CA Lecture 2: Data Representation I

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CA Lecture 2 Data Representation (I)

• Fixed Point Numbers
• (Chapter 2 of textbook)

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Data Representation

• Computer operations may or may not be
  arithmetic.
• Before being manipulated, all kind of
  information must be represented by patterns of
  1s and 0s.
• Data representation is about how the
  information should be represented in a machine.

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Data Representation

• Different kinds of representations fixed point
  numbers, floating point numbers, and characters.
• There are limits imposed on the accuracy and
  range of numeric calculations by the finite
  nature of the data representations.

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Fixed Point Numbers

• Fixed point is one method to store numbers in a
  computer system.
• In such a system, each number has exactly the
  same number of digits, and the point is always
  in the same place/position.
• In computer, the position of binary point is
  not stored anywhere when representing real
  numbers.
• Usually, fixed point numbers are used to
  represent integers.

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Range and Precision in Fixed Point Numbers

• A fixed point representation can be
  characterized by the range of expressible numbers
  and the precision.
• Range is the distance between the largest and
  smallest numbers.
• Precision is the distance between two adjacent
  numbers on a number line.
• Error is 1/2 of the difference between two
  adjoining numbers.

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Range and Precision in Fixed Point Numbers Cont.

• Using only two digits of precision for signed
  base 10 numbers, with the decimal point placed on
  the far right.
• The range (interval between lowest and highest
  numbers) is -99, 99 and the precision
  (distance between successive numbers) is 1.
• The maximum error, which is the difference
  between the value of a real number and the
  closest representable number, is 1/2 the
  precision. For this case, the error is 1/2 X 1
  0.5.

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Range and Precision in Fixed Point Numbers Cont.

• A fixed-point unsigned decimal example using 3
  digits and the decimal point placed 2 digits from
  the right. What is the range, precision and
  error?
• Range is 0.00, 9.99
• Precision is 0.01
• Error is 0.005

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Range and Precision in Fixed Point Numbers Cont.

• A fixed-point unsigned decimal example using 3
  digits and the decimal point placed on the far
  right. What is the range, precision and error?
• Range is 000, 999
• Precision is 1.0
• Error is 0.5

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Range and Precision in Fixed Point Numbers Cont.

• A fixed-point unsigned decimal example using 3
  digits and the decimal point placed on the far
  left. What is the range, precision and error?
• Range is .000,.999
• Precision is .001
• Error is .0005

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Range and Precision in Fixed Point Numbers Cont.

• Trade off between range and precision.
• Range and precision are finite in the
  implementation of the architecture but are
  infinite in the real world.
• Overflow will occur when a number is outside
  the range of the number system.

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Radix Number Systems

• The base, or radix, of a number system defines
  the range of possible values that a digit may
  have.
• Weighted position code is the general form for
  determining the decimal value of a number in a
  base k fixed point number system.

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Radix Number Systems Cont.

• Weighted position code of k base number system

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Convert base 2 to base 10

• What is (1001)2 in base 10?
• Answer 1 x 23 0 x 22 0 x 21 1 x 20
• 8 1 910

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Radix Number Systems Cont.

• Polynomial method is a way to convert a number
  from an arbitrary base into a base 10 number.
• The bit that carries the most weight is called
  the most significant bit (MSB), and the bit that
  carries the least weight is called the least
  significant bit (LSB).

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Radix Number Systems Cont.

• Base 8 (octal) and base 16 (hexadecimal) radix,
  are related to base 2 (binary).
• Every 3 binary digits make up 1 octal digit
• (010110)2 (010)2(110)2 (2)8(6)8 (26)8
• Every 4 binary digits make up 1 hexadecimal
  digit
• (10110110)2 (1011)2(0110)2 (B)16(6)16
  (B6)16

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Conversion from base 10 to any base

• Converting the integer part of a fixed point
  number the remainder method.
• Converting the fractional part of a fixed point
  number the multiplication method.
• Both methods utilize the general polynomial
  form of a binary number.

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Base Conversion with the Remainder Method
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Base Conversion with the Multiplication Method
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Non-terminating Base 2 Fraction

• We cant always convert a terminating base 10
  fraction into an equivalent terminating base 2
  fraction. Convert (0.2)10 to base 2

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Simple Unsigned Binary Addition

• Two binary numbers A and B are added from right
  to left, creating a sum and a carry in each bit
  position.
• In each bit position, there are four cases if
  we ignore carry
• 0 0, 0 1, 1 0, 1 1.

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Simple Unsigned Binary Subtraction

• What is (101)2 (010)2 ?
• Answer is (011)2
• What is (11001)2 (01110)2 ?
• Answer is (01011)2

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Signed Fixed Point Numbers

• For an 8-bit representation, there are 28 256
  possible bit patterns. These bit patterns can
  represent both positive and negative numbers. We
  can assign half of the bit patterns to negative
  numbers and half of the bit patterns to positive
  numbers.
• 4 different methods to represent signed fixed
  point numbers (we will cover) are
• Signed Magnitude, Ones Complement
• Twos Complement, Excess (Biased)

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Signed Magnitude

• Use the leftmost bit to represent the sign, 0
  for and 1 for .
• The remaining bits contain the absolute
  magnitude.

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Signed Magnitude

• Two bit patterns are used to represent a single
  number zero (0 and 0).
• Thus for 8-bit, signed magnitude can only
  represent 255 different numbers even though there
  are 256 bit patterns.

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Ones Complement

• Convert all of the 1s in the number to 0s, and
  all of the 0s to 1s.
• In general, N-bit numbers are assigned to
  binary counter values in the obvious way as
  integers from 0 to 2N-1-1, and then the negative
  numbers are assigned in reverse order.
• The term ones complement refers to the fact
  that negating a number in this format is
  accomplished by simply complementing the bit
  pattern (inverting each bit).

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Ones Complement
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Ones Complement

• The leftmost bit is 0 for positive numbers.
• The leftmost bit is 1 for negative numbers.
• Two bit patterns are used to represent a single
  number zero (0 and 0).
• Thus for 8-bit, ones complement can only
  represent 255 different numbers even though there
  are 256 bit patterns.

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Twos Complement

• The twos complement is formed in a way similar
  to forming the ones complement complement all
  of the bits in the number, then add 1 and discard
  any possible carry-out occurred at the MSB of the
  number.

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Twos Complement cont.

• Note that the leftmost bit is 0 for positive
  numbers, and is 1 for negative numbers.
• There is an equal number of positive and negative
  numbers provided zero is considered to be a
  positive number, as the sign bit of zero is 0.
  The positive number starts at 0, but the negative
  numbers start at 1, thus the magnitude of the
  most negative number is one greater than the
  magnitude of the most positive number.
• Thus there is no positive number to match 128
  for 8-bit representation of fixed point 2s
  complement.

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Fixed Point Numbers Excess Representation

• The leftmost bit is the sign (usually 1
  positive, 0 negative). Positive and negative
  representations of a number are obtained by
  adding a bias to the twos complement
  representation. This goes both ways, converting
  between positive and negative numbers. The effect
  is that numerically smaller numbers have smaller
  bit patterns, simplifying comparisons for
  floating point exponents.

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Excess Representation
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Excess Representation cont.

• Another way to convert decimals to excess
  representation
• add the excess (128)10 to the original number,
  and then create the unsigned binary version.
• E.g. for (12)10 in excess 128 representation
• Compute (128 12 140)10 and produce the
  unsigned binary pattern (10001100)2.
• For (-12)10 in excess 128 representation
• Compute (128 -12 116)10 and produce the
  unsigned binary pattern (01110100)2.

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3-Bit Signed Integer Representations
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Tutorial One Questions

• 1. What is the decimal equivalent of the largest
  unsigned binary number that can be obtained with
• a. 8 bits b. 32 bits c. n bits
• 2. Convert the following unsigned binary numbers
  to decimal
• a. 111010 b. 10101111.101 c. 110110110
• 3. Convert the following decimal numbers to
  unsigned binary
• a. 1950 b. 1011 c. 134 d. 2001

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Tutorial One Questions Cont.

• 4. Convert the following numbers with the
  indicated bases to decimal
• a. (12101)3 b. (4231)5 c. (A89)12
• 5. Convert the following decimal numbers to the
  indicated bases
• a. 7562.45 to octal
• c. 175.175 to binary
• 6. Perform the following conversions
• a. (764.7)8 to hexadecimal b. (F6D.C)16 to octal
  
• c. (147.5)8 to base 4

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Tutorial One Questions Cont.

• 7. Represent the decimal values 26 and -123 as
  signed, 10-bit numbers in the following binary
  formats
• a. Signed magnitude b. 1s complement
• c. 2s complement
• 8. What is the range of numbers for the
  2s-complement system, using n-bit
  representation?
• 9. Compute the sum of 9 and 5 using binary
  numbers in 4-bit binary formats.

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Tutorial One Questions Cont.

• 10. How many bits are required to represent the
  following decimal numbers as unsigned binary
  integers?
• 12 b. 147
• 11. Convert the following decimal numbers to
  8-bit signed-magnitude representation
• 23 b. 23 c. 48
• 12. Convert the following decimal numbers to
  8-bit twos complement representation
• 23 b. 23 c. 48

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Tutorial One Questions Cont.

• 13. Convert the following decimal numbers to
  3-bit excess 4 representation
• 3 b. 3 c. 2
• 14. Convert the following decimal numbers to
  8-bit excess 128 representation
• 3 b. 3 c. 25