Decimal Number System

Decimal Number System

When we write decimal (base 10) numbers, we use a positional system. Each digit is multiplied by an appropriate power of 10 depending on position in the number:

For example:

                     843 = (8*­_10­­2) + (4*₁₀1) + (3*₁₀0)

                            = 8*100 + 4*10 + 3*1

                            = 800 + 40 + 3

In the decimal number system, there are ten possible values that can appear in each digit position, and so there are ten numerals the familiar zero though nine (0,1,2,3,4,5,6,7,8,9). In a positional national system, the number base is called the radix. Thus, the base 10 system that we normally use has a radix of 10. The term radix and base can be use interchangeably. When writing numbers in a radix other then ten, or where the radix isn’t clear form the context, it is customary to specify the radix using a subscript. Thus, in a case where the radix isn’t understood, decimal numbers would be written like this:

127₁₀ 11₁₀ 5673₁₀

Generally, the radix will be understood form the context and specification is left off.

 Binary Number System

 The binary number is also a positional notation numbering system, but in this case, the base is not ten, but is instead two. Each digit position in a binary number represents a power of two. So, when write a binary number, each binary digit is multiplied by an appropriate power of 2 based on the position in the number.

For example:

                  101101 = 1*₂5 + 0*₂4 + 1*₂3 + 1*₂2 + 0*₂1 + 1*₂0

                               = 1*32 + 0*16 + 1*8 + 1*4 + 0*2 + 1*1

                               = 32 + 8 + 4 + 1

  In the binary number system, there are only two possible values that can appear in each digit position rather than the ten that can appear in a decimal number. Only the numerals 0 and 1 are used in binary numbers. The term ‘bit’ is a contraction of the words “binary” and “digit”, and when talking about binary numbers, it is often necessary to talk of the number of bits used to store or represent the number. This merely describes the number of binary digits that would be required to write the number. The number in the above examples is a 6 bit number.

The following are some additional example of binary numbers.

                      101101₂   11₂   10110₂

 Octal (base 8) Number System

The octal number system is used to represent binary numbers in a form which is more readable to human beings. Instead of using only two symbols and many digits, eight separate symbols 0 to 7 are used to represent groups of three binary digits. One difficulty with octal however is that most computers tend to use word lengths which are multiples of four rather than three bits. As a result data word cannot generally be expressed as an exact number of octal digits.

Hexadecimal (Base 16) Number System

 In the hexadecimal number system the number 0-9 are represented in their normal way, but numbers 10-15 are represent by the letters A-F respectively. Each hexadecimal digit is thus equivalent to four binary digits and a word of eight bits (byte) can be expressed exactly by to hex digits.