n301/Encoding n301.tplt Chapter 4 The Hardware World Binary number system Base 2 instead of 10

2 digits, 0 and 1
BIT (Binary digIT)

Binary number system 2 All information stored internally in binary form

letters, intergers, decimal numbers, graphic images
external/internal encoding/decoding

Internal representation letters A = 01000001 integers 65 = 01000001 different information but same internal code computer needs to know the data type to know how to interpret (assign meaning) COMPUTER ARITHMETIC:Place Values number system with five symbols: 0, 1, 2, 3, and 4 How would we count 7 coins? 1 stack & 2 coins COMPUTER ARITHMETIC:Place Values 2 How would 24 coins look? Or how about 25 coins? COMPUTER ARITHMETIC:Place Values 3 numerical representaion scheme:

4 = stack-of-stacks
1 = 1 stack of 5
3 = single coins

COMPUTER ARITHMETIC:Place Values 4 This idea of giving symbols different values that depends on where they are writen and using zeros to fill the empty positions is called the place-value system. COMPUTER ARITHMETIC:Place Values 5 The stack size of number system is called its base COMPUTER ARITHMETIC:Place Values 6 What number system (base) did we design? COMPUTER ARITHMETIC:The Base-2 System - Binary Binary Addition:

ZERO         F        0
ONE          T        1
TWO         TF       10
THREE       TT       11
FOUR       TFF      100
FIVE       ???      ???

COMPUTER ARITHMETIC:The Base-2 System - Binary 2 Binary Subtraction:

     24 - 5  =>     ?? + 4 = 24
Ex:  10101 - 11 =>  ????? 
                    +  11
                    10101

COMPUTER ARITHMETIC:The Base-2 System - Binary 3 Multiplication:

COMPUTER ARITHMETIC:Base₁₀ => Base₂ Conversion Ex: 1024₁₀; 6786₁₀ COMPUTER ARITHMETIC:Base₁₀ => Base₂ Conversion 2 Negative Numbers in Binary Three representation schemes are used:

Signed Magnitude
Left most bit is the sign bit (0 => + & 1 => -).
Remaining bits hold absolute magnitude.

COMPUTER ARITHMETIC:Base₁₀ => Base₂ Conversion 3

Ex:     2₁₀ => 0000 0010₂
         -2₁₀ => 1000 0010₂



COMPUTER ARITHMETIC:Base₁₀ => Base₂ Conversion 4
One's Compliment
Sign bit is similar to (1).  The magnitude is complimented.



COMPUTER ARITHMETIC:Base₁₀ => Base₂ Conversion 5
Ex:     4₁₀ => 0000 0100₂
         -4₁₀ => 1111 1011₂



COMPUTER ARITHMETIC:Base₁₀ => Base₂ Conversion 6
Two's Compliment
Sign bit same as (1).
Magnitude is complimented first and a "1" is added to the complimented digits.



COMPUTER ARITHMETIC:Base₁₀ => Base₂ Conversion 7
Ex:     7₁₀ => 0000 0111₂
         -7₁₀ => 1111 1001₂



COMPUTER ARITHMETIC:Base₁₀ => Base₂ Conversion 8
Ex:     7 + -3 A - B = A + ~B + 1



COMPUTER ARITHMETIC: Octal Number System
Base=8
8 Symbols: {0,1,2,3,4,5,6,7}



COMPUTER ARITHMETIC: Octal Number System - Octal to Decimal
(a_n-1a_n-2 ... a₁a₀)₈ = (a_n-1x8^n-1+a_n-2x8^n-2+ ... +a₁x8¹+a₀x8⁰)₁₀
Example:
(127)₈ = (1x 8² + 2 x 8¹ + x 8⁰) = (64 + 16 +)₁₀ = (87)₁₀



COMPUTER ARITHMETIC: Octal Number System - Decimal to Octal
Repeated division by 8.
(Similar to principle to generate binary codes).
Example: (213)₁₀ = (what)₈?
1. 213/8 = 26(quotient),5(remainder) => lowest octal number = 5
2. 26/8 = 3(quitient), 2(remainder) => second octal digit = 2
3. 3/8 = 0(quotient), 3(remainder) => third octal digit = 3
Stop, since quotient = 0
Hence, (213)₁₀ = (325)₈



COMPUTER ARITHMETIC: Octal Number System - Octal to Binary
Expand each octal digit to 3 binary bits.
Example: (725)₈ = (111|010|101)₂



COMPUTER ARITHMETIC: Octal Number System - Binary to Octal
Combine every 3 bits into one octal digit.
Example: (110|010|011)₂ = (623)₈



COMPUTER ARITHMETIC: Hexadecimal Number System
Base = 16
16 symbols: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A(=10), B(=11), C(=12), D(=13), E(=14), F(=15)}



COMPUTER ARITHMETIC: Hexadecimal Number System - Hexdecimal to Decimal
(a_n-1a_n-2 ... a₁a₀)₁₆ = (a_n-1x16^n-1+a_n-2x16^n-2+ ... +a₁x16¹+a₀x16⁰)₁₀
Example:
(1C7)₁₆ = 1 16² + 12 x 16¹ + 7 x 16⁰)₁₀ = (256 + 197 + 7)₁₀ = (455)₁₀



COMPUTER ARITHMETIC: Hexadecimal Number System - Decimal to Hexdecimal
Repeated division by 16.
(similar in principle to generating binary and octal codes).
Example: (829)₁₀ = (what)₁₆?1. 829/16 = 51(quotient),13 = D(remainder) =>
     lowest octal number = D
2. 51/16 = 3(quitient), 3(remainder) => second octal digit = 3
3. 3/16 = 0(quotient), 3(remainder) => third octal digit = 3
Stop, since quotient = 0
Hence, (829)₁₀ = (33D)₁₆



COMPUTER ARITHMETIC: Hexadecimal Number System - Hexadecimal to Binary
Expand each hexadecimal digit to 4 binary bits.
Eaxple: (E29)₁₆ = (1110|0100|1001)₂



COMPUTER ARITHMETIC: Hexadecimal Number System - Binary to Hexadecimal
Combine every 4 bits into one hexadecimal digit.
Eample: (0101|1111|1010|0110)₂ = (5FA6)₁₆