Binary

1
Binary

• What is really in the mind of a computer?
• Binary, bits (binary digits)
• Everything is encoded in binary
• How does a computer remember pictures?
• Documents? Smells?

2
Binary versus Decimal

• Binary representation uses just digits 0, 1
• Consider a binary odometer on a car
• Converting binary to decimal and vice versa
• What is 23 in binary?
• What is 101010 in decimal?
• Think in powers of two
• KB1024 bytes? the debate continues

3
Grouping Binary Digits

• Groups of bits represent data, instructions
• Numbers typically stored as 32 bits
• ASCII
• standard encoding of characters
• all characters encoded as eight bits
• example a is the number 01000001
• Unicode created to address large character sets
• Unicode uses 16 bits

4
Bits and Switches

• A bit has two possible values one or zero
• The bit is a logical entity a number
• A bit has a corresponding physical entity
• In computers, this is an electrical switch
• Consider a light switch
• Each electrical switch has two states
• it is set to pass current (like a 1 for a bit)
• it is set to not pass current (like a 0 for a
  bit)

5
Switches to Gates

• Combine switches in logical ways via gates
• Types AND, OR, NOT, etc.
• AND is 1 when both inputs are 1
• OR is 1 when either inputs are 1
• NOT is opposite of single input
• Each type has a corresponding graphical symbol

6
Gates
7
One More Gate XOR

• XOR (exclusive or) gate
• Results in 0 when inputs are same
• Results in 1 when inputs are different
• Useful in complementing an input bit

8
Gates to Circuits

• A circuit board designed using many gates
• Circuits often depicted graphically using gate
  symbols
• Example the one-bit adder circuit
• Consider adding two one-bit numbers A, B
• The result is a two-bit number C S where S is
  the sum of the bits and C is the carry bit

9
One Bit Adder
A
OR
S
AND
NOT
B
C
AND
10
Circuits to Larger Circuits

• Begin with circuit design
• Then establish means to manufacture it
• Another circuit designer then uses it as a part
• Smaller designs used to make medium designs
• Medium designs used to make larger designs
• Chips and circuit boards are nothing more than
  very large designs in very small packages

11
Boolean Logic

• Boolean logic is a branch of mathematics
• A Boolean variable has only one of two values
• true (corresponding to a 1 for a bit)
• false (corresponding to a 0 for a bit)
• Three-way correspondence among binary digits,
  electrical switches, Boolean variables
• Gives a mathematical model of computation
• Revisit the half-adder with this in mind

12
Boolean Expression for Adder
A
OR
S
AND
NOT
B
C
AND
S (A or B) and (not C) C A and B
13
Majority Function

• Returns 1 when two of three inputs are 1
• Boolean expression
• M ABC ABC ABC ABC
• Ways to represent this function
• boolean expression
• circuit diagram
• truth table

14
Circuit Truth Table
15
Translation Practice

• Translate to a circuit diagram1) F A B2) F
  A B C3) F AB AC4) F AB AB5) F
  ABC A6) F (AB) C7) F (AB C)

16
Important Number Systems

• We use decimal system, base 10
• Computers use binary system, base 2
• Sometimes computer science uses
• octal system, base 8
• hexadecimal system, base 16
• Trick for hexadecimal is using letters A through
  F for the digits 10 through 15

17
Examples of Octal, Hexadecimal
18
Conversion Practice

• Convert to hex 1111 1010 0010 0001
• Convert to octal 7EDC81A0
• Convert to hex, octal 0011 0000 1110 1101
• Convert to decimal 1010.1001
• Convert to binary 27.25

19
Data Types

• Need a plan to represent the basic types
• whole s, strings, booleans, real s
• How much space for each?
• How to represent each?
• Floating point is especially tricky
• IEEE 754 standardized three formats
• single (32 bits), double (64), extended (80)

20
Real Numbers as 10YY
21
IEEE Single, Double Formats
22
IEEE Single Precision

• Bits split into sign bit, exponent, fraction
• Value (-1)(sign bit) x fraction-1 x 2(exp
  127)
• Consider 0.01 1 x 2(-2)
• sign bit 0, fraction 1-1, exp -2127
• Can also represent special numbers
• infinity 1s in exponent field 0s in fraction
• NaN (Not a Number) for illegal results

23
Converting to IEEE Format

• Convert this 0.1
• Normalize 1.0 x 2-1
• Convert exponent1.0 x 2-1 127 1.0 x 2126
• Fill in the three parts (sign, exp,
  fraction)sign 0exponent 01111110fraction
  00000000000000000000000

24
Binary Digits and Real Numbers

• 1/2 0.5
• 1/4 0.25
• 1/8 0.125
• 1/16 0.0625
• 1/32 0.03125
• 1/64 0.015625
• 1/128 0.0078125
• 1/256 0.00390625
• 1/512 0.001953125

25
Floating Point Practice

• Put these numbers in binary form
• Then in IEEE Single Precision format
• 0.5
• 1
• 1.5
• 0.01
• 7.1

26
Binary Arithmetic

• How do we add binary numbers?
• Same as decimal allow for a carry digit
• For subtraction, need concept of negative
• What is a negative binary number?
• Use high-order bit to indicate sign
• How to convert a number to its negative?
• Use twos complement

27
Twos Complement

• Basic method is flip bits and add one, e.g.
• flip bits of 0011 (decimal 3) and get 1100
• then add one giving 1101 (-3 in 2s comp)
• Without 2s complement, a 4-bit number (0000 thru
  1111) can represent 0 thru 15
• With 2s complement, a 4-bit number can represent
  -8 (as 1000) thru 7 (as 0111)
• Converting negative decimal to binary

28
Twos Complement

• 0000 0
• 0001 1
• 0010 2
• 0011 3
• 0100 4
• 0101 5
• 0110 6
• 0111 7

1000 -8 1001 -7 1010 -6 1011 -5 1100
-4 1101 -3 1110 -2 1111 -1
29
Converting to 2s Complement

• Consider -3 in decimal
• Convert to binary
• Get 3 in binary 0011
• Flip the bits 1100
• Add one 1100 1 1101
• -3 in decimal 1101 in binary

30
Subtracting with 2s Complement

• Consider 9 - 3 00001001 - 00000011
• Convert number to subtract using 2s complement
  -00000011 11111101
• Then add the result to the first number
  0000100111111101-------------- 100000110
• Throw away the carry out from the sign bit
• Final answer 00000110 6

31
Overflow

• What about a 4-bit addition 0001 0111?
• Result is 1000, but 1 7 -8? I dont think so!
• And so begins the need for overflow handling
• Overflow is always detectable, but not always
  handled left to software to decide
• Detected when carry into sign bit not same as
  carry out of sign bit