Number System

We have seen that inside a computer, data is stored in a format that cannot be easily read by human beings. This is why I/O interfaces are required. Every computer stored numbers, letters and other special characters in a coded form. Before going into the details of the codes we must understand the number system.
Number system is of two types

1. Positional
2. non-positional

Non positional number system

In early days, human beings counted on fingers were not adequate, stone s, pebbles or sticks were used to indicate values. This method used an additive approach or the non-positional number system. In this approach we have symbols such as I for 1, II for 2, IIIII for 5 etc. Each symbol represents the same value regardless of its position in the number and the symbols are simply added to find out the value of a particular number. Since it is very difficult to perform arithmetic with such a number system, positional number systems were developed.

Positional number systems

In positional number system, there are only a few symbols called digits, and these symbols represent different values depending on the position they occupy in the number. The value of each digit in such a number is determined by three considerations

1. the digit itself
2. the position of the digit in the number
3. The base of the number system (where base is defined as the total number of digits available in the number system).

The number system that we use in our daily life is called decimal number system. In this system, the base is equal to 10 because there are altogether 10 symbols 0,1,2,3,4,5,6,7,8,9 used in the system.

Binary number system

 

The binary number system is exactly like the decimal system except that the base is 2 instead of 10. We have only two symbols or digits 0 and 1 that can be used in this number system. Note that the largest single digit is 1 (one less than the base). Again, each position in a binary number represents a power of the base (2). Ass system, the rightmost position is the units (2⁰) position, the second position from the right is the 2's (2¹) position and proceeding in this way we have 4's (2²), 8's (2³) position, 16's (2⁴) position, and so on. Thus the decimal equivalent of binary number 10101 (written as 10101₂) is

(1 x 2⁴) + (0 x 2³) +(1 x 2²) +(0 x 2¹) +(1 x 2⁰) = 16 + 0 + 4 + 0 +1 =21

In order to be specific about which system we are referring to, it is common practice to indicate the base as a subscript. Thus we write:

10101₂= 21₁₀

Binary digit or bit

"Binary digit" is often referred to by the common abbreviation "bit”. Thus, a bit in computer terminology means either a 0 or a 1. A binary number consisting of n bit is called an n-bit number. Following Table lists all the 3-bit numbers of along with their decimal equivalent. It may be seen that a 3-nit number can have one of the 8 values ranging from 0 to 7. In fact it can be shown that any decimal number in the range of 0 to 2ⁿ - 1 can be represented in the binary form as an n-bit number.

Binary	Decimal
000	0
001	1
010	2
011	3
100	4
101	5
110	6
111	7

3-bit number with their decimal values

Every computer stores numbers, letters and other special characters in binary form. There are several occasions when computer professionals have to know the raw data contained in a computer’s memory. A common way of looking at the contents of a computer’s memory is to print out the memory contents on the line printer. This is called as memory dump. Because of the quantity of printout that would be required in a memory dump of binary digits and the lack of digit variety (0 and 1 only) two number system are used as short cut notation.

1. octal
2. hexa decimal

Octal number system

In the octal number system the base is 8. So in this system there are only eight symbols or digits:0,1,2,3,4,5,6,7 (8 and 9 do not exist in this system). Here the largest digit is 7. Each position in an octal number represents a power of the base (8). Thus the decimal equivalent of the octal number 2057 (written as 2057) is:

(2 x 8³) + (0 x 8²) +(5 x 8¹) +(7 x 8⁰) =1024 + 0 + 40 + 7 =1071

So we have,

2057₈= 10711₁₀

You can see that since there are only 8 digits in the octal number system, so 3 bits (23 = 8) are sufficient to represent any octal number in binary.

Hexadecimal number system

The hexadecimal number system is one with a base of 16. The base of 16 suggests choices of 16 symbols or character digits. The first 10 are the digits from 0 to 9 i.e. 0,1,2,3,4,5,6,7,8,9. The remaining six digits are denoted by A, B, C, D, E, F representing 10, 11, 12, 13, 14, 15 respectively. In the hexadecimal number system, therefore, the letter A has a decimal equivalent value of 10 and the hexadecimal F has a decimal equivalent value of 15. The largest single digit is F (15). Again, each position in a hexadecimal system represents a power of the base (16). Thus the decimal equivalent of the hexadecimal number 1AF (written as (1AF₁₆) is:

=(1 x 16²) + (A x 16¹) +(F x 16⁰)

= (1 x 256) + (10 x 16) +(15 x 1)

=256 + 160 + 15 =431

Thus, 1AF₁₆=4311₁₀

Since there are only 16 digits in the hexadecimal number system, so 4 bits (2⁴ =16) are sufficient to represent any hexadecimal number in binary.