Number Representation

1
Lecture 3

• Number Representation

2
Decimal and Binary

• Decimal system
• It is basically intuitive. The power of the base
  of system, which in decimal is 10.
• The first position is ,
• the second position is ,
• and the third position is .

3
Decimal and Binary

• 243 3 x 4 x 2 x ,
• it is very intuitional.
• Decimal system

4
Decimal and Binary

• Binary system
• Whereas the decimal system is based on 10,
  the binary system is based on 2.
• In position table, each position is double the
  previous position. Again, this is because the
  base of the system is 2.

5
Decimal and Binary

• Why is
• 243 1 x 1 1 x 2 0 x 4 0 x 8 1 x 16
  
• 1 x 32 1 x 64 1 x 128 ?
• Binary system

6
Conversions

• The integer and fraction portions of the number
  are handled separately.
• Radix divide technique convert the integer
  portion.
• Radix multiply technique convert the fraction
  portion.

7
Conversions

• Radix divide technique
• Divide the given integer successively by the
  required radix, noting the remainder at each
  step.
• The quotient at each step becomes the new
  dividend for subsequent division.

8
Conversions

• Stop the division process when the quotient
  becomes zero.
• Collect the remainders from each step (last to
  first) and place them left to right to form the
  required number.

9
Conversions

• Integers
• Binary to decimal conversion
• Since each binary bit can be only 0 or 1,
  the
• result will be either 0 or the value of
  the weight.
• After multiplying all the digits, add the
  results.

10
Conversions

• 1x 0x 1x 1x 0x 1x
• 1x1 0x2 1x4 1x8 0x16 1x32
• 45
• Binary to decimal conversion

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Conversions

• Decimal to binary conversion
• To convert from decimal to binary, use
  repetitive division.
• Decimal to binary conversion

12
Conversions

• Decimal to binary conversion

13
Conversions

• Decimal to Octal conversion

14
Conversions

• Decimal to Hexadecimal

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Conversions

• Radix multiply technique
• Successively multiply the given fraction by the
  required base, noting the integer portion of the
  product at each step.
• Use the fractional part of the product as the
  multiplicand for subsequent steps.

16
Conversions

• Stop when the fraction either reaches 0 or
  recurs.
• Collect the integer digits at each step from
  first to last and arrange them left to right.

Decimal to binary conversion
17
Conversions

• Fraction
• Decimal to binary conversion

18
Conversions

• Decimal to Octal and Decimal to Hexadecimal
• 
  

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Conversions

• Integer and fraction

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Integer representation

• An integer can be positive or negative.
• A negative integer ranges from negative infinity
  to 0 a positive integer ranges from 0 to
  positive infinity.
• Range of integers

21
Integer representation

• However, no computer can store all the integers
  in the range. To do so would require an infinite
  number of bits, which means a computer with
  infinite storage capability.
• To use computer memory more efficiently, two
  broad categories of integer representation have
  been developed
• unsigned integers
• signed integers.

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Integer representation

• Signed integers may also be represented in three
  distinct ways
• Taxonomy of integers

23
Integer representation

• Unsigned integers format
• An unsigned integer is an integer without a sign.
  Its range is between 0 and positive infinity.

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Integer representation

• The following defines the range of unsigned
  integers in a computer, where N is the number of
  bits allocated to represent one unsigned integer
  
• Range 0 ( -1 )
• shows two common ranges for computers today.

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Integer representation

• Representation
• Storing unsigned integers is a straightforward
  process as outlined in the following steps
• The number is changed to binary.
• If the number of bits is less than N, 0s are
  added to the left of the binary number so that
  there is a total of N bits.

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Integer representation

• unsigned integers are stored in two different
  computers
• 8-bit allocation
• 16-bit allocation.
• Note that decimal 258 and decimal 24,760 cannot
  be stored in a computer using 8-bit allocation
  for an unsigned integer.

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Integer representation

• Decimal 1,245,678 cannot be stored in either of
  these computers.
• The condition called overflow occurs
• Storing unsigned integers in two different
  computer

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Integer representation

• Overflow
• If you try to store an unsigned integer such as
  256 in an 8-bit memory location, you get a
  condition called overflow.

8-bit
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Integer representation

• Applications
• Counting When you count, you do not need
  negative numbers. You start counting from 1
  (sometimes 0) and go up.
• Addressing Some computer languages store the
  address of a memory location inside memory
  location. Addresses are positive numbers starting
  from 0.

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Integer representation

• Sign-and-magnitude format
• The following defines the range of sign-and
  -magnitude integers in a computer, where N is the
  number of bits allocated to represent one
  sign-and -magnitude integer
• Range - ( -1 ) ( -1 )

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Integer representation

• There two 0s in sign-and-magnitude
  representation
• 0 ? 00000000
• -0 ? 10000000

Range of sign-and-magnitude integer
32
Integer representation

• Representation
• Storing sign-and-magnitude integers is a
  straightforward process
• The number is changed to binary the sign is
  ignored.
• If the number of bits is less than N -1, 0s are
  added to the left of number so that here is a
  total of N-1 bits.

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Integer representation

• The leftmost bit defines the sign of the number.
  If it is 0, the number is positive. If it is 1,
  the number is negative.
• Ex. 8 bit
• Range 127 -127

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Integer representation

• Interpretation
• How do you interpret a sign-and-magnitude binary
  representation in decimal?
• Ignore the first (leftmost) bit.
• Change the N-1 bits from binary to decimal
• Attach a or a sign to the number based on the
  leftmost bit.
• EX.

10010101 represents -21 01111111 represents
127
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Integer representation

• Ones complement format
• The range of ones complement integers in a
  computer, where N is the number of bits allocated
  to represent a ones complement integer
• Range - ( -1 ) ( -1 )

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Integer representation

• There two 0s in ones complement representation
• 0 ? 00000000
• -0 ? 11111111

Range of ones complement integer
37
Integer representation

• Representation
• Sorting ones complement integers requires the
  following steps
• The number is changed to binary the sign is
  ignored.
• 0s are added to the left of the number to make a
  total of N bits.

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Integer representation

• If the sign is positive, no more action is
  needed. If the sign is negative, every bit is
  completed (changed from 0 to 1 or from 1 to 0).
• Ex. 8 bit
• Decimal Binary number 1s complement
  
• 1 ? 00000001 ? 00000001
• 2 ? 00000010 ? 00000010
• -1 ? 00000001 ? 11111110
• -2 ? 00000010 ? 11111101
• (ignore sign)

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Integer representation

• In ones complement representation, the leftmost
  bit defines the sign of the number. If it is 0,
  the number is positive. If it is 1, the number is
  negative.
• Decimal 1s complement Decimal 1s
  complement

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Integer representation

• Twos complement format
• Twos complement is most common, the most
  important, and the most widely used
  representation of integers today.
• The range of twos complement integers in a
  computer, where N is the number of bits allocated
  for a twos complement integer
• Range - ( ) ( -1 )

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Integer representation

• Note that in this system,there is only one 0 and
  that the beginning of the range is 1 less than
  that of ones complement.
• 0 and -0 ? 00000000

Range of twos complement integer
42
Integer representation

• Representation
• Storing twos complement requires the following
  steps
• The number is changed to binary the sign is
  ignored.
• If the number of bits is less than N, 0s are
  added to the left of the number so that there is
  a total of N bits.

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Integer representation

• If the sign is positive, no more action is
  needed. If the sign is negative, leave all the
  rightmost 0s and the first 1 unchanged.
  Complement the rest of the bits.
• Ex. 8 bit
• Decimal Binary number 2s complement
  
• 1 ? 00000001 ? 00000001
• 2 ? 00000010 ? 00000010
• -1 ? 00000001 ? 11111111
• -2 ? 00000010 ? 11111110
• (ignore sign)

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Integer representation

• Summary

Summary of integer representation
45
Floating-point representation

• Floating point
• Floating-point is a system of arithmetic in which
  the decimal (or other radix) point is allowed to
  float as computations are performed.

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Floating-point representation

• Normalization
• There is a standard representation for
  floating-point numbers. It is normalization, the
  moving of the decimal point so that there is only
  one digit to the left of the decimal point.
• Ex.
• 245.17 2.4517 x
• x

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Floating-point representation

• IEEE standard
• The Institute of Electrical and Electronic
  Engineers (IEEE) has defined three standards for
  floating-point numbers two are used to store
  numbers in memory (single precision).

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Floating-point representation

• Single-precision
• It is a computer numbering format that occupies
  one storage location in computer memory at a
  given address.
• Note that the number inside the boxes
• is the number of bits for each field.

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Floating-point representation

• Single-precision representation
• The procedure for storing a normalized
  floating-point number in memory using
  single-precision format is as follows
• Store the sign as 0 (positive) or 1(negative).
• Store the exponent (power of 2) as Excess 127.
• Store the mantissa as an unsigned integer.

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Floating-point representation

• Ex. 111.000011 - x 1.11000011
• Step 1 - ? Sign 1
• Step 2 Exponent 2 127
• Step 3 0.11000011
• ? Mantissa 11000011000000000000000

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Floating-point representation

• Double precision
• It is a computer numbering format that occupies
  two storage locations in computer memory at
  address and address1.
• Note that the number inside the boxes
• is the number of bits for each field.

52
Floating-point representation

• Double-precision representation
• The procedure for storing a normalized
  floating-point number in memory using
  single-precision format is as follows
• Store the sign as 0 (positive) or 1(negative).
• Store the exponent (power of 2) as Excess_1023.
• Store the mantissa as an unsigned integer.

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Floating-point representation

• Ex. -0.75 - -1.1 x
• Step 1 - ? Sign 1
• Step 2 Exponent -1 1023
• Step 3 0.1
• ? Mantissa 1000000000
• 52 bits

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Addition

• One's complement arithmeticthe carry generated
  from the sign bit is added to the LSB of the
  result to complete the addition.
• Ex.

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Addition
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Addition

• Two's complement arithmetic
• EX.
• Here the sign-magnitude and twos complement
  representations are the same, since both numbers
  are positive.

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Addition

• Here the negative number is represented in the
  
• complement form.
• The sign bits are also included in the addition
  process.
• There is a carry from the sign bit position,
  which is ignored.

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Addition

• no carry is generated from the MSB during the
  addition.
• the result is negative since the sign bit is 1.
• the result is in the complement form and must be
  complemented to obtain the sign-magnitude
  representation.

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Addition