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.  
The operation is "addition."  "Addition" is the same as ordinary 
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 
comments made above about sets, operations, and rules.

- 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/