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Lecture 10

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

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  1. Lecture 10 Binary numbers – for fractional numbers

  2. 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 !!!

  3. 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

  4. 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

  5. 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

  6. 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

  7. 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

  8. 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

  9. _ _ _ _ _ _ _ _ _ _ __ _ _ _ _ • 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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