Lec 4- Numbering Systems

BCS101- Computer Science Fundamentals Fall 2024 / 2025 Lec 4: Numbering Systems Dr. Esraa Mosleh  Agenda Differentiate between Decimal, Binary, Octal, Hexadecimal number Systems. Convert a number between bases....

BCS101- Computer Science Fundamentals Fall 2024 / 2025 Lec 4: Numbering Systems Dr. Esraa Mosleh  Agenda Differentiate between Decimal, Binary, Octal, Hexadecimal number Systems. Convert a number between bases. 2  Introduction to Numbering Systems Decimal System :We are all familiar with the decimal number system (Base 10). Some other numbering systems that we will work with are: Binary → Base 2 ‫النظام الثنائي‬ Octal → Base 8 ‫النظام الثماني‬ Hexadecimal → Base 16 ‫النظام السداسي عشر‬ List of Numbers  Significant Digits Binary: 11101101 Most significant digit Least significant digit Hexadecimal: 1D63A7A Most significant digit Least significant digit  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  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 0-8  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 1 0 0 0 0 1 12 STOP when quotient equals 0  Decimal to Binary Conversion Example1 : convert 25 to binary 10 Solution = 2510 = ?2 25 = 2 12 balance 1 LSB 12 = 2 6 balance 0 = 6 3 balance 0 2 = 3 1 balance 1 2 = 12 0 balance 1 MSB... Answer = 110012  Binary to Decimal Conversion The easiest method for converting a binary number to its decimal equivalent is to use the Multiplication Algorithm First 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 Conversion 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 2: convert 110112 to decimal value Solution: 1 1 0 1 1 24 23 22 21 20 = 16+8+2+1 = 2710  Binary to Decimal Conversion (cont.) Example 3 : Convert 101101012 to decimal value Solution: 1 0 1 1 0 1 0 1 27 26 25 24 23 22 21 20 = = 128 + 32 + 16 + 4 + 1 18110 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.  Octal Number System Also known as the Base 8 System Uses digits 0 - 7 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  Decimal to Octal Conversion 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 4: convert 35910 to Decimal Value Solve = 35910 = ?8 = 359 8 44 balance 7 LSB = 44 8 5 balance 4 = 85 0 balance 5 MSB... Answer = 5478  Octal to Decimal Conversion Convert from octal to decimal by multiplying each octal digit by its positional weight. Example 5: Convert 1638 to decimal value Solve = 1 x (82 ) + 6 x (81 ) + 3 x (80 ) = 1 x 64 + 6 x 8 + 3 x 1 = 11510 Example 6: Convert 3338 to decimal value Solve = 3 x (82 ) + 3 x (81 ) + 3 x (80 ) = 3 x 64 + 3 x 8 + 3 x 1 = 21910  Octal to Binary Conversion Convert from octal to binary by converting each octal digit to a three bit binary equivalent Octal digit 0 1 2 3 4 5 6 7 Binary 000 001 010 011 100 101 110 111 Equivalent  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. Octal to Binary Conversion (cont.)  Binary to Octal Conversion Number 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. Binary to Octal Conversion (cont.)  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  Decimal to Hexadecimal Conversion Example: Convert 83010 to its hexadecimal equivalent: 830 / 16 = 51 R14 51 / 16 = 3 R3 = E in Hex 3 / 16 = 0 R3 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  Binary to Hexadecimal Conversion The easiest method for converting binary to hexadecimal is to use a substitution code Each hex number is converted to 4 binary digits  Binary Arithmetic The binary addition facts 0-26  Binary Arithmetic (cont.) Binary 00010 2 Equivalent addition + 01011 + 11 Decimal addition 01101 13 Thanks for your kind listening