Agenda

1
Agenda

• Data Representation
• Purpose
• Numbering Systems
• Decimal
• Binary
• Conversions
• Decimal to Binary
• Binary to Decimal

2
Agenda

• How Computers Store Data
• Numbering Systems
• Octal
• Hexadecimal
• Conversions
• Decimal to Binary
• Binary to Decimal

3
Data Representation Game

• Groups
• Number Systems
• Roman Numerals,
• Chinese,
• Decimal,
• Binary,
• Blocks,
• Hexadecimal,
• Octal

4
How Computers Store Data

• Computers are electronic devices that store data
  in a series of switches (circuits) which are
  either on or off.
• The digit 1 indicates that the switch is ON
• The digit 0 indicates that the switch is OFF
• Because there are two possible ways in which to
  represent this type of digit, we call this system
  as a binary number system

5
How Computers Store Data

• In order to have the computer store various
  alphabetical (upper and lower case) characters as
  well as numbers and special characters, one
  binary digit is not sufficient.
• Computers will use a series of binary digits
  (referred as bits) to represent a character (byte)

6
How Computers Store Data

• How to determine total number of character
  possibilities for a binary numbering system
• 2n
• Where n represents the number of binary digits
  or bits

7
Various Encoding Schemes
8
Various Encoding Schemes
9
Why Study Numbering Systems?

• Humans are most comfortable with the decimal
  Numbering System (digits containing 10
  possibilities).
• Unfortunately, humans are not comfortable (or
  efficient) working with binary numbers (as stored
  in computers).
• To compromise, humans may need to learn binary,
  octal hexadecimal numbering systems

10
Purpose of Data Representation

• It is important to understand the various methods
  that computers interpret characters and numbers.
• Programmers need to know, from time-to-time how
  to translate or interpret data represented by
  computers

11
Numbering Systems

• Decimal (Base 10)
• numbering system for humans
• Binary (Base 2)
• numbering system for computers
• Octal(Base 8)/ Hexadecimal(Base16)
• Shorthand method of representing binary numbers

12
Decimal Numbers (Base 10)

• The decimal system is a positional system. Each
  digit represents a quantity via its position in
  the number. This system is best understood by
  humans.
• The numbers are represented as sums of the powers
  of ten (hence decimal system)

13
Decimal Numbers (Base 10)

• Look at number 3572 as sums of powers of 10
• 3x103 3x1000 3000
• 5x102 5x100 500
• 7x101 7x10 70
• 2x100 2x1 2
• 3,572

14
Decimal Numbers (Base 10)

• When referring to the number as a decimal system
  (base 10), the number is displayed as normal
• eg. 3,572

15
Binary Numbers (Base 2)

• The binary system is a system based on two
  numbers
• 0 and 1
• The numbers are represented as sums of the powers
  of two
• (i.e. Combinations of 0 or 1)

16
Binary Numbers (Base 2)

• Lets take the binary number 110101. What does
  this binary number represent as a number that
  humans would understand?
• We could to convert binary number to a decimal
  number

17
Rules to Convert Binary Number to Decimal Number

• Steps
• Reading left to right, multiply placeholder (1 or
  0) by the corresponding power of 2

18
Convert Binary to Decimal

• 110101
• 1x25 1 x 32 32
• 1x24 1 x 16 16
• 0x23 0 x 8 0
• 1x22 1 x 4 4
• 0x21 0 x 2 0
• 1x20 1 x 1 1
• Decimal 53

19
Binary Numbers (Base 2)

• When referring to the number as a binary (base
  2), the number is displayed with the subscript 2
  to not confuse number with decimal
• eg.
• 10 (Decimal)
• 102 (Binary) 1x21 0x20 2

20
Definitions
Quotient
Divisor
Dividend
- Divisor x Quotient
Remainder
21
Converting Decimal to BinaryMethod 1

• Steps
• 1. Divide the number by 2 and save the
• quotient for the next step. Write the
• remainder on the side
• 2. If the quotient in step 1 is 0, stop
• 3. If the quotient in step 1 is not 0, repeat
• step 1, using the quotient as the new number
• Obtain the answer by reading the remainders in
  the inverse order in which they are produced

22
How to convert Decimal to BinaryMethod 2

• Divisor is 2 (binary is base 2)
• Dividend is the DECIMAL number to be converted
• Remainder is THE CURRENT DIGIT
• Quotient is the dividend for the next largest
  digit.

23
Converting Decimal to BinaryMethod 1

• Convert 25 to binary
• Calculation Quotient Remainder
• 25 / 2 12 1
• 12 / 2 6 0
• 6 / 2 3 0
• 3 / 2 1 1
• 1 / 2 0 1
• 0 / 2

24
Converting Decimal to BinaryMethod 1

• Convert 25 to binary
• Calculation Quotient Remainder
• 25 / 2 12 1
• 12 / 2 6 0
• 6 / 2 3 0
• 3 / 2 1 1
• 1 / 2 0 1
• (Reading up) 25 110012

Remember Read upwards!!
25
How to convert Decimal to BinaryMethod 2

• Divisor is 2
• Dividend is 25
• Quotient is 12
• Remainder is 1
• Therefore the rightmost digit is 1

26
How to convert Decimal to BinaryMethod 2
12
25
2
- 24
1
27
How to convert Decimal to BinaryMethod 2

• A second division takes place
• New Dividend is 12 (The quotient from previous
  slide - step 1)

28
How to convert Decimal to BinaryMethod 2
12
12
25
2
2
- 24
1
29
How to convert Decimal to BinaryMethod 2

• Divisor is 2
• New Dividend is 12 (The quotient from step 1)
• Quotient is 6
• Remainder is 0
• Therefore the next digit is 0

30
How to convert Decimal to BinaryMethod 2
6
12
25
2
12
2
- 24
- 12
1
0
31
How to convert Decimal to BinaryMethod 2
3
12
6
6
25
2
2
12
- 24
- 12
- 6
1
0
0
32
How to convert Decimal to BinaryMethod 2
1
12
3
6
3
6
25
12
2
- 6
- 24
- 12
- 2
1
0
0
1
33
How to convert Decimal to BinaryMethod 2
0
12
3
1
6
3 - 2
1
6
25
12
2
- 6
- 24
- 12
- 0
1
0
1
0
1
34
How to convert Decimal to BinaryMethod 2
0
12
3
1
6
3 - 2
1
6
25
12
2
- 6
- 0
- 24
- 12
1
0
1
0
1
Final value of Decimal 25 in Binary is 110012
35
How to convert Decimal to BinaryMethod 2

• Check
• 110012 1 x 16 1 x 8 1
• 16 8 1
• 25

NOTE Method 2 can be used instead of Method 1
in all the examples which follow.
36
Octal Numbers (Base 8)

• The octal system is a system based on eight
  numbers
• 0, 1, 2, 3, 4, 5, 6, 7
• The numbers are represented as sums of the powers
  of eight

37
Octal to Decimal Conversion

• Lets take the octal number 7268. What does this
  octal number represent as a number that humans
  would understand?
• We could convert octal number to a decimal number

38
Rules to Convert Octal Number to Decimal Number

• Steps
• Reading left to right, multiply placeholder (0
  through 7) by the corresponding power of 8

39
Convert Octal to Decimal

• 7268
• 7x82 7 x 64 448
• 2x81 2 x 8 16
• 6x80 6 x 1 6
• Decimal 470

40
Convert Decimal to Octal

• We convert from decimal to octal by a procedure
  similar to converting decimal to binary
• Instead of successively dividing by 2, we
  successively divide by 8

41
Convert Decimal to Octal

• Steps
• 1. Divide the number by 8 and save the
• quotient for the next step. Write the
• remainder on the side
• 2. If the quotient in step 1 is 0, stop
• 3. If the quotient in step 1 is not 0, repeat
• step 1, using the quotient as the new number
• Obtain the answer by reading the remainders in
  the inverse order in which they are produced

42
Converting Decimal to Octal

• Convert 79510 to Octal
• Calculation Quotient Remainder
• 795 / 8 99 3
• 99 / 8 12 3
• 12 / 8 1 4
• 1 / 8 0 1
• (Reading up) 79510 14338

Remember Read upwards!!
43
Hexadecimal Numbers (Base 16)

• The octal system is a system based on sixteen
  numbers
• 0,1,2,3,4,5,6,7,8,9, A,B,C,D,E,F
• The numbers are represented as sums of the powers
  of sixteen

44
Hexadecimal to Decimal Conversion

• Lets take the hexadecimal number 90FC16. What
  does this hexadecimal number represent as a
  number that humans would understand?
• We could convert hexadecimal number to a decimal
  number

45
Rules to Convert Hexadecimal Number to Decimal
Number

• Steps
• Reading left to right, multiply placeholder (0
  through F) by the corresponding power of 16
• Remember
• A10, B11, C12, D13, E14, F15

46
Convert Hexadecimal to Decimal

• 90FC16
• 9x163 9 x 4096 36864
• 0x162 0 x 256 0
• 15x161 15 x 16 240
• 12x160 12 x 1 12
• Decimal 3711610

47
Convert Decimal to Hexadecimal

• We convert from decimal to hexadecimal by a
  procedure similar to converting decimal to binary
• Instead of successively dividing by 2, we
  successively divide by 16

48
Convert Decimal to Hexadecimal

• Steps
• 1. Divide the number by 16 and save the
• quotient for the next step. Write the
• remainder on the side
• 2. If the quotient in step 1 is 0, stop
• 3. If the quotient in step 1 is not 0, repeat
• step 1, using the quotient as the new number
• Obtain the answer by reading the remainders in
  the inverse order in which they are produced

49
Converting Decimal to Hexadecimal

• Convert 42910 to Hexadecimal
• Calculation Quotient Remainder
• 429 / 16 26 D
• 26 / 16 1 A
• 1 / 16 0 1
• 0 / 16
• (Reading up) 42910 1AD16

Remember Read upwards!!
50
Numbering System Shortcuts

• It is very simple to convert binary numbers to
  octal or hexadecimal numbers since 8 is 23, and
  16 is 24
• In other words
• 1 Octal digit 3 binary digits
• 1 Hex digit - 4 binary digits

51
Binary to Octal

• Notice the Pattern
• Largest 3 digit binary is 111
• 1 octal digit will represent a 3 digit binary
  number
• Highest Octal digit is 7
• Therefore 1112 78

52
Binary to Octal

• Relationship
• Octal Binary
• 0 000
• 1 001
• 2 010
• 3 011
• 4 100
• 5 101
• 6 110
• 7 111

Does this table look familiar?
53
Practical Example

• applying octal values of rwx the chmod command
  (e.g., chmod 751).
• chmod 777 - 111 111 111 - rwx rwx rwx
• chmod 755 - 111 101 101 - rwx r-x r-x
• chmod 711 - 111 001 001 - rwx --x --x
• chmod 644 - 110 100 100 - rw- r-- r--

54
Binary to Hexadecimal

• Notice the Pattern
• Largest 4 digit binary is 1111
• 1 hex digit will represent a 4 digit binary
  number
• Highest hex digit is F
• Therefore 11112 F16

55
Binary to Hexadecimal

• Relationship
• Hexadecimal Binary Hexadecimal Binary
• 0 0000 8 1000
• 1 0001 9 1001
• 2 0010 A 1010
• 3 0011 B 1011
• 4 0100 C 1100
• 5 0101 D 1101
• 6 0110 E 1110
• 7 0111 F 1111

56
Convert Hex to Binary

• Steps
• Convert Hex number to groups of powers of 2.
• Convert to Binary number (Remember to drop
  leading zeros for first set of binary numbers -
  i.e. first left set)

57
Convert Hex to Binary

• 11F616
• 1 1 F
  6
• 000(1) 000(1) (8)(4)(2)(1) 0(4)(2)0
• 1 0001 1111 0110
• 10001111101102

Drop Leading zeros
000
58
Convert Binary to Hex

• Steps
• Separate into 4 bit groups starting from the
  right.
• Calculate decimal equivalent (in placeholders in
  powers of 2)
• Convert to Hexadecimal number

59
Convert Binary to Hex

• 10001111101102
• 1 0001 1111 0110
• 0001 0001 1111 0110
• 1 1 (8)(4)(2)(1) 0(4)(2)0
• 1 1 15 6
• 11F616

60
Converting Octal to Hexadecimal

• The easiest method to convert between Octal and
  Hexadecimal is to convert to binary as an
  intermediate step
• Regroup binary in groups of 4 for hexadecimal and
  3 for octal

61
Storing Numbers

• Numeric information (stored as a non negative
  number) is often store in a computer in binary.
• Eg.
• 1 byte (0 - 255 numbers)
• 2 bytes (0 - 65535)
• 4 bytes (0-4294967295)

62
Data Representation

• So far, we have learned how to convert between
  decimal, binary, octal and hexadecimal numbering
  systems and have performed non-decimal
  arithmetic.
• One of the main purposes of this topic is to
  represent numbers as binary code

63
Word Size

• For PCs 8 bits is used to represent characters,
  but large number may require more than just 8
  bits.
• The number of bits that can be received,
  processed and transferred from the CPU is
  referred to as the word size
• PCs have 16 and 32 bit word sizes
• Mainframes have 64 bit word sizes

64
Storing Numbers

• Numeric information (stored as a non negative
  number) is often stored in a computer in binary.
• Eg.
• 1 byte (0 - 255 numbers)
• 2 bytes (0 - 65535)
• 4 bytes (0-4294967295)

65
Data Formats

• Unsigned Binary
• Data stored as a binary number, with no way to
  express a negative quantity

66
Unsigned Numbers

• If the sign of the number is not important (ie.
  The number is not negative) then it is referred
  to as an unsigned number.
• Decimal Number Ranges
• 8 bits 0 - 255
• 16 bits 0 - 65535
• 32 bits 0 - 4294967295

67
Data Formats

• Signed Binary
• Data stored as a binary number, but using a
  leading zero to represent a positive number, and
  the twos complement of a binary number for a
  negative number

68
Signed Numbers

• If the sign of the number does matter (ie. The
  number is positive or negative) then it is
  referred to as an signed number.
• Decimal Number Ranges
• 8 bits -128 to 127
• 16 bits -32768 to 32767
• 32 bits -214743648 to 2147483647

69
How to Store a Negative Number as Binary?

• You should know how to represent an unsigned
  number or value as binary code.
• How is a negative number stored as binary?
• By using a mathematical construct referred to as
  Twos Complement

70
Adding / Subtracting Binary Numbers

• Addition
• 0 0 0
• 0 1 1, 1 0 1
• 1 1 10
• Subtraction
• 0 - 0 0
• 1 - 1 0, 1 - 0 1
• 0 - 1 (Must borrow from next placeholder)
• Therefore 10 - 1 1

71
Twos Complement

• Simple method of subtracting two binary numbers
  by adding.
• Two Complement
• Flip binary numbers (0 becomes 1, visa versa)
• then add 1
• Result becomes negative
• Therefore, short-hand method of representing
  negative integers

72
Twos Complement

• By performing the mathematical construct Twos
  Complement, a binary number can be stored as its
  opposite (to the computer, this represents a
  negative number)
• Steps
• Flip the digits of the binary number (ie. 1
  becomes a 0, and a 0 becomes a 1)
• add 12

73
Twos Complement Example

• Assuming a computer with a 16 bit word size, take
  the decimal number 48
• Binary Number 00000000001100002
• Flip Digits (Bits) 11111111110011112
• Add 12 12
• Twos Complement 11111111110100002

To the computer the binary number represents -48!
This wont work on your calculator...
74
Twos Complement

• In order to store a signed number as binary in
  your computer, the first digit in the binary
  number is used to represent the sign of the
  number.
• A binary number beginning with a 0 represents a
  positive number
• A binary number beginning with a 1 represents a
  negative number

75
Twos Complement

• Twos Complement also is used to simplify the
  mathematical operations of a computer
• The number of circuits (processes) are fewer for
  adding numbers as opposed to subtracting numbers.
• Therefore, to simplify the process (and save
  money) manufacturers prefer to add a positive and
  negative number as opposed to subtraction!

76
Twos Complement Exercise

• Find the Twos Complement of 110012

77
Twos Complement Exercise

• Determine whether the following 8-bit binary
  numbers are positive or negative
• 100011112
• 000011002
• 111111112