Although number systems are in everyday use in millions of different ways, we never really stop to consider the fundamental nature of what it is we are actually doing when we perform simple arithmetic. In computing and control applications it is essential to understand number system in order to relate to the computing operations being performed. Numbers are the natural
root language of all computers.
A number system defines a set of values used to represent quantity. We talk about the number of people attending class, the number of modules taken per student, and also use numbers to represent grades achieved by students in tests. Quantifying values and items in relation to each other is helpful for us to make sense of our environment. We do this at an early age; figuring out if we have more toys to play with, more presents, more sweets and so on. The study of number
systems is not just limited to computers. We apply numbers every day, and knowing how numbers work will give us an insight into how a computer manipulates and stores numbers. Mankind through the ages has used signs or symbols to represent numbers. The early forms were straight lines or groups of lines, much like as depicted in the film Robinson Crusoe, where a group of six vertical lines with a diagonal line across represented one week. Its difficult representing large or very small
numbers using such a graphical approach. As early as 3400BC in Egypt and 3000BC in Mesopotamia, they developed a symbol to represent the unit 10. This was a major advance, because it reduced the number of symbols required. For instance, 12 could be represented as a 10 and two units (three symbols instead of 12 that was required previously). The Romans devised a number system which could represent all the numbers from 1 to 1,000,000 using only seven symbols
I = 1 ,V = 5 ,X = 10 ,L = 50 ,C = 100 ,D = 500 ,M = 1000
A small bar placed above a symbol indicates the number is multiplied by 1000. The number system in most common use today is the Arabic system. It was first developed by the Hindus and was used as early as the 3rd century BC. The introduction of the symbol 0, used to indicate the positional value of digits was very important. We thus became familiar with the concept of groups of units, tens of units, hundreds of units, thousands of units and so on. In number systems, its often
helpful to think of recurring sets, where a set of values is repeated over and over again.
Considering the decimal number system, it has a set of values which range from 0 to 9. This basic set is repeated over and over, creating large numbers. Note how the set of values 0 to 9 is repeated, and for each repeat, the column to the left is incremented (from 0 to 1, then 2). Each increase in value occurs, till the value of the largest number in the set is reached (9), at which stage the next value is the smallest in the set (0) and a new value is
generated in the left column (ie, the next value after 9 is 10).
09, 10 - 19, 20 - 29, 30 -39 etc
We always write the digit with the largest value on the left of the number
The base value of a number system is the number of different values the set has before repeating itself. For example, decimal has a base of ten values, 0 to 9.
Binary = 2 (0, 1)
Octal = 8 (0 - 7)
Decimal = 10 (0 - 9)
Duodecimal = 12 (used for some purposes by the Romans)
Hexadecimal = 16 (0 - 9, A-F)
Vigesimal = 20 (used by the Mayans)
Sexagesimal = 60 (used by the Babylonians)
The weighting factor is the multiplier value applied to each column position of the number. For instance, decimal has a weighting factor of TEN, in that each column to the left indicates a multiplication value increase of 10 over the previous column on the right, ie; each column move to the left increases in a multiply factor of 10.
the number 321 becomes
1 * 100 = 1 * 1 = 1
2 * 101 = 2 * 10 = 20
3 * 102 = 3 * 100 = 300
adding these gives 321