Numbers

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville VIC 3010 Australia
aram@unimelb.edu.au

and

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 13 July 2009

Numbers

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 ,

$\frac{a}{b}$, where $a$ is an integer and $b$ is an integer which is not equal to 0

GREAT for addition, subtraction, multiplication, and division,

BUT not so great for finding$\sqrt{2}$ =????, ... and so we discovered the real numbers

which are all of the possible decimal expansions.

Examples:

$\pi =\mathrm{3.1415926...},$

$e=\mathrm{2.71828...},$

$\sqrt{2}=1.414\dots ,$

$\frac{1}{3}=\mathrm{0.333...},$

$\frac{1}{8}=0.125=\mathrm{0.125000000000...},$

$10=\mathrm{10.0000...},$

GREAT for addition, subtraction, multiplication and division,

BUT not so great for finding$\sqrt{-9}=\mathrm{????},$

...and so we discovered the complex numbers,

$a+bi$, where $a$ and $b$ are real numbers and $i=\sqrt{-1}.$

Examples of complex numbers:

$3+4i,$

$7+9i,$

$3.2+6.7i$,

$5+0i=5$,

$0+10i=10i,$

$\pi +0i=\pi$,

$\frac{1}{3}+\frac{2}{6}i=\frac{1}{3}+\frac{1}{3}i,$

$\sqrt{7}+\sqrt{2}i.$

and ${\left(3i\right)}^2=3^2{i}^2=9i^2=-9$. So $\sqrt{-9}=3i$.

GREAT! We now have

• Addition

$(3+4i)+(7+9i)=10+13i.$

• Subtraction

$(3+4i)+(7+9i)=3-7+4i-9i=-4-5i.$

• Multiplication

• Division

• Square roots

• Graphing

• Factoring

$x^2+5=(x+\sqrt{5}i)(x-\sqrt{5}i),$

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 $(x-u_1)(x-u_2)(x-u_3)\dots (x-u_n)$, where $u_1,u_2,u_3,\dots ,u_n$ are some complex numbers.

References

[C08] W. Chen Fundamentals of Analysis, 100 pp. (web edition, 2008). Download from http://rutherglen.ics.mq.edu.au/wchen/lnfafolder/lnfa.html