Computer Math

1
Computer Math

• CPS120
• Introduction to Computer Science
• Lecture 7

2
Memory Units

• 1 nibble
• 1 byte
• 1 word
• 1 long word
• 1 quad word
• 1 octa-word

• 4 consecutive bits
• 8 consecutive bits
• 2 consecutive bytes
• 4 consecutive bytes
• 8 consecutive bytes
• 16 consecutive bytes

3
Larger Units of Memory

• 1 Kilobyte
• 1 Megabyte
• 1 Gigabyte
• 1 Terabyte
• 1 Petabyte
• 1 Exabyte

• 1024 bytes
• 106 bytes
• 109 bytes
• 1012 bytes
• 1015 bytes
• 1018 bytes

32 Mb 32103 Kb 32 103 1024 bytes
32,768,000 bytes
4
Representing Data

• The computer knows the type of data stored in a
  particular location from the context in which the
  data are being used
• i.e. individual bytes, a word, a longword, etc
• 01100011 01100101 01000100 01000000
• Bytes 99(10, 101 (10, 68 (10, 64(10
• Two byte words 24,445 (10 and 17,472 (10
• Longword 1,667,580,992 (10

5
Alphanumeric Codes

• American Standard Code for Information
  Interchange (ASCII)
• 7-bit code
• Since the unit of storage is a bit, all ASCII
  codes are represented by 8 bits, with a zero in
  the most significant digit
• H e l l o W o r l d
• 48 65 6C 6C 6F 20 57 6F 72 6C 64
• Extended Binary Coded Decimal Interchange Code
  (EBCDIC)

6
Number Systems

• We use the DECIMAL (10 system
• Computers use BINARY (2 or some shorthand for it
  like OCTAL (8 or HEXADECIMAL (16

7
Codes

• Given any positive integer base (RADIX) N, there
  are N different individual symbols that can be
  used to write numbers in the system. The value of
  these symbols range from 0 to N-1
• All systems we use in computing are positional
  systems
• 495 400 90 5

8
Conversions
9
Decimal Equivalents

• Assuming the bits are unsigned, the decimal value
  represented by the bits of a byte can be
  calculated as follows
• Number the bits beginning on the right using
  superscripts beginning with 0 and increasing as
  you move left
• Note 20, by definition is 1
• Use each superscript as an exponent of a power of
  2
• Multiply the value of each bit by its
  corresponding power of 2
• Add the products obtained

10
Horners Method

• Another procedure to calculate the decimal
  equivalent of a binary number
• Note This method works with any base
• Horners Method
• Step 1 Start with the first digit on the left
• Step 2 Multiply it by the base
• Step 3 Add the next digit
• Step 4 Multiply the sum by the base
• Step 5 Continue the process until you add the
  last digit

11
Binary to Hex

• Step 1 Form four-bit groups beginning from the
  rightmost bit of the binary number
• If the last group (at the leftmost position) has
  less than four bits, add extra zeros to the left
  of the group to make it a four-bit group
• 0110011110101010100111 becomes
• 0001 1001 1110 1010 1010 0111
• Step 2 Replace each four-bit group by its
  hexadecimal equivalent
• 19EAA7(16

12
Converting Decimal to Other Bases

• Step 1 Divide the number by the base you are
  converting to (r)
• Step 2 Successively divide the quotients by (r)
  until a zero quotient is obtained
• Step 3 The decimal equivalent is obtained by
  writing the remainders of the successive division
  in the opposite order in which they were obtained
• Know as modulus arithmetic
• Step 4 Verify the result by multiplying it out

13
Addition Subtraction Terms

• A B
• A is the augend
• B is the addend
• C D
• C is the minuend
• D is the subtrahend

14
Addition Rules All Bases

• Addition
• Step 1 Add a column of numbers
• Step 2 Determine if there is a single symbol for
  the result
• Step 3 If so, write it and go to the next
  column. If not, write the accompanying number
  and carry the appropriate value to the next column

15
Subtraction Rules All Bases

• Step1 Start with the rightmost column, if the
  column of the minuend is greater than that of the
  subtrahend, do the subtraction, if not
• Step 2 Borrow one unit from the digit to the
  left of the once being processed
• The borrowed unit is equal to borrowing the
  radix
• Step 4 Decrease the column form which you
  borrowed by one
• Step 3 Subtract the subtrahend from the minuend
  and go to the next column

16
Addition of Binary Numbers

• Rules for adding or subtracting very similar to
  the ones in decimal system
• Limited to only two digits
• 0 0 0
• 0 1 1
• 1 0 1
• 1 1 0 carry 1

17
Addition Subtraction of Hex

• Due to the propensity for errors in binary, it is
  preferable to carry out arithmetic in hexadecimal
  and convert back to binary
• If we need to borrow in hex, we borrow 16
• It is convenient to think in decimal and then
  translate the results back to hex

18
Representing Signed Numbers

• Remember, all numeric data is represented inside
  the computer as 1s and 0s
• Arithmetic operations, particularly subtraction
  raise the possibility that the result might be
  negative
• Any numerical convention needs to differentiate
  two basic elements of any given number, its sign
  and its magnitude
• Conventions
• Sign-magnitude
• Twos complement
• Ones complement

19
Representing Negatives

• It is necessary to choose one of the bits of the
  basic unit as a sign bit
• Usually the leftmost bit
• By convention, 0 is positive and 1 is negative
• Positive values have the same representation in
  all conventions
• However, in order to interpret the content of any
  memory location correctly, it necessary to know
  the convention being used used for negative
  numbers

20
Comparing the Conventions
21
Sign-Magnitude

• For a basic unit of N bits, the leftmost bit is
  used exclusively to represent the sign
• The remaining (N-1) bits are used for the
  magnitude
• The range of number represented in this
  convention is 2 N1 to 2 N-1 -1

22
Sign-magnitude Operations

• Addition of two numbers in sign-magnitude is
  carried out using the usual conventions of binary
  arithmetic
• If both numbers are the same sign, we add their
  magnitude and copy the same sign
• If different signs, determine which number has
  the larger magnitude and subtract the other from
  it. The sign of the result is the sign of the
  operand with the larger magnitude
• If the result is outside the bounds of 2 n1 to
  2 n-1 1, an overflow results

23
Twos Complement Convention

• A positive number is represented using a
  procedure similar to sign-magnitude
• To express a negative number
• Express the absolute value of the number in
  binary
• Change all the zeros to ones and all the ones to
  zeros (called complementing the bits)
• Add one to the number obtained in Step 2
• The range of negative numbers is one larger than
  the range of positive numbers
• Given a negative number, to find its positive
  counterpart, use steps 2 3 above

24
Twos Complement Operations

• Addition
• Treat the numbers as unsigned integers
• The sign bit is treated as any other number
• Ignore any carry on the leftmost position
• Subtraction
• Treat the numbers as unsigned integers
• If a "borrow" is necessary in the leftmost place,
  borrow as if there were another invisible
  one-bit to the left of the minuend

25
Overflows in Twos Complement

• The range of values in twos-complement is 2
  n1 to 2 n-1 1
• Results outside this band are overflows
• In all overflow conditions, the sign of the
  result of the operation is different than that of
  the operands
• If the operands are positive, the result is
  negative
• If the operands are negative, the result is
  positive

26
Ones Complement

• Devised to make the addition of two numbers with
  different signs the same as two numbers with the
  same sign
• Positive numbers are represented in the usual way
• For negatives
• STEP 1 Start with the binary representation of
  the absolute value
• STEP 2 Complement all of its bits

27
One's Complement Operations

• Treat the sign bit as any other bit
• For addition, carry out of the leftmost bit is
  added to the rightmost bit end-around carry

28
Binary Alphanumeric Codes

• A binary code is a group of n bits that assume up
  to 2n distinct combinations of 1s and 0s with
  each combination representing one element of the
  set that is being coded- i.e. permutations
• With two bits we can form a set of four elements
• With three bits we can represent 8 elements
• With four bits we can represent 16 elements

29
Weighted Codes

• A sequence of binary digits representing a
  decimal digit is called a code word
• The code with weights 8, 4, 2, 1 is know as the
  Binary-Coded-Decimal (BDC) code
• Another code could be weighted 2-4-1-2
• Some codes are not unique in representing decimal
  numbers
• These codes cannot be used interchangeably
• The correct code is self-complementing
• The value of 9-N can be obtained by complementing
  the bits of the code

30
Transmission Errors

• When binary data is transmitted, there is a
  possibility of an error in transmission due to
  equipment failure or "noise"
• Bits change from 0 to 1 or vice-versa

31
Categorizing Coding Schemes

• The number of bits that have to change within a
  byte before it becomes invalid characterizes the
  code
• Single-error-detecting code
• To detect single errors have occurred we use an
  added parity check bit makes each byte either
  even or odd
• Two-error-detecting code

32
Transformations

• The minimum distance of a code is the number of
  bits that need to change in a code word to result
  another valid code word
• Some codes are self-correcting (error-correcting
  code)

33
Error Detection Even Parity

• Bytes Transmitted
• 01100011
• 11100001
• 01110100
• 11110011
• 00000101 Parity Block
• B
• I
• T

• Bytes Received
• 01100011
• 11100001
• 11111100
• 11110011
• 00000101 Parity Block
• B
• I
• T

34
Hamming Code

• This method of multiple-parity checking can be
  used to provide multiple-error detection