Binary Numbers

1
Binary Numbers
2
Outcome

• Familiar with the binary system
• Binary to Decimal and decimal to binary
• Arithmetic and logic operation in binary system
• Logic gates
• Half Adder and Full Adder
• Hexadecimal system

3
Reading

• Goldsmiths Study guide
• mathematics for computing
• http//www.math.grin.edu/rebelsky/Courses/152/97F
  /Readings/student-binary
• http//en.wikipedia.org/wiki/Binary_numeral_system
  
• http//www.cut-the-knot.org/do_you_know/BinaryHist
  ory.shtml
• http//www.binarymath.info/

4
The Decimal Number System (cont)

• The decimal number system is also known as base
  10. The values of the positions are calculated
  by taking 10 to some power.
• Why is the base 10 for decimal numbers?
• Because we use 10 digits, the digits 0 through 9.

5
The Binary Number System base 2

• The decimal number system is a positional number
  system with a base 10.
• Example 5623
• 5623 5000 600 20 3 5 x 103 6 x102
  2 x 101 3 x 100

5000 600 20 3
5 x 103 6 x102 2 x 101 3 x 100
6
The Binary Number System

• The binary number system is also known as base 2.
  The values of the positions are calculated by
  taking 2 to some power.
• Why is the base 2 for binary numbers?
• Because we use 2 digits, the digits 0 and 1.

7
The Decimal Number System - base 10

• The decimal number system is a positional number
  system with a base 10.
• Example 1011
• 10112 1000 000 10 1 1 x 23 0 x22
  1 x 21 1 x 20 1110

1000 000 10 1
1 x 23 0x22 1 x 21 1x 20
8
Why Bits (Binary Digits)?

• Computers are built using digital circuits
• Inputs and outputs can have only two values
• True (high voltage) or false (low voltage)
• Represented as 1 and 0
• Can represent many kinds of information
• Boolean (true or false)
• Numbers (23, 79, )
• Characters (a, z, ) ASCII, UNICODE
• Pixels
• Sound
• Can manipulate in many ways
• Read and write
• Logical operations
• Arithmetic

9
Base 10 and Base 2

• Base 10
• Each digit represents a power of 10
• 5417310 5 x 104 4 x 103 1 x 102 7 x 101
  3 x 100
• Base 2
• Each bit represents a power of 2
• 101012 1 x 24 0 x 23 1 x 22 0 x 21 1 x
  20 2110

10
Converting from Binary to Decimal

• 1 0 0 1 1 0 1 1 X 20 1
• 26 25 24 23 22 21 20 0 X 21 0
• 1 X 22 4
• 20 1 1 X 23 8
• 21 2 0 X 24
  0
• 22 4 0 X
  25 0
• 23 8 1 X 26 64
• 24 16
  7710
• 25 32
• 26 64

11
Converting from Binary to Decimal (cont)

• Practice conversions
• Binary Decimal
• 110010
• 100111
• 1010101

12
Converting From Decimal to Binary (cont)

• Make a list of the binary place values up to the
  number being converted.
• Perform successive divisions by 2, placing the
  remainder of 0 or 1 in each of the positions from
  right to left.
• Continue until the quotient is zero.
• Example 4210
• 25 24 23 22
  21 20
• 32 16 8 4 2
  1
• 1 0 1 0 1
  0
• 42/2 21 and R 0
• 21/2 10 and R 1
• 10/2 5 and R 0
• 5/2 2 and R 1
• 2/2 1 and R 0
• 1/2 0 and R 1
• 4210 1010102

13
Example 1710

• We repeatedly divide the decimal number by 2
  and keep remainders
• 17/2 8 and R 1
• 8/2 4 and R 0
• 4/2 2 and R 0
• 2/2 1 and R 0
• 1/2 0 and R 1
• The binary number representing 17 is 10001

14
Converting From Decimal to Binary (cont)

• Practice conversions
• Decimal Binary
• 59
• 82
• 175

15
Fractional Numbers

• Decimal
• 456.7810 4 x 102 5 x 101 6 x 100 7 x
  10-18 x 10-2
• Binary
• 1011.112 1 x 23 0 x 22 1 x 21 1 x 20 1
  x 2-1 1 x 2-2
• 8 0 2
  1 1/2 ¼
• 11 0.5 0.25 11.7510

16
Binary Fractional to decimal nNumbers (cont)

• Example1
• 1011.112 1 x 23 0 x 22 1 x 21
  1 x 20 1 x 2-1 1 x 2-2
• 8 0 2 1
  1/2 ¼
• 11 0.5 0.25 11.7510
• Example 2
• 111.112 1 x 22 1 x
  21 1 x 20 1 x 2-1 1 x 2-2
• 4 2 1
  1/2 ¼ 7.7510
• Example3 11.0112 1 x 21 1 x 20 0x 2-1
  1 x 2-2 1 x 2-3
• 2
  1 0 ¼ 1/8 3.37510

17
Fractional numbers

• Examples 7.7510 (?)2
• Conversion of the integer part same as before
  repeated division by 2
• 7 / 2 3 (Q), 1 (R) ? 3 / 2 1 (Q), 1 (R) ?
  1 / 2 0 (Q), 1 (R) 710 1112
• Conversion of the fractional part perform a
  repeated multiplication by 2 and extract the
  integer part of the result
• 0.75 x 2 1.50 ? extract 1
• 0.5 x 2 1.0 ? extract 1 0.7510
  0.112
• 0.0 ? stop
• ? Combine the results from integer and
  fractional part, 7.7510 111.112
• How about choose some of
• Examples try 5.625

4
2
1
1/2
1/4
1/8
0.25
0.125
0.5
18
Fractional Numbers (cont.)

• Exercise 3 Convert (0.8125)10 to its binary
  form

Solution 0.8125 x 2 1.625 ? extract 1 0.625
x 2 1.25 ? extract 1 0.25 x 2
0.5 ? extract 0 0.5 x 2 1.0
? extract 1 0.0
? stop ? (0.8125)10 (0.1101)2
19
Representing fraction with error

• Example Convert (0.6)10 to its binary form

0.6 x 2 1.2 ? extract 1 0.2 x 2 0.4 ?
extract 0 0.4 x 2 0.8 ? extract 0 0.8 x 2
1.6 ? extract 1 0.6 x 2 ? (0.6)10
(0.1001 1001 1001 )2
20
Fractional Numbers (cont.)

• Errors
• One source of error in the computations is due to
  back and forth conversions between decimal and
  binary formats
• Example (0.6)10 (0.6)10 1.210
• Since (0.6)10 (0.1001 1001 1001 )2
• Lets assume a 8-bit representation (0.6)10 (0
  .1001 1001)2 , therefore
• 0.6 0.10011001
• 0.6 ? 0.10011001
• 1.00110010
• Lets reconvert to decimal system
• (1.00110010)b 1 x 20 0 x 2-1 0 x 2-2 1 x
  2-3 1 x 2-4 0 x 2-5 0 x 2-6 1 x 2-7 0 x
  2-8
• 1 1/8 1/16 1/128
  1.1953125
• ? Error 1.2 1.1953125
• 0.0046875

21
Bits, Bytes, and Words

• A bit is a single binary digit (a 1 or 0).
• A byte is 8 bits
• A word is 32 bits or 4 bytes
• Long word 8 bytes 64 bits
• Quad word 16 bytes 128 bits
• Programming languages use these standard number
  of bits when organizing data storage and access.

22
Adding Two Integers Base 10

• From right to left, we add each pair of digits
• We write the sum, and add the carry to the next
  column

0 1 1 0 0 1 Sum Carry
1 9 8 2 6 4 Sum Carry
2 1
6 1
4 0
0 1
0 1
1 0
23
Example

• 10011110
  1101111
• 111
  1101
• --------------------
  -------------------
• 101 0 0 101 1111100

24
Binary subtraction
25
Binary subtraction (Cont)
26
Binary subtraction (Cont)
27
Exercise

• 10010101 - 11011 ?
• 10000001 - 111 ?

28
Binary Multiplication
1 0 0 0 two 8ten multiplicand
1 0 0 1 two 9ten multiplier __________
__ 1 0 0 0 0 0 0
0 partial products 0 0 0 0 1 0 0
0 ____________ 1 0 0 1 0 0 0two 72ten
29
Binary Division
1 3 Quotient 1 1 / 1 4 7 Divisor
/ Dividend 1 1 3 7 Partial
remainder 3 3 4
Remainder
0 0 0 0 1 1 0 1 1 0 1 1 / 1 0 0 1 0
0 1 1 1 0 1 1 0 0
1 1 1 0 1 0 1 1
0 0 1 1 1 1 1 0 1
1 1 0 0
30
Bitwise Operators Shift Left/Right

• Shift left (ltlt) Multiply by powers of 2
• Shift some of bits to the left, filling the
  blanks with 0
• Shift right (gtgt) Divide by powers of 2
• Shift some of bits to the right
• For unsigned integer, fill in blanks with 0
• What about signed integers? Varies across
  machines
• Can vary from one machine to another!

0
0
1
1
0
1
0
0
53
0
0
0
0
1
1
0
1
53gtgt2
31
Boolean Algebra to Logic Gates

• Logic circuits are built from components called
  logic gates.
• The logic gates correspond to Boolean operations
  , , .
• Binary operations have two inputs, unary has one

OR
AND
NOT
32
AND
A
Logic Gate
AB
Truth Table
B
A B AB
0 0 0
0 1 0
1 0 0
1 1 1
A
B
Series Circuit
AB
33
OR
A
Logic Gate
AB
Truth Table
B
A B AB
0 0 0
0 1 1
1 0 1
1 1 1
A
Parallel Circuit
B
AB
34
NOT
A
Logic Gate (also called an inverter)
A or A
Truth Table
a A
0 1
1 0
35
n-input Gates

• Because and are binary operations, they can
  be cascaded together to OR or AND multiple inputs.

A
A
B
ABC
ABC
B
C
A
A
B
ABC
ABC
B
C
C
36
NAND and NOR Gates

• NAND and NOR gates can greatly simplify circuit
  diagrams. As we will see, can you use these
  gates wherever you could use AND, OR, and NOT.

A B A?B
0 0 1
0 1 1
1 0 1
1 1 0
NAND
A B A?B
0 0 1
0 1 0
1 0 0
1 1 0
NOR
37
XOR and XNOR Gates

• XOR is used to choose between two mutually
  exclusive inputs. Unlike OR, XOR is true only
  when one input or the other is true, not both.

A B A?B
0 0 0
0 1 1
1 0 1
1 1 0
XOR
A B A B
0 0 1
0 1 0
1 0 0
1 1 1
XNOR
38
Binary Sums and Carries

• a b Sum a b Carry
• 0 0 0 0 0 0
• 0 1 1 0 1 0
• 1 0 1 1 0 0
• 1 1 0 1 1 1

XOR
AND
0100 0101
69
0110 0111
103
1010 1100
172
39
Design Hardware Bit by Bit

• Adding two bits
• a b half_sum carry_out
• 0 0 0 0
• 0 1 1 0
• 1 0 1 0
• 1 1 0 1
• Half-adder circuit

40
Half Adder (1-bit)
A
B
A B S(um) C(arry)
0 0 0 0
0 1 1 0
1 0 1 0
1 1 0 1
S
Half Adder
C
41
Half Adder (1-bit)
A B S(um) C(arry)
0 0 0 0
0 1 1 0
1 0 1 0
1 1 0 1
42
Full Adder
A
B
Cin A B S(um) Cout
0 0 0 0 0
0 0 1 1 0
0 1 0 1 0
0 1 1 0 1
1 0 0 1 0
1 0 1 0 1
1 1 0 0 1
1 1 1 1 1
S
Full Adder
Carry In (Cin)
Cout
43
Full Adder
H.A.
H.A.
44
Full Adder
45
4-bit Ripple Adder using Full Adder
A0
B0
S0
46
Working with Large Numbers

• 0 1 0 1 0 0 0 0 1 0 1 0 0 1 1 1 ?
• Humans cant work well with binary numbers there
  are too many digits to deal with.
• Memory addresses and other data can be quite
  large. Therefore, we sometimes use the
  hexadecimal number system.

47
The Hexadecimal Number System

• The hexadecimal number system is also known as
  base 16. The values of the positions are
  calculated by taking 16 to some power.
• Why is the base 16 for hexadecimal numbers ?
• Because we use 16 symbols, the digits 0 and 1 and
  the letters A through F.

48
The Hexadecimal Number System (cont)

• Binary Decimal Hexadecimal Binary
  Decimal Hexadecimal
• 0 0 0
  1010 10 A
• 1 1 1
  1011 11 B
• 10 2 2
  1100 12 C
• 11 3 3
  1101 13 D
• 100 4 4
  1110 14 E
• 101 5 5
  1111 15 F
• 110 6 6
• 111 7 7
• 1000 8 8
• 1001 9 9

49
The Hexadecimal Number System (cont)

• Example of a hexadecimal number and the values of
  the positions
• 3 C 8 B 0 5 1
• 166 165 164 163 162 161 160

50
Example of Equivalent Numbers

• Binary 1 0 1 0 0 0 0 1 0 1 0 0 1 1 12
• Decimal 2064710
• Hexadecimal 50A716
• Notice how the number of digits gets smaller as
  the base increases.

51
Summary

• Convert binary to decimal
• Decimal to binary
• Binary operation
• Logic gates
• Use of logic gates to perform binary operations
• Half adder
• Full adder
• The need of Hexadecimal Hexadecimal

52
Next lecture (Data representation)

• Put this all together
• negative and positive integer representation
• unsigned notation
• Signed notation
• Excess notation
• Tows complement notation
• Floating point representation
• Single and double precision
• Character, colour and sound representation