Graphics Reference
In-Depth Information
2.2 Prime Numbers
A natural number that can be divided only by 1 and itself, without leaving
a remainder, is called a
prime number
. Examples are
.
There are 25 primes less than 100, 168 primes less than 1000 and 455 052 512
primes less than 10 000 000 000. The
fundamental theory of arithmetic
states,
'Any positive integer (other than 1) can be written as the product of prime
numbers in one and only one way.' For example, 25 = 5
{
2
,
3
,
5
,
7
,
11
,
13
,
17
}
13; 27 =
3
×
3
×
3; 28 = 2
×
2
×
7; 29 = 29; 30 = 2
×
3
×
5 and 92 365 = 5
×
7
×
7
×
13
×
29.
In 1742 Christian Goldbach conjectured that every even integer greater
than 2 could be written as the sum of two primes:
×
5; 26 = 2
×
4=2+2
14 = 11 + 3
18 = 11 + 7
,
etc
.
No one has ever found an exception to this conjecture, and no one has ever
proved it.
Although prime numbers are enigmatic and have taxed the brains of
the greatest mathematicians, unfortunately they play no part in computer
graphics!
2.3 Integers
Integers
include negative numbers, as follows:
{
...
−
3
,
−
2
,
−
1
,
0
,
1
,
2
,
3
,
4
,...
}
.
2.4 Rational Numbers
Rational
or
fractional
nu
mb
ers are numbers that can be represented as a
fraction. For example, 2
,
√
16
,
0
.
25 are rational numbers because
2=
4
2
,
√
16 = 4 =
8
2
,
0
.
25 =
1
4
Some rational numbers can be stored accurately inside a computer,
but many others can only be stored approximately. For example, 4
/
3=
1
.
333 333
...
produces an infinite sequence of threes and has to be truncated
when stored as a binary number.
2.5 Irrational Numbers
Irrational numbers
cannot be represented as fractions. Examples are
√
2=
1
.
414 213 562
...,π
=3
.
141 592 65
...
and e = 2
.
718 281 828
...
Such numbers
