### Number Systems in Computer Architecture :

There are four number systems of possible interest to the computer programmer: decimal, binary, octal, and hexadecimal.  Each system is characterized by its base or radix, always given in decimal, and the set of permissible digits.

1. Decimal Number System :

The Decimal system is composed of 10 symbols or numbers. These 10 symbols are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Using these symbols as digits of a number,we can express any quantity. In decimal system, the base or radix is 10 as it has 10 digits. Even though the decimal system has only 10 symbols, any number of any magnitude can be expressed by using the system of positional weighting. For example, in a decimal system the number (245)10 signifies that :

• The subscript 10 represents that the number in the bracket is decimal number.
• The left most number "2" is the Most Significant Digit.
• The right most number "5" is the Least Significant Digit.

The weight of each number is calculated by multiplying the individual number with the positional weight of that number. The positional weight of a number is calculated by giving the weight to an individual number moving from right to left as unit, Ten, Hundred, Thousand and so on.

245 = 2x102 + 4x101 + 5x100

Similarly the weightage for the decimal number having fractional part can be calculated is same manner. The number positioned next to the decimal point (.) is having the weightage as 10-1, 10-2,  10-3 and so on. The below example shows the enumeration for the number  (4235.234)10

4235.234 =  2x103 + 2x102 + 3x101 + 5x100 .  2x10-1 + 3x10-2 + 4x10-3

2. Binary Number System :

In the Binary number system, there are only two digits or symbols i.e. 0 and 1. As binary number systems has two digits or symbols its base or radix is 2. With the combination of these two binary numbers, these bits can be used to represent decimal number of any magnitude.

In the binary number system each bit has the weightage depending upon its position. The right most bit (least significant bit) has the weightage of 1, next bit will have the weightage double of the first bit i.e. 2 like wise the weightage of the bits goes on getting doubled from the first one while moving towards the left side and finally all weightages are added up which represents decimal equivalent of that binary number. The below table shows the counting sequence of four bit binary number from 0000 to 1111 along with weightage of individual bit and its decimal equivalent.

3. Octal Number System :

The Octal number system has a base or radix of eight, meaning that it has eight possible digits as: 0, 1, 2, 3,4, 5, 6 and 7. Octal Numbers are represented by three bits in binary numbers because the highest digit in octal is 7 which can be represented by minimum of three digits as (111)2.

The hexadecimal number has the base or radix of 16. Thus, it has 16 possible digit symbols from 0 through 9 plus the letters A, B, C, D, E and F as the 16 digit symbols. Hexadecimal numbers are represented by four binary digits because the highest hexadecimal digit is F which is represented by four bit binary number as (1111)2.

The below table shows the comparative count values among all the number system :

Memory structure in Computer :

Bit : A bit is the smallest unit of data in a computer. BIT is short for Binary Digit. A bit has a single binary value, either 0 or 1.

Nibble : A group of 4 bits is called nibble.

Byte :  A byte is a unit of data which represents eight binary digits long. A byte is the smallest unit, which can represent a data item or a character.

1 byte = 8 bits
28 = 256

Word : A word is a group of fixed number of bits processed as a unit. The size of a word varies from one computer to another, depending on the CPU. For computers with a 16-bit CPU, a word is 16 bits (2 bytes). In modern systems, a word can be 32 bits (4 bytes) or as long as 64 bits (8 bytes).

1 word = 4 bytes
232 = 4294967296

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