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Lecture 10 - PowerPoint PPT Presentation

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

Binary numbers – for fractional numbers

• 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 !!!

• 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

• 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

• 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

• Note: Terminating decimal fractions may not have terminating binary representation.

• Try converting each of the following

• .75

• .1875

• .2

• .6

• 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

• 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

_ _ _ _ _ _ _ _ _ _ __ _ _ _ _

• 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