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### Number Systems: Two Points of View

```
Date: 06/30/98 at 10:11:06
From: Liz
Subject: Number systems

What are the number systems? Our teacher just gave it to us for
research and I have no idea what it is.
```

```
Date: 06/30/98 at 12:10:13
From: Doctor Rob
Subject: Re: Number systems

A number system is a set of numbers and some operations that satisfy
certain rules.  One of the rules is that the operations on the numbers
must yield results that are back in the set.

An example of a number system is the natural numbers (positive
integers, 1, 2, 3, and so on) with addition and multiplication.
They satisfy laws called the Associative Law of Addition, the
Commutative Law of Addition, the Associative Law of Multiplication,
the Commutative Law of Multiplication, and the Distributive Law, and
there is an identity for multiplication (namely 1).

Different number systems satisfy different sets of rules. Here is
another number system. The set is the twelve numbers 1 through 12.
addtion, except if the sum is greater than 12, subtract 12 from the
sum to give the result.  Thus 4 + 3 = 7, so 4 "+" 3 = 7, too.
On the other hand, 9 + 11 = 20, but 20 > 12, so instead 9 "+" 11 = 8.

This is called Clock Arithmetic. Do you see why? This number system
satisfies the Associative Law, the Commutative Law, the Identity Law
(12 is the identity element!), and the Negative Law.

Notice that the first example had two operations, while the second had
only one.

There are many other number systems you are familiar with: the
integers (positive, negative, and zero), fractions, real numbers, and
so on. There are lots more you would not be familiar with, but will
encounter as you progress through more and more advanced mathematics:
complex numbers, groups, rings, fields, algebras, modules, vector
spaces, quaternions, surreal numbers, and so on. All fit in with the

- Doctor Rob, The Math Forum
http://mathforum.org/dr.math/
```

```
Date: 07/02/98 at 07:27:13
From: Doctor Margaret
Subject: Re: Number systems

Hello Liz,

Thanks for your question. Number systems are the different ways that
different civilizations use to write about and do mathematics.

One of the most simple number systems is the one your computer uses.
It is called a binary system and consists of two numbers, 0 for off
and 1 for on. This system has a base of 2. Our own number system has a
base of ten and is called the decimal system. There are many other
systems that were used by various cultures through history. The
Babylonians used a system with 12 as a base.

Another interesting part of number systems is to see how various
cultures wrote their numbers. The Egyptians used hieroglyphics, the
Sumerians used cuneiform writing and the Romans used letters like XXIX
which means 29.

I =  1
IX =  9 - because the "I" - which is equal to 1 - appears to the
X = 10   left of the X, which is 10; if "I" had appeared to the
XI = 11   right, it would equal 11.

I hope this short example gives you an idea about what's out there.

The best way to find out about different number systems is to research
the history of a culture. The ancient ones are the most fun. You might
also want to find out how a Chinese abacus works.  That is also part
of a number system.

Happy research!

- Doctor Margaret, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
Elementary Math History/Biography
High School Discrete Mathematics
High School History/Biography
High School Sets
Middle School History/Biography