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Computer Science 101 Binary Systems Humans Decimal Numbers (base 10) Sign-Magnitude (-324) Decimal Fractions (23.27) Letters for text Computers Binary Numbers (base 2) Two’s complement and sign-magnitude Binary fractions and floating point ASCII codes for characters (A65) Why binary? Information is stored in computer via voltage levels. Using decimal would require 10 distinct and reliable levels for each digit. This is not feasible with reasonable reliability and financial constraints. Everything in computer is stored using binary: numbers, text, programs, pictures, sounds, videos, ... How can that be? Everything in computer is numbers, text, programs, ... Everything in computer is numbers, text, programs, ... Everything in computer is numbers, text, programs, ... Everything in computer is numbers, text, programs, ... stored using binary: pictures, sounds, videos, stored using binary: pictures, sounds, videos, stored using binary: pictures, sounds, videos, stored using binary: pictures, sounds, videos, Transistor A transistor is an electronic switch Basic unit of modern computer storage Two steady states based on voltage levels Say, 500 million transistors on a chip 1 cm2 Change states in billionth of sec Solid state Morse Code Morse Code Tree Decimal: Non-negatives Base 10 Uses decimal digits: 0,1,2,3,4,5,6,7,8,9 Positional System - position gives power of the base Example: 3845 = 3x103 + 8x102 + 4x101 + 5x100 Positions: …543210 Binary: Non-negatives Base 2 Uses binary digits (bits): 0,1 Positional system Example: 1101 = 1x23 + 1x22 + 0x21 + 1x20 Conversions External (Human) 25 A Internal (Computer) 11001 01000001 Humans want to see and enter numbers in decimal. Computers must store and compute with bits. Binary to Decimal Conversion Algorithm: • Expand binary number using positional scheme. • Perform computation using decimal arithmetic. Example: 110012 1x24 + 1x23 + 0x22 + 0x21 + 1x20 = 24 + 23 + 20 = 16 + 8 + 1 = 2510 Decimal to Binary - Algorithm 1 Algorithm: While N 0 do Set N to N/2 (whole part) Record the remainder (1 or 0) end-of-loop Set A to remainders in reverse order Decimal to binary - Example Example: Convert 32410 to binary N Rem N Rem 324 162 0 5 0 81 0 2 1 40 1 1 0 20 0 0 1 10 0 32410 = 1010001002 Decimal to Binary - Algorithm 2 Algorithm: Set A to 0 (all bits 0) While N 0 do Find largest P with 2P N Set bit in position P of A to 1 Set N to N - 2P end-of-loop Decimal to binary - Example Example: Convert 32410 to binary N Power P A 324 68 4 0 256 64 4 32410 = 1010001002 8 6 2 100000000 101000000 101000100 Binary Addition One bit numbers: + 0 1 0 | 0 1 1 | 1 10 Example 1111 1 110101 (53) + 101101 (45) 1100010 (98) Octal Numbers Base 8 Digits 0,1,2,3,4,5,6,7 Number does not have so many digits as binary Easy to convert to and from binary Often used by people who need to see the internal representation of data, programs, etc. Octal Conversions Octal to Binary • Simply convert each octal digit to a three bit binary number. • Example: 5368 = 101 011 1102 Binary to Octal • Starting at right, group into 3 bit sections • Convert each group to an octal digit • Example 110111111010102 = 011 011 111 101 010 = 337528 Hexadecimal Base 16 Digits 0,…,9,A,B,C,D,E,F Hexadecimal Binary • Just like Octal, only use 4 bits per digit. Example: 98C316 = 1001 1000 1100 00112 Example 110100111010112 = 0011 0100 1110 1011 = 34EB Python example