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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
- If you are using this method, what happens when you add the positive numbers 7 and 2?
- 0111 + 0010 = 1001 !!!

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

- 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

- 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

- Note: Terminating decimal fractions may not have terminating binary representation.
- Try converting each of the following
- .75
- .1875
- .2
- .6

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

- 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

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