Slide 1: THE NUMBER SYSTEMS
INTRODUCTION: INTRODUCTION Every computer stores numbers, letters and special characters in coded form. ASCII BINARY DECIMAL OCTAL HEXADECIMAL FRACTIONAL CONVERSIONS
ASCII: ASCII American Standard Code for Information Interchange ASCII is a code for representing English characters as numbers, with each letter assigned a number from 0 to 127. For example, the ASCII code for uppercase M is 77. Most computers use ASCII codes to represent text, which makes it possible to transfer data from one computer to another. Text files stored in ASCII format are sometimes called ASCII files. Text editors and word processors are usually capable of storing data in ASCII format . Numeric data files or Executable programs are never stored in ASCII format.
DECIMAL System: DECIMAL System Decimal system is a way of writing numbers. Any number, from huge quantities to tiny fractions, can be written in the decimal system using only the ten basic symbols 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0. The value of any of these symbols dep ends on the place it occupies in the number. Example : 2586 10 = (2X1000)+(5X100)+(8X10)+(6X1) or 2X10 3 + 5X10 2 + 8X10 1 + 6X10 0
BINARY System: BINARY System A method of representing numbers in which only the digits 0 and 1 are used. Successive units are powers of 2 . The first ten numbers in binary notation, corresponding to the numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 in decimal notation, are 0, 1, 10, 11, 100, 101, 110, 111, 1000, and 1001. The decimal equivalent of a binary number can be calculated by adding together each digit multiplied by its power of 2; for example, the binary number 1011010 corresponds to (1 × 2 6 ) + (0 × 2 5 ) + (1 × 2 4 ) + (1 × 2 3 ) + (0 × 2 2 ) + (1 × 2 1 ) + (0 × 2 0 ) = 64 + 0 + 16 + 8 + 0 + 2 + 0 = 90 in the decimal system.
OCTAL System: OCTAL System The octal, or base 8, number system is a common system used with computers. Because of its relationship with the binary system. It is useful in programming some types of computers. One octal digit is the equivalent value of three binary digits. Example : 562 8 = 5X8 2 + 6X8 1 + 2X8 0 = 320 + 48 + 2 = 370 10
HEXADECIMAL System: HEXADECIMAL System It is number system having a base 16; the symbols for the numbers 0--9 are the same as those used in the decimal system, and the numbers 10--15 are usually represented by the letters A--F. The system is used as a convenient way of representing the internal binary code of a computer. Example : 2A3B 16 = 2x16 3 + A(10)x16 2 + 3x16 1 + B(11)X16 0 = 8192 + 2560 + 48+ 11 = 10811 10
FRACTIONAL System: FRACTIONAL System The decimal system uses powers of 10 to determine the value of a position. The binary system uses powers of 2 to determine the value of a position. All numbers or values to the left of the radix point are whole numbers, and all numbers to the right of the radix point are fractional numbers. Example : Binary no. 101.1 2 = 1x2 2 + 0x2 1 + 1x2 0 + 1x2 -1 = 4 + 0 + 1 +.5 =5.5 10
CONVERSIONS: CONVERSIONS Decimal to Binary Number We will take the number 29 and divide it by 2 until the answer reaches 0 and where there is a remainder we write one (1) where there is no remainder we write zero (0). 4706 10 =100101100010 2 29 10 = 11101 2
CONVERSIONS: CONVERSIONS Decimal to Octal Number 2502 10 = 4706 8
CONVERSIONS: CONVERSIONS Decimal to Hexadecimal Number 428 10 = 1AC 16
CONVERSIONS: CONVERSIONS Binary to Decimal Number 11001 2 = 1x2 4 + 1x2 3 + 0x2 2 + 0x2 1 + 1x2 0 = 16 + 8 + 0 + 0 + 1 = 251 10 10101010 2 = 1x2 7 + 0x2 6 + 1x2 5 + 0x2 4 + 1x2 3 + 0x2 2 + 1x2 1 + 0x2 0 = 128 + 0 + 32 + 0+ 8 + 0 + 2 + 0 = 170 10
CONVERSIONS: CONVERSIONS Octal and Hexadecimal to Decimal Number Octal : 562 8 = 5x8 2 + 6x8 1 + 2x8 0 = 320 + 48 + 2 = 370 10 Hexadecimal : 2A3B 16 = 2x16 3 + A(10)x16 2 + 3x16 1 + B(11)X16 0 = 8192 + 2560 + 48+ 11 = 10811 10
CONVERSIONS: CONVERSIONS Fractional to Decimal Number Binary to Decimal: 110.101 2 = 1x2 2 + 1x2 1 + 0x2 0 + 1x2 -1 + 0x2 -2 + 1x2 -3 = 4 + 2 + 0 + .5 + 0 + .125 =6.625 10 Octal to Decimal: 127.548 8 = 1x8 2 + 2x8 1 + 7x8 0 + 5x8 -1 + 4x8 -2 + 8x8 -3 = 64 + 16 + 7 + 5/8 + 4/64 = 87 + 0.625 + 0.0625 = 87.6875 10 Hexadecimal to Decimal : 2B.C4 16 = 2x16 1 + B(11)x16 0 + C(12)x16 -1 + 4x16 -2 = 32 + 11 + 12/16 + 4/256 = 43 + 0.75 + 0.015625 = 43.765652 10
Thank You!!!: Thank You!!! By : Pulkit Jain