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Lecture 10. Binary numbers – for fractional numbers. Negative Integers. One method of representing negative numbers is called Sign-Magnitude Give up the leftmost bit to be used for a “sign” A 1 in the leftmost bit means the number is negative. 1011 is -3

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lecture 10

Lecture 10

Binary numbers – for fractional numbers

negative integers
Negative Integers
  • One method of representing negative numbers is called Sign-Magnitude
    • Give up the leftmost bit to be used for a “sign”
    • A 1 in the leftmost bit means the number is negative.
    • 1011 is -3
    • If you are using this method, what happens when you add the positive numbers 7 and 2?
    • 0111 + 0010 = 1001 !!!
example with 3 bits
example with 3 bits
  • In a three bit system, we could get the following representations:
  • 000 = 0
  • 001 = 1
  • 010 = 2
  • 011 = 3
  • 100 = -0
  • 101 = -1
  • 110 = -2
  • 111 = -3
twos complement
Twos complement
  • This is the sign magnitude method and gives all the numbers from –(23-1 - 1) to (23-1 - 1) or -22 – 1 to 22 – 1. But it seems wasteful to have two ways of writing zero.
  • Instead computer scientists have invented two’s complement. When a number has a one in the left most location, it is translated by taking its “complement”, adding 1 and attaching a -1
fractional numbers
Fractional numbers
  • 1011.101 What does this correspond to in base 10?
  • The place values to the right of the decimal continue to have smaller place values given by 2 raised to a negative exponent
  • 0.1 in binary is 2-1 = .5
  • 0.01 in binary is 2-2 = .25
  • 0.001 in binary is 2-3 = .125
  • 0.0001 in binary is 2-4 = .0625
  • 0.00001 in binary is 2-5 = .03125
  • 0.000001 in binary is 2-6 = .015625
  • 0.0000001 in binary is 2-7 = .0078125
converting fractional decimal numbers to binary
Converting fractional decimal numbers to binary
  • Note: Terminating decimal fractions may not have terminating binary representation.
  • Try converting each of the following
    • .75
    • .1875
    • .2
    • .6
fractional numbers1
Fractional numbers
  • 1011.101 What does this correspond to in base 10?
  • The place values to the right of the decimal continue to have smaller place values given by 2 raised to a negative exponent
  • 0.1 in binary is 2-1 = .5
  • 0.01 in binary is 2-2 = .25
  • 0.001 in binary is 2-3 = .125
  • 0.0001 in binary is 2-4 = .0625
  • 0.00001 in binary is 2-5 = .03125
  • 0.000001 in binary is 2-6 = .015625
  • 0.0000001 in binary is 2-7 = .0078125
storing the floating point binary number in the computer
Storing the Floating Point Binary Number in the Computer
  • There is of course no decimal point – only 0’s and 1’s. We will have to decide on a code for storing these numbers.
  • Actual computers use 32 or 64 bits to store a floating point number. But to work by hand, we will assume that we have a “baby” computer that only uses 16 bits.
  • Recall that 235.67 can be written as .23567x103
    • 23567 is the Mantissa
    • 3 is the Exponent
slide11
_ _ _ _ _ _ _ _ _ _ __ _ _ _ _
  • The first 10 bits are for the Mantissa with the first bit being the sign
  • The next six are for the exponent with the first one being the sign.
  • 1101.011 is written as .1101011x24
  • The mantissa is positive so the sign is 0. We have nine more bits for the mantissa so we add 00 to the end getting 0110101100
  • The exponent is also positive and we have 4 bits to represent the number 4 giving 000100
  • The complete 16 bits: 0110101100 000100
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