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I The Natural Numbers
The natural (or counting) numbers are 1,2,3,4,5,
1
,
2
,
3
,
4
,
5
,
etc. There are infinitely many natural numbers. The set of natural numbers, {1,2,3,4,5,...}
{
1
,
2
,
3
,
4
,
5
,
...
}
, is sometimes written N
N
for short.
The whole numbers are the natural numbers together with 0
0
.
(Note: a few textbooks disagree and say the natural numbers include 0
0
.)
The sum of any two natural numbers is also a natural number (for example, 4+2000=2004
4
+
2000
=
2004
), and the product of any two natural numbers is a natural number (The real numbers is the set of numbers containing all of the rational numbers and all of the irrational numbers. The real numbers are “all the numbers” on the number line. There are infinitely many real numbers just as there are infinitely many numbers in each of the other sets of numbers. But, it can be proved that the infinity of the real numbers is a bigger infinity.
The "smaller", or countable infinity of the integers and rationals is sometimes called ℵ0
ℵ
0
(alef-naught), and the uncountable infinity of the reals is called ℵ1
ℵ
1
(alef-one).
There are even "bigger" infinities, but you should take a set theory class for that!
The Complex Numbers The complex numbers are the set {a+bi
a
+
b
i
| a
a
and b
b
are real numbers}, where i
i
is the imaginary unit, −1‾‾‾√
−
1
. (click here for more on imaginary numbers and operations with complex numbers).
The complex numbers include the set of real numbers. The real numbers, in the complex system, are written in the form a+0i=a
a
+
0
i
=
a
. a real number.
This set is sometimes written as C
C
for short. The set of complex numbers is important because for any polynomial p(x)
p
(
x
)
with real number coefficients, all the solutions of p(x)=0
p
(
x
)
=
0
will be in C
C
.
Beyond...
There are even "bigger" sets of numbers used by mathematicians. The quaternions, discovered by William H. Hamilton in 1845
1845
, form a number system with three different imaginary units!
Is this ok?