Todays Material

1
Todays Material

• Number Systems
• Decimal
• Binary
• Octal
• Hexadecimal
• Conversion between number systems
• Arithmetic Operations
• Representation of Signed Numbers
• Sign-Magnitude
• Ones Complement
• Twos Complement

2
Decimal Numbers

• Most commonly used numbering system
• Based on the number 10
• called the radix or basis of the system
• In any numerical system, the basis tells us how
  many different symbols there are in the system
• In decimal system, there are 10 symbols
• 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 each called a digit

3
Number Systems

• In general, given any positive integer radix
  (base) N, there are N different individual
  symbols that can be used to write numbers in the
  system
• The value of these symbols ranges from 0 to N-1
• According to this definition
• Decimal (radix 10) has 10 symbols
• 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
• Binary (radix 2) has 2 symbols
• 0, 1
• Octal (radix 8) has 8 symbols
• 0, 1, 2, 3, 4, 5, 6, 7
• Hexadecimal (radix 16) has 16 symbols
• 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A(10), B(11),
  C(12), D(13), E(14), F(15)

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Notation

• To indicate the basis on which a given number is
  written, we will use a subscript at the lower
  right side of the number
• 10012
• 0110112
• 4568
• 2278
• 12356410
• 129010
• 1016
• 2C116
• FF16

5
Positional Systems

• Binary, octal, decimal and hexadecimal systems
  are called positional systems
• That is, the value represented by a symbol in the
  numerical representation of a number depends on
  its position in the number
• E.g., digit 4 in decimal number 478 represents
  400 units whereas digit 4 in decimal number 842
  represents 40 units.
• As a side note, Roman numerals is not a
  positional system
• Consider the roman numeral XXII
• position of X or I has no significance.
• Decimal equivalent XXII 22

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Calculating the Value of a Symbol

• In positional systems, the value of any symbol in
  a number can be calculated as follows
• Number the digits from right to left using
  superscripts. Begin with 0 as the superscript of
  the rightmost symbol, and increase superscripts
  by 1 as you move from left to right
• Use each superscript to a power of the basis
• Multiply the symbols own value in decimal by its
  corresponding power of the basis
• Sum all products together to calculate the
  decimal equivalent of the entire number

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Example I Decimal Number

• Whats the value of each digit in 324510
• (1) Number digits using superscripts
• 33 22 41 50
• (2-3) Use superscripts to form powers of the
  basis and multiply the digits own value, in
  decimal, by its corresponding power
• The value of 3 in 3245 is equal to 3103 31000
  3000
• The value of 2 in 3245 is equal to 2102 2100
  200
• The value of 4 in 3245 is equal to 4101 410
  40
• The value of 5 in 3245 is equal to 5100 51
  5
• (4) Adding the decimal equivalent of each digit,
  we calculate the decimal equivalent of the number
• 3000 200 40 5 3245

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Example II Binary Number

• Whats the value of each digit in 110012
• (1) Number digits using superscripts
• 14 13 02 01 10
• (2-3) Use superscripts to form powers of the
  basis and multiply the digits own value, in
  decimal, by its corresponding power
• The value of 1 in 11001 is equal to 124 116
  16
• The value of 1 in 11001 is equal to 123 18
  8
• The value of 0 in 11001 is equal to 022 04
  0
• The value of 0 in 11001 is equal to 021 02
  0
• The value of 1 in 11001 is equal to 120 11
  1
• (4) Adding the decimal equivalent of each digit,
  we calculate the decimal equivalent of the number
• 16 8 0 0 1 25

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Example III Octal Number

• Whats the value of each digit in 137028
• (1) Number digits using superscripts
• 14 33 72 01 20
• (2-3) Use superscripts to form powers of the
  basis and multiply the digits own value, in
  decimal, by its corresponding power
• The value of 1 in 13702 is equal to 184 14096
  4096
• The value of 3 in 13702 is equal to 383 3512
  1536
• The value of 7 in 13702 is equal to 782 764
  448
• The value of 0 in 13702 is equal to 081 08
  0
• The value of 2in 13702 is equal to 280 21
  2
• (4) Adding the decimal equivalent of each digit,
  we calculate the decimal equivalent of the number
• 4096 1536 448 0 2 6082

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Example IV Hexadecimal Number

• Whats the value of each digit in 3D8F16
• (1) Number digits using superscripts
• 33 D2 81 F0
• (2-3) Use superscripts to form powers of the
  basis and multiply the digits own value, in
  decimal, by its corresponding power
• The value of 3 in 3D8F is equal to 3163 34096
  12288
• The value of D in 3D8F is equal to D162 13256
  3328
• The value of 8 in 3D8F is equal to 8161 816
  128
• The value of F in 3D8F is equal to F160 151
  15
• (4) Adding the decimal equivalent of each digit,
  we calculate the decimal equivalent of the number
• 12288 3328 128 15 15759

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Conversion Between Binary, Octal, Hex

• Here is the representation for the first 15
  decimal numbers in binary, octal, hex, decimal

Hex
Decimal
Binary
Octal
12
Hex-2-Binary

• Given a hex number, we can find its binary
  equivalent by replacing each hex symbol by its
  binary equivalent
• Whats the binary equivalent of hex number AF3C4?
• A 10 1010
• F 15 1111
• 3 0011
• C 12 1100
• 4 0100
• A F 3 C 4
• 1010 1111 0011 1100 0100

13
Binary-2-Hex

• To convert binary numbers to their hex
  equivalent, we reverse the process
• Form 4-bit groups beginning from the rightmost
  bit of the binary number. If the last group at
  the leftmost position has less than 4 bits, add
  extra zeros to the left of the bits in this group
  to make it a 4-bit group
• Replace each 4-bit group by its hex equivalent

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Binary-2-Hex (cont)

• Whats the hex equivalent of the following binary
  number 1101011010110
• 11 1010 1101 0110
• Add 0s to left of 11 to make it 0011
• We have 0011 1010 1101 0110
• 3 A D 6

15
Octal-2-Binary

• Given an octal number, we can find its binary
  equivalent by replacing each octal symbol by its
  binary equivalent
• Whats the binary equivalent of octal number
  23754?
• 2 010
• 3 011
• 7 111
• 5 101
• 4 100
• 2 3 7 5 4
• 010 011 111 101 100

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Binary-2-Octal

• To convert binary numbers to their octal
  equivalent, we reverse the process
• Form 3-bit groups beginning from the rightmost
  bit of the binary number. If the last group at
  the leftmost position has less than 3 bits, add
  extra zeros to the left of the bits in this group
  to make it a 3-bit group
• Replace each 3-bit group by its octal equivalent

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Binary-2-Octal Example

• Whats the hex equivalent of the following binary
  number 1101011010110
• 11 101 011 010 110
• Add a 0 to the left of 11 to make it 011
• We have 011 101 011 010 110
• 3 5 3 2 6

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Converting Decimal to Other Bases

• Conversion of a given decimal number to another
  integer basis r (rgt0) is carried out by initially
  diving the number by r, and the successively
  dividing the quotients by until a zero quotient
  is obtained
• Decimal equivalent is obtained by writing the
  remainders of the successive divisions in the
  opposite order to that in which they were
  obtained

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Decimal-2-Binary Example

• Whats the binary equivalent of decimal 46
• Number Quotient when Remainder
• Dividing by 2
• 46 23 0
• 23 11 1
• 11 5 1
• 5 2 1
• 2 1 0
• 1 0 1

• Binary number is 101110
• Verify this by converting it to decimal
• 125 024 123 122 121 020
  32842 46

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Decimal-2-Octal Example

• Whats the octal equivalent of decimal 46
• Number Quotient when Remainder
• Dividing by 8
• 46 5 6
• 5 0 5
• Octal number is 56
• Verify this by converting it to decimal
• 581 680 406 46

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Decimal-2-Hex Example

• Whats the hex equivalent of decimal 46
• Number Quotient when Remainder
• Dividing by 16
• 46 2 14E
• 2 0 2
• Hex number is 2E
• Verify this by converting it to decimal
• 2161 E160 3214 46

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Rules for Forming Numbers

• Given a positional number system, how do we form
  consecutive numbers higher than that represented
  by the systems largest symbol?
• Consider decimal number system
• 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
• Then how do we proceed?
• After writing all single-digit numbers, we form
  all 2-digit combinations beginning with 1. Then
  we form all 2-digit combinations beginning with 2
  and so on until we reach 99.
• After exhausting all 2-digit numbers, we start
  forming 3-digit combinations, then all 4-digit
  combinations and so on.

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Rules for Forming Numbers (cont)

• We follow a similar strategy in binary, octal and
  hex number systems
• Binary Base 2
• 0, 1, 10, 11, 100, 101, 110, 111, 1000, 1001,
  1010, 1011, ..
• Octal Base 8
• 0, 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15,
  16, 17, 20, ..
• Hex Base 16
• 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F,
  10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 1A, 1B,
  1C, 1D, 1E, 1F, 20, 21..

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Arithmetic Operations (Addition)

• Arithmetic operations in binary, octal and hex
  systems follow similar rules to the decimal
  system
• In decimal, how do we add 342 and 485?

25 7 (no carry)
7
28 12-102 (carry 1)
2
134 8 (no carry)
8

• Final result is 82710

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Addition of Binary Numbers

• Add the following binary numbers
• 1101 (13)
• 0111 (7)

• 0 0 0
• 0 1 1
• 1 0 1
• 1 1 0 with a carry of 1

110 (carry 1)
0
0
1010 (carry 1)
1111 (carry 1)
1
1100 (carry 1)
0
carry
1

• Final result is 101002 2010

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Addition of Octal Numbers

• Add the following octal numbers
• 3241
• 276

167 (no carry)
7
3
4711-83 (carry 1)
1225 (no carry)
5
300 (no carry)
3

• Final result is 35378

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Addition of Hex Numbers

• Add the following hex numbers
• 1A23
• 7C28

3811B (no carry)
B
4
224 (no carry)
AC22-166 (carry 1)
6
1179 (no carry)
9

• Final result is 964B16

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Arithmetic Operations (Subtraction)

• In decimal, subtract 982 from 4015?

5-23B (no borrow)
3
11-83 (borrowed 10 from the left digit)
3
9-90 (borrowed 10 from the left digit, then
gave 1 to the right digit)
0
3-03 (gave 1 to the right digit)
3

• Final result is 303310

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Subtraction in other bases

• Subtraction in other bases is very similar to
  decimal
• When borrowing from the left digit, we borrow as
  big a number as the size of the radix
• This means that
• In binary, we borrow 2
• In octal, we borrow 8
• In hexadecimal, we borrow 16

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Subtraction of Binary Numbers

• Subtract 1112 (7) from 1000102 (34)

0
0
1
0
1
0
1
1
1
2-11 (borrowed 2 from the left digit)
1
1
2-11 (borrowed 2 from the left digit)
1-10 (borrowed 2 from the left digit,
gave 1 to the right digit)
0
1-01 (borrowed 2 from the left digit,
gave 1 to the right digit)
1
1
1-01 (borrowed 2 from the left digit,
gave 1 to the right digit)

• Final result is 110112 2710

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Subtraction of Octal Numbers

• Subtract 2768 from 35378

7-61 (no borrow)
1
4
83-74 (borrowed 8 from the left digit)
4-22 (gave 1 to the right digit)
2
3-00 (no borrow)
3

• Final result is 32418

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Subtraction of Hex Numbers

• Subtract 8C3 from 1F00A

1
A-37 (no borrow)
7
4
16-C(12)4 (borrowed 16 from the left digit)
F(15)-87 (borrowed 16 from the left digit,
gave 1 to the right digit)
7
E-0E (gave 1 to the right digit)
E
1-01
1

• Final result is 1E74716

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Hint on Arithmetic Operations

• Arithmetic operations in binary is error prone
  for humans since we have to deal with lots of 0s
  and 1s
• So when you are asked to work with binary
  numbers, first convert the binary number to
  hexadecimal, do the operation and convert back to
  binary representation

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Representing Signed Numbers

• Until now we have assumed that all numbers are
  unsigned, i.e., positive
• How do we represent both positive and negative
  numbers with n bits?
• General convention is to use one bit as the sign
  bit and the remaining n-1 bits as the magnitude
• Leftmost bit represents the sign
• 0 means a positive number, 1 means a negative
  number
• Remaining n-1 bits represents the magnitude

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Signed Number Conventions

• There are 3 conventions used to represent signed
  numbers
• Sign-and-Magnitude convention
• Ones Complement convention
• Twos Complement convention

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Sign-and-Magnitude Convention

• Given a basic unit of n bits, the leftmost bit
  represent the sign
• The remaining n-1 bits represent the magnitude
• The range of values that can be represented in
  this convention ranges from -2n-11, 2n-1-1
• E.g. How do we represent -49 in 8 bits?
• Bits 0-6 would represent the magnitude
• Bit 7 would be the sign 1
• Whats the binary value of 49? 0110001
• So, 49 would be 00110001, -49 would be 10110001

37
Twos Complement Convention

• Most popular convention
• Positive numbers are represented similar to sign
  and magnitude
• To represent a negative number do the following
• (1) Express the absolute value of the number in
  binary
• (2) Change all 0s to 1 and all 1s to 0
  (complement the number)
• (3) Add one(1) to the binary number in step 2
• The range of values that can be represented in
  this convention ranges from -2n-1, 2n-1-1

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Twos Complement Convention (Ex)

• How do we represent -49 in 8 bits?
• Step1 49 00110001
• Step2 Complement 11001110
• Step 3 Add 1 11001111 CF
• How do we represent -1?
• Step1 1 00000001
• Step2 Complement 11111110
• Step3 Add 1 11111111 FF

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Twos Complement Convention (Cont)

• To find the positive equivalent of a negative
  number represented in 2s complement, simple
  apply steps 2 3
• E.g. Given the negative number 11001111, whats
  the positive equivalent?
• Step 2 Complement 00110000
• Step 3 Add 1 00110001 49
• E.g. Given 11111111, whats the positive
  equivalent?
• Step2 Complement 00000000
• Step 3 Add 1 00000001

40
Ones Complement Convention

• Positive numbers are represented similar to sign
  and magnitude
• To represent a negative number do the following
• (1) Express the absolute value of the number in
  binary
• (2) Change all 0s to 1 and all 1s to 0
  (complement the number)
• The range of values that can be represented in
  this convention ranges from -2n-11, 2n-1-1

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Ones Complement Convention (Ex)

• How do we represent -49 in 8 bits?
• Step1 49 00110001
• Step2 Complement 11001110 CE
• How do we represent -1?
• Step1 1 00000001
• Step2 Complement 11111110 FD

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Representation of Some numbers in Different
Conventions

• Notice that 0 has 2 representations in
  sign-magnitude and 1s complement conventions!
• This complicates ALU design. So modern machines
  use 2s complement to represent signed numbers

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Summary

• Computers represent numbers in binary, which uses
  0s and 1s to represent numbers
• Since dealing with binary numbers is difficult
  for humans, we can use octal or hexadecimal
  number systems to represent binary numbers
• Signed numbers are represented in twos
  complement convention in modern architectures,
  which has a single representation for each number

44
Resources

• There is a good interactive tutorial on
  http//courses.cs.vt.edu/csonline/NumberSystems/L
  essons/index.html