Last updates: 13 July 2009
At some point humankind wanted to count things and so we discovered the positive integers,
0, 1, 2, 3, 4, 5, 6, ...
GREAT for counting something,
BUT what if you don't have anything? ... How do we talk about nothing, null, zilch?
... and so we discovered the nonnegative integers,
0, 1, 2, 3, 4, 5, 6, ...
GREAT for adding
5 + 3 = 8, 0 + 10 = 10, 21 + 37 = 48,
BUT not so great for subtraction,
5 − 3 = 2, 2 − 0 = 2, 12 − 34 =???.
...and so we discovered the integers ...,
−3, −2, −1, 0, 1, 2, 3, ...
GREAT for adding subtracting and multiplying,
3 · 6 = 18, −3 · 2 = −6, 0 · 7 = 0,
BUT not so great if you only want part of the sausage, ...and so we discovered the rational numbers ,
, where is an integer and is an integer which is not equal to 0
GREAT for addition, subtraction, multiplication, and division,
BUT not so great for finding =????, ... and so we discovered the real numbers
which are all of the possible decimal expansions.
GREAT for addition, subtraction, multiplication and division,
BUT not so great for finding
...and so we discovered the complex numbers,
, where and are real numbers and
,
,
,
and . So .
GREAT! We now have
Finally we have The Fundamental Theorem of Algebra, which is one reason why the complex number system is “the right” number system to use. It says that any polynomial can be factored completely as , where are some complex numbers.
[C08] W. Chen Fundamentals of Analysis, 100 pp. (web edition, 2008). Download from http://rutherglen.ics.mq.edu.au/wchen/lnfafolder/lnfa.html