Chapter1: Number Systems Prepared by: Department of Preparatory year Edited by: Lec. GHADER R. KURDi

Objectives Understand the concept of number systems. Describe the decimal, binary, hexadecimal and octal system. Convert a number in binary, octal or hexadecimal to a number in the decimal system. Convert a number in the decimal system to a number in binary, octal and hexadecimal. Convert a number in binary to octal, hexadecimal and vice versa.

Introduction Data is a combination of Numbers, Characters and special characters. The data or Information should be in the form machine readable and understandable. Data has to be represented in the form of electronic pulses. The data has to be converted into electronic pulses and each pulse should be identified with a code.

Introduction Every character, special character and keystrokes have numerical equivalent (ASCII code). Thus using this equivalent, the data can be interchanged into numeric format. For this numeric conversions we use number systems. Each number system has radix or Base number , Which indicates the number of digit in that number system.

The decimal system Base (Radix): 10 Decimal digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 Examples:

The binary system Base (Radix): 2 Decimal digits: 0 (absence of pulses) , 1 (presence of pulse) Examples:

The octal system Base (Radix): 8 Decimal digits: 0, 1, 2, 3, 4, 5, 6, 7 Examples:

The hexadecimal system Base (Radix): 16 Hexadecimal digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A-10, B-11, C-12, D-13, E-14, F-15 Examples:

Quantities/Counting (1 of 3) Decimal Binary Octal Hexa- decimal 1 2 10 3 11 4 100 5 101 6 110 7 111

Quantities/Counting (2 of 3) Decimal Binary Octal Hexa- decimal 8 1000 10 9 1001 11 1010 12 A 1011 13 B 1100 14 C 1101 15 D 1110 16 E 1111 17 F

Quantities/Counting (3 of 3) Decimal Binary Octal Hexa- decimal 16 10000 20 10 17 10001 21 11 18 10010 22 12 19 10011 23 13 10100 24 14 10101 25 15 10110 26 10111 27 Etc.

Exercise – true or false... Number Decimal Binary Octal Hexa- decimal 100001 33 1110108 1AF 11021 67S

Exercise – true or false... Number Decimal Binary Octal Hexa- decimal 100001 √ 33 × 1110108 1AF 11021 67S

Summary of the Number Systems Base Symbols Used by humans? Used in computers? Decimal 10 0, 1, … 9 Yes No Binary 2 0, 1 Octal 8 0, 1, … 7 Hexa- decimal 16 0, 1, … 9, A, B, … F

Conversion Among Bases The possibilities: Decimal Octal Binary Hexadecimal

Quick Example 2510 = 110012 = 318 = 1916 Base (radix)

Decimal to Decimal (just for fun) Octal Binary Hexadecimal Next slide…

Decimal to Decimal Technique: Multiply each bit by 10n, where n is the “weight” of the bit The weight is the position of the bit, starting from 0 on the right Position 6th 5th 4th 3rd 2nd 1st Weight 105 104 103 102 101 100 Add the results

Weight 12510 => 5 x 100 = 5 + 2 x 101 = 20 + 1 x 102 = 100 125 Base

Binary to Decimal Decimal Octal Binary Hexadecimal

Binary to Decimal Technique: Multiply each bit by 2n, where n is the “weight” of the bit The weight is the position of the bit, starting from 0 on the right Position 6th 5th 4th 3rd 2nd 1st Weight 25 24 23 22 21 20 Add the results

Example Bit “0” 1010112 => 1 x 20 = 1 + 1 x 21 = 2 + 0 x 22 = 0 + 1 x 23 = 8 + 0 x 24 = 0 + 1 x 25 = 32 4310

Octal to Decimal Decimal Octal Binary Hexadecimal

Octal to Decimal Technique Multiply each bit by 8n, where n is the “weight” of the bit The weight is the position of the bit, starting from 0 on the right Position 6th 5th 4th 3rd 2nd 1st Weight 85 84 83 82 81 80 Add the results

Example 7248 => 4 x 80 = 4 x 1 = 4 + 2 x 81 = 2 x 8 = 16 + 46810

Hexadecimal to Decimal Octal Binary Hexadecimal

Hexadecimal to Decimal Technique Multiply each bit by 16n, where n is the “weight” of the bit The weight is the position of the bit, starting from 0 on the right Position 6th 5th 4th 3rd 2nd 1st Weight 165 164 163 162 161 160 Add the results

Example ABC16 => C x 160 = 12 x 1 = 12 + B x 161 = 11 x 16 = 176 + A x 162 = 10 x 256 = 2560 274810

Decimal to Binary Decimal Octal Binary Hexadecimal

Decimal to Binary Technique Divide by two (the base), keep track of the remainder until the value is 0 First remainder is bit 0 (LSB, least-significant bit) Second remainder is bit 1 Etc.

Example 2 125 62 1 12510 = ?2 2 31 0 2 15 1 2 7 1 2 3 1 2 1 1 2 0 1 12510 = 11111012

Decimal to Octal Decimal Octal Binary Hexadecimal

Decimal to Octal Technique Divide by 8 Keep track of the remainder

Example 123410 = ?8 8 1234 154 2 8 19 2 8 2 3 8 0 2 123410 = 23228

Decimal to Hexadecimal Octal Binary Hexadecimal

Decimal to Hexadecimal Technique Divide by 16 Keep track of the remainder

Example 123410 = ?16 16 1234 77 2 16 4 13 = D 0 4 123410 = 4D216

Octal to Binary Decimal Octal Binary Hexadecimal

Octal to Binary Technique Convert each octal digit to a 3-bit equivalent binary representation Note: The octal system has a concept of representing three binary digits as one octal numbers, thereby reducing the number of digits of binary number still maintaining the concept of binary system.

Example 7058 = ?2 7 0 5 111 000 101 7058 = 1110001012

Hexadecimal to Binary Decimal Octal Binary Hexadecimal

Hexadecimal to Binary Technique Convert each hexadecimal digit to a 4-bit equivalent binary representation Note: the hexadecimal system has a concept of representing four binary digits as one hexadecimal number.

Example 10AF16 = ?2 1 0 A F 0001 0000 1010 1111 10AF16 = 00010000101011112

Binary to Octal Decimal Octal Binary Hexadecimal

Binary to Octal Technique Group the digits of the given number into 3's starting from the right if the number of bits is not evenly divisible by 3 then add 0's at the most significant end Write down one octal digit for each group

Example 10110101112 = ?8 1 011 010 111 1 3 2 7 10110101112 = 13278

Binary to Hexadecimal Decimal Octal Binary Hexadecimal

Binary to Hexadecimal Technique Group bits in fours, starting on right Convert to hexadecimal digits

Example 10101110112 = ?16 10 1011 1011 B B 10101110112 = 2BB16

Octal to Hexadecimal Decimal Octal Binary Hexadecimal

Octal to Hexadecimal Technique Octal Binary Hex Use binary as an intermediary Octal Binary Hex

Example 10768 = ?16 1 0 7 6 001 000 111 110 2 3 E 10768 = 23E16

Hexadecimal to Octal Decimal Octal Binary Hexadecimal

Hexadecimal to Octal Technique Hex Binary Octal Use binary as an intermediary Hex Binary Octal

Example 1F0C16 = ?8 1 F 0 C 0001 1111 0000 1100 1 7 4 1 4 1F0C16 = 174148

Exercise – Convert ... Decimal Binary Octal Hexa- decimal 33 1110101 703 1AF Don’t use a calculator!

Binary fractions The decimal value is calculated in the same way as for non-fractional numbers, the exponents are now negative

Example Bit “0” 101.1102 => 1 x 22 = 4 + 0 x 21 = 0 + 1 x 20 = 1 + 1 x 2-1 = .5 + 1 x 2-2 = .25 + 0 x 2-3 = 0 + 5.7510

Summary of Conversion Methods Conversion Among Bases Conversion from the binary system to octal, hexadecimal and vice versa. Conversion from the decimal system to binary, octal or hexadecimal Conversion from binary, octal or hexadecimal to the decimal system

Chapter2: Arithmetic operations in Binary System

Introduction The basic arithmetic operations which can be performed rising binary number system are: Addition Subtraction Multiplication Division

Binary Addition A B A + B 1 10 “two”

Binary Addition Two n-bit values Add individual bits Propagate carries E.g., 1 1 1 1 1 1 10101 21 + 11001 + 25 101110 46 1101 13 + 1111 + 15 11100 28

Binary Subtraction A B A - B 1 With one barrow

Binary Subtraction Two n-bit values Subtract individual bits If you must subtract a one from a zero, you need to “borrow” from the left E.g., 111101 61 -100101 - 37 011000 24 101010 42 - 11001 - 25 10001 17

Binary Multiplication A  B 1

Multiplication Binary, two n-bit values As with decimal values E.g., 1011 x 11 1011 1011 100001 11 x 3 33 1110 x 1011 1110 1110 0000 1110 10011010

Binary Division A B A / B Not acceptable 1

Division Binary, two n-bit values As with decimal values E.g., 010 10 100 10 000 00 00 00101 101 11001 101 00101 101 000

Exercises 1011 + 1101 11001010 – 10011011 1000 * 1001 1001010 / 1000