signed , unsigned and fractional binary numbers

8/2/2019 Signed , Unsigned and Fractional Binary Numbers

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R E P O R T O F :

E Z E K I E L A . D E L E O N

B S E C E – 0 3

Signed , Unsigned and

Fractional Binary Numbers



Signed and Unsigned Binary Numbers

The value of a decimal number is based on threefactors.

The type of symbol used

The position of the symbol with respect to a fixedreference (decimal point).

The use of zero.




Example:

Based on the three factors listed earlier, let's look at what the decimal number 702 represents.

7 0 2 .

The digit "2" in the ones positional

value column represent the absolute value of 2.

Decimal

OnesTensHundreds

The "0" in the tens positional value column

means there are no tens, so the absolute value is 0.

The digit "7" in the hundreds

positional column means there areseven hundreds.

therefore, the absolute value of all

three digits is 702.




An Unsigned binary number, this form of binary representation is known as unsigned integers

because it can only represent integers that are positive.

When the numeric representation is also allowed toinclude negative integers, its is known as signed integers. The positive values are stored as simple

binary numbers, but negative numbers are stored intwo complement form.



Things to remember:

For signed integers that use two’scomplement notation, the MSB isassigned the role of indicating the signof the number. An MSB of 1 means that

the number is negative; an MSB of 0means that the number is positive (orzero).





The MSB in this case is approximately termed thesign bit.

The procedure for determining the

decimal equivalent of a two’scomplement binary number is exactly the same as the one we have used for

unsigned integer, except that the sign of the MSB is negative.




b3 b2 b1 b0

MSB LSB

1 0 0 1

MSB LSB

= b 3∙2 3 + b2 ∙22 + b1∙21 + b0∙20

= 1∙2 3 + 0 ∙22 + 0∙21 + 1∙20 = 8 + 1 = 9

A 4- bit binary number where b0 is

the least significant bit (LSB) and b 3 isthe most significant

bit (MSB)

-23

22

21

20

Sign bit:0 = positive number1 = negative number

= b 3∙( -2 3 )+ b2 ∙22 + b1∙21 + b0∙20

= 1∙( -2 3 ) + b2 ∙22 + b1∙21 + b0∙20

= -8 + 1 = -7

Calculating the decimal equivalent of a 4- bit binary number in unsigned integerformat

Calculating the decimal equivalent of a 4-bitbinary number in signed integer format usingtwo’s complement. Observe the sign of the MSB is

negative




In Fractional Binary numbers, we must move the binary point toward the left so that some (or all) of the bits are to the right of the binary point.

b7 b6 b 5 b4 b 3 b2 b1 b0

. Assumedlocation of binary point

Signed integers: -27 26 25 24 23 22 21 20

Unsigned integers: -27

26

25

24

23

22

21

20

Location of binary point for an 8- bit binary number in two’s complementsigned integer and unsigned integer formats.




.Location of binary point

b7 b6 b 5 b4 b 3 b2 b1 b0

MSB LSB

b7 b6 b 5 b4 b 3 b2 b1 b0

MSB LSB

.Location of binary point

Sign bit

Location of binary point for an 8-bit binary number. The upper part shows the unsignedfraction format. The lower part shows the

signed fraction format.




.Location of binary point(unsigned fraction)

b7 b6 b 5 b4 b 3 b2 b1 b0

MSB LSB

b7 b6 b 5 b4 b 3 b2 b1 b0

MSB LSB

.Location of binary point(signed fraction)

2-1 2-2 2-3 2-4 2-5 2-6 2-7 2-8

0.5 0.25 0.125 0.0068125 0.00390625

0.5 0.250.015625 0.0078125

-20 2-1 2-2 2-3 2-4 2-5 2-6 2-7

1

To determine thedecimal equivalent of a binary fraction,

multipl the binary fraction, multiply the binary digit that is n places to the right of the decimal point by

2

-n

, or 1/2

n

. Theupper part shows theunsigned fraction.The lower part showsthe signed fractionformat.

signed , unsigned and fractional binary numbers

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