Numbering Systems

Introduction to Numbering Systems Decimal System We are all familiar with the decimal number system (Base 10). Some other number systems that we will work with are: Binary  Base 2النظام الثنائي Octal  Base 8النظام الثماني Hexadecimal  Base 16النظام السداسي عشر

Characteristics of Numbering Systems The digits are consecutive.الارقام متسلسلة The number of digits is equal to the size of the base. عدد الأرقام = الأساس Zero is always the first digit.الصفر هو دائما اول رقم The base number is never a digit.الأساس ليس ضمن العناصر When 1 is added to the largest digit, a sum of zero and a carry of one results .عند اضافة واحد لأكبر رقم نحصل علي صفر ونحمل واحد Numeric values are determined by the implicit positional values of the digits.تعتمد قيمة العدد على موضع الرقم

Most significant digit Least significant digit Significant Digits Binary: 11101101 Most significant digit Least significant digit Hexadecimal: 1D63A7A

Binary System

Binary Number System Also called the “Base 2 system” The binary number system is used to model the series of electrical signals computers use to represent information 0 represents the no voltage or an off state 1 represents the presence of voltage or an on state

Binary Decimal

Decimal to Binary Conversion The easiest way to convert a decimal number to its binary equivalent is to use the Division Algorithm This method repeatedly divides a decimal number by 2 and records the quotient and remainder  The remainder digits (a sequence of zeros and ones) form the binary equivalent in least significant to most significant digit sequence

An algorithm for finding the binary representation of a positive integer

Division Algorithm Convert 67 to its binary equivalent: 6710 = x2 Step 1: 67 / 2 = 33 R 1 Divide 67 by 2. Record quotient in next row Step 2: 33 / 2 = 16 R 1 Again divide by 2; record quotient in next row Step 3: 16 / 2 = 8 R 0 Repeat again Step 4: 8 / 2 = 4 R 0 Repeat again Step 5: 4 / 2 = 2 R 0 Repeat again Step 6: 2 / 2 = 1 R 0 Repeat again Step 7: 1 / 2 = 0 R 1 STOP when quotient equals 0 1 0 0 0 0 1 12

Binary to Decimal Conversion The easiest method for converting a binary number to its decimal equivalent is to use the Multiplication Algorithm Multiply the binary digits by increasing powers of two, starting from the right Then, to find the decimal number equivalent, sum those products

Multiplication Algorithm Convert (10101101)2 to its decimal equivalent: Binary 1 0 1 0 1 1 0 1 Positional Values x x x x x x x x 27 26 25 24 23 22 21 20 Products 128 + 32 + 8 + 4 + 1 17310

BINARY TO DECIMAL CONVERTION Any binary number can be converted to its decimal equivalent simply by summing together the weights of the various positions in the binary number which contain 1. Example 1: convert 110112 to decimal value Solve: 1 1 1 1 = 16+8+2+1 =

Convert 101101012 to decimal value Example 2 : Convert 101101012 to decimal value Solve: 1 1 1 1 1 = 128 + 32 + 16 + 4 + 1 = You should noticed the method is find the weights (i.e., powers of 2) for each bit position that contains 1, and then to add them up.

DECIMAL TO BINARY CONVERTION Example : convert 2510 to binary Solve = 2510 = ?2 25 2 = 12 balance 1 LSB 12 2 = 6 balance 0 6 2 = 3 balance 0 3 2 = 1 balance 1 1 2 = 0 balance 1 MSB . . . Answer = 110012

Octal System

Octal Number System Also known as the Base 8 System Uses digits 0 - 7 Readily converts to binary Groups of three (binary) digits can be used to represent each octal digit Also uses multiplication and division algorithms for conversion to and from base 10

Octal Decimal

OCTAL TO DECIMAL CONVERTION Convert from octal to decimal by multiplying each octal digit by its positional weight. Example 1: Convert 1638 to decimal value Solve = = 1 x 64 + 6 x 8 + 1 x 1 = 11510 Example 2: Convert 3338 to decimal value Solve = = 3 x 64 + 3 x 8 + 3 x 1 = 21910

DECIMAL TO OCTAL CONVERTION Convert from decimal to octal by using the repeated division method used for decimal to binary conversion. Divide the decimal number by 8 The first remainder is the LSB and the last is the MSB. Example : convert 35910 to Decimal Value Solve = 35910 = ?8 359 8 = 44 balance 7 LSB 44 8 = 5 balance 4 5 8 = 0 balance 5 MSB . . . Answer = 5478

Octal Binary

OCTAL TO BINARY CONVERTION Convert from octal to binary by converting each octal digit to a three bit binary equivalent Octal digit 1 2 3 4 5 6 7 Binary Equivalent 000 001 010 011 100 101 110 111 Convert from binary to octal by grouping bits in threes starting with the LSB. Each group is then converted to the octal equivalent Leading zeros can be added to the left of the MSB to fill out the last group.

BINARY TO OCTAL CONVERSION Can be converted by grouping the binary bit in group of three starting from LSB Octal is a base-8 system and equal to two the power of three, so a digit in Octal is equal to three digit in binary system.

Hexadecimal System

Hexadecimal Number System Base 16 system Uses digits 0-9 & letters A,B,C,D,E,F Groups of four bits represent each base 16 digit

Hexadecimal Decimal

Decimal to Hexadecimal Conversion Convert 83010 to its hexadecimal equivalent: 830 / 16 = 51 R14 51 / 16 = 3 R3 3 / 16 = 0 R3 = E in Hex 33E16

Hexadecimal to Decimal Conversion Convert 3B4F16 to its decimal equivalent: Hex Digits 3 B 4 F x x x x Positional Values 163 162 161 160 Products 12288 +2816 + 64 +15 15,18310

Hexadecimal Binary

Binary to Hexadecimal Conversion The easiest method for converting binary to hexadecimal is to use a substitution code Each hex number converts to 4 binary digits

Binary Arithmetic

The binary addition facts

Binary Arithmetic The individual digits of a binary number are referred to as bits Each bit represents a power of two 01011 = 0 • 24 + 1 • 23 + 0 • 22 + 1 • 21 + 1 • 20 = 11 00010 = 0 • 24 + 0 • 23 + 0 • 22 + 1 • 21 + 0 • 20 = 2 00010 + 01011 01101 2 + 11 13 Equivalent decimal addition Binary addition