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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 stored using
binary: numbers, text, programs, pictures, sounds, videos, ...
Everything in computer is stored using binary: numbers, text, programs, pictures, sounds, videos, ...
Everything in computer is stored using binary: numbers, text, programs, pictures, sounds, videos, ...
Everything in computer is stored using binary: numbers, text, programs, 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
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 Internal(Human) (Computer) 25 11001 A 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 doSet N to N/2 (whole part)Record the remainder (1 or
0)end-of-loopSet A to remainders in reverse
order
Decimal to binary - Example
Example: Convert 32410 to binary N Rem N Rem324162 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 NSet bit in position P of A to
1Set N to N - 2P
end-of-loop
Decimal to binary - Example
Example: Convert 32410 to binary N Power P A 324 256 8 100000000 68 64 6 101000000 4 4 2 101000100
0
32410 = 1010001002
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
Example110100111010112 = 0011 0100 1110 1011
= 34EB