Decimal fraction
Decimal, also called denary, is the base 10 numeral system, which uses symbols 0-9 (called digits).
Decimal is the predominant numeral system used by humans, though some cultures do or did use other number systems. This is almost certainly because humans have ten fingers; digit is also the anatomical term referring to fingers and toes. The system is called Arabic numerals by Europeans and Hindi numerals by Arabs, after the people they got it from.
Computers use other number systems and so, sometimes, do people using computers.
Binary, octal and hexadecimal are common, because they are easy to implement on a computer.
Humans' use of decimal gave rise to binary-coded decimal in the context of computers.
Fractions
A decimal fraction is a fraction where the denominator is a power of ten.
Decimal fractions usually expressed without a denominator, the decimal point being inserted into the numerator at a position corresponsing to the power of ten of the denominator. E.g. 8/10, 833/100, 83/1000, 8/10000 and 80/10000 are expressed thus: 0.8, 8.33, 0.083, 0.0008 & 0.008.
Note it is usual, for a decimal number which is less than one, to have a leading zero, and that trailling zeroes after the decimal point are not necessary although, in science, engineering and statistics, they can be retained to show a level of confidence in the accuracy of the number: 0.080 showing an error of 1 part in a thousand, and 0,08 showing an error of 1 in a 100.
Ten is the product of the first and third prime number and is one greater than the square of the second prime number. This leads to plenty of simple decimal fractions:
- 1/2 = 0.5
- 1/3 = 0.3333 recurring
- 1/4 = 0.25
- 1/5 = 0.2
- 1/6 = 0.16666 recurring
- 1/8 = 0.125
- 1/9 = 0.1111 recurring
Prime factors other than 3 may give longer repeating sequences, see for instance 7, 11, 13.
Decimal representation of the real numbers
Every real number has a decimal representation, with certain numbers having two. For rational numbers which can be written as p/(2a5b) (i.e. the only prime factors in denominator are 2 and 5), there is a terminating decimal representation. For instance 1/1=1, -1/2=-0.5, 3/5=0.6, 3/25=0.12 and 1306/1250=1.0448. Such numbers are the only real numbers which don't have a unique decimal representation, as they can also be written as a representation that has a recurring 9. For instance 1=0.99999..., -1/2=-0.499999..., etc.
Rational numbers p/q with prime factors in the denominator other than 2 and 5 (when reduced to simplest terms) have a unique recurring (or repeating) decimal representation. This can be seen to be a consequence of the long division algorithm, in that there are only (q-1) possible nonzero remainders on division by q, so that the recurring pattern will have a period less than q-1. For instance to find 3/7 by long division:
.4 2 8 5 7 1 4 ...
--------------------
7 ) 3.0 0 0 0 0 0 0 0
2 8 30/7 = 4 r 2
-----
2 0
1 4 20/7 = 2 r 6
-----
6 0
5 6 60/7 = 8 r 4
-----
4 0
3 5 40/7 = 5 r 5
-----
5 0
4 9 50/7 = 7 r 1
-----
1 0
7 10/7 = 1 r 3
-----
3 0
2 8 30/7 = 4 r 2 (again)
-----
2 0
etc
The converse to this observation is that every recurring decimal represents a rational number p/q. This is a consequence of the fact the recurring part of a decimal representation is, in fact, an infinite geometric series which will sum to a rational number. For instance,
This leaves the irrational numbers. They also have unique decimal representation, and can be characterised as the numbers whose decimal representations neither terminate nor recur.
Naturally, the same trichotomy holds for other base-n numeral systems:
- Terminating representation: rational where the denominator divides some nk
- Recurring representation: other rational
- Non-terminating, non-recurring representation: irrational
and a version of this even holds for irrational-base numeration systems, such as golden mean base representation.
See also
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