The representation of numbers
Generally, a number N(b) in a fixed, positive, integral base b is represented in positional notation following the rule: the positional digits ai, from a number, are integers, such that they satisfies the constraint 0≤ai≤(b-1) for i=... -2,-1,0,1,2, ... n. The value of each position in a number is known as its position coefficient or weight. For example:
346(10)=3*102+4*101+6*100=300+40+6=346(10) or the year 1999(10)
A simple decimal weighting table is
... 103 102 101 100 . 10-1 10-2 10-3 ...
In general, in a base b system the successive digit positions, from left to right, have the weights
... b3 b2 b1 b0 . b-1 b-2 b-3 ...
↑ radix position
The symbol "." is called radix point.
The portion of the number to the right of the radix is called the fraction part of the number, and the portion to the left is called the integral part. In the decimal numbering system, the radix point is called the decimal point.
Integer. Generally an integer is represented as [±]n,
where n∈N. Given a base b
and an integer number N.
Definition. A sequence of digits anan-1...a1a0 is said to be a representation of the number N in the base b (in b number system) if and only if it satisfies the properties:
- the digits an,an-1, ..., a1,a0 are natural numbers and satisfies the
constraint:
0<=ai<= (b-1), i=0,1,2, ..., n
- we have the equality:
N=an*bn+an-1*bn-1+ ... + a0*b0.
For a given number N we have a representation, and only one, in a base b.
Real. Generally a real number is represented as [±]n.m, where n∈N and m∈N. Given a real number R and a base b.
Definition. A sequence of digits anan-1...a1a0a-1a-2...a-m is said to be a representation in the base b if and only if:
- the digits an,an-1, ..., a1,a0,a-1,a-2, ..., a-m are natural numbers and satisfy the property:
0<=ai<= b-1, i=-m, ...,-1,0,1,2, ..., n
- we don't have a range j such that:
aj=aj-1-aj-2=b-1
- we have the equality:
N=an*bn+an-1*bn-1+... + a0*b0 +a-1*b-1+a-2*b-2 +... +a-m*b-m.
If the real number R satisfies the proprieties just outlined, by definition we write (express) this number as:
anan-1 ... a1a0 v a-1a-2 ... a-m
where v is the decimal point (, or .). In this sequence an is said to be the most significant digit and a-m is said to be the most unsignificant digit.
For a given number R we have a representation, and only one, in a base b, that can be represented in one of the following formats:
1)
anan-1an-2…a0a-1a-2…a-m
2) an*bn+an-1*bn-1+... + a0*b0 +a-1*b-1+a-2*b-2 +... +a-m*b-m
3)
