1/1
Term.
1/11
Repeat
1/21
1/31
1/41
1/2
1/12
1/22
1/32
1/42
1/3
1/13
1/23
1/33
1/43
1/4
1/14
1/24
1/34
1/44
1/5
1/15
1/25
1/35
1/45
1/6
1/16
1/26
1/36
1/46
1/7
1/17
1/27
1/37
1/47
1/8
1/18
1/28
1/38
1/48
1/9
1/19
1/29
1/39
1/49
1/10
1/20
1/30
1/40
1/50

4. How to convert Repeating Decimal to Rational
A terminating decimal can be easily displayed as a rational expression by multiplying the decimal by the inverse of the place value the end of the decimal occupies.
A repeating decimal takes a little more work, but the place value still plays a vital role.

5. How to convert Repeating Decimal to Rational (continued)
1000r – r = 3132
Eg. 999r = 3132 and r = 3132/999 = 116/37
By multiplying the decimal by the place value inverse of the repeating point and manipulating the data as shown, it is possible to obtain a rational representation of the repeating decimal.