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Negative Number Representation
- Sign Magnitude
Sign magnitude is a very simple representation of negative numbers. In sign magnitude the first bit is dedicated to represent the sign and hence it is called sign bit.
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Sign bit ‘1’ represents negative sign.
Sign bit ‘0’ represents positive sign.
In sign magnitude representation of a n – bit number, the first bit will represent sign and rest n-1 bits represent magnitude of number. Ricoh aficio sp 3510dn user manual.
For example,
- +25 = 011001
Where 11001 = 25
And 0 for ‘+’
- -25 = 111001
Where 11001 = 25
And 1 for ‘-‘.
Range of number represented by sign magnitude method = -(2n-1-1) to +(2n-1-1) (for n bit number)
But there is one problem in sign magnitude and that is we have two representations of 0
– 0 = 100000
- 2’s complement method
To represent a negative number in this form, first we need to take the 1’s complement of the number represented in simple positive binary form and then add 1 to it.
For example:
(8)10 = (1000)2
1’s complement of 1000 = 0111
Adding 1 to it, 0111 + 1 = 1000
So, (-8)10 = (1000)2
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Please don’t get confused with (8)10 =1000 and (-8)10=1000 as with 4 bits, we can’t represent a positive number more than 7. So, 1000 is representing -8 only.
Range of number represented by 2’s complement = (-2n-1 to 2n-1 – 1)
Floating point representation of numbers
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- 32-bit representation floating point numbers IEEE standard
Normalization
- Floating point numbers are usually normalized
- Exponent is adjusted so that leading bit (MSB) of mantissa is 1
- Since it is always 1 there is no need to store it
- Scientific notation where numbers are normalized to give a single digit before the decimal point like in decimal system e.g. 3.123 x 103
For example, we represent 3.625 in 32 bit format.
Changing 3 in binary=11
How to download instagram on mac computer. Changing .625 in binary
Writing in binary exponent form
3.625=11.101 X 20
On normalizing
11.101 X 20=1.1101 X 21
On biasing exponent = 127 + 1 = 128
(128)10=(10000000) 2
For getting significand
Digits after decimal = 1101
Expanding to 23 bit = 11010000000000000000000
Setting sign bit
As it is a positive number, sign bit = 0
Itools. Finally we arrange according to representation
- 64-bit representation floating point numbers IEEE standard
Again we follow the same procedure upto normalization. After that, we add 1023 to bias the exponent.
For example, we represent -3.625 in 64 bit format.
Changing 3 in binary = 11
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Changing .625 in binary
Writing in binary exponent form
3.625 = 11.101 X 20
On normalizing
11.101 X 20 = 1.1101 X 21
On biasing exponent 1023 + 1 = 1024
(1024)10 = (10000000000)2
So 11 bit exponent = 10000000000
52 bit significand = 110100000000 …………. making total 52 bits
Setting sign bit = 1 (number is negative)
So, final representation
1 10000000000 110100000000 …………. making total 52 bits by adding further 0’s
Converting floating point into decimal
Let’s convert a FP number into decimal
1 01111100 11000000000000000000000
The decimal value of an IEEE number is given by the formula:
(1 -2s) * (1 + f) * 2( e – bias )
where Expandrive mac serial.
- s, f and e fields are taken as decimal here.
- (1 -2s) is 1 or -1, depending upon sign bit 0 and 1
- add an implicit 1 to the significand (fraction field f), as in formula
Again, the bias is either 127 or 1023, for single or double precision respectively.
First convert each individual field to decimal.
- The sign bit s is 1
- The e field contains 01111100 = (124)10
- The mantissa is 0.11000 … = (0.75)10
Putting these values in formula
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(1 – 2) * (1 + 0.75) * 2124 – 127 = ( – 1.75 * 2-3 ) = – 0.21875
This article has been contributed by Anuj Batham.
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