decimal fractions

[sharooz]

AoA, Sir, plz explain why some decimal fractions cannot be finitely represented in binary system.

[zaasmi]

@engr-ali said in decimal fractions:

Assallamulikum Sir, Conversion (59)10 into binary 59/2=28-1 ? 28/2=14-0 14/2=7-0 7/2=3-1 3/2=1-1 (111001) ?

Dear @Engr-Ali ,

$(59){10} = (111011){2}$

The mistake you have made is in the first step where you divided $59/2$. It has to be $29-1$ instead of $28-1$.

Best wishes

[Engr Ali]

Assallamulikum Sir, Conversion (59)10 into binary 59/2=28-1 ? 28/2=14-0 14/2=7-0 7/2=3-1 3/2=1-1 (111001) ?

[zaasmi]

@farrukh said in decimal fractions:

sir i am good understand the Binary equivalent of the decimal fraction but binary to Hexadecimal change it??

Dear student,

It’s good to know that you understand conversion from binary to decimal. Now for conversion from binary to hexadecimal, you can use indirect method (recommended) or a direct method.

In indirect method, you will first convert binary number to decimal number and then the decimal number to hexadecimal number.

In direct method, you will consider the binary number in a group of 4 digits start from the right-most digit and then for each 4-digit binary you will write its equivalent hexadecimal using a conversion table. The following webpage will help you further Lear more

Best wishes

[Farrukh]

sir i am good understand the Binary equivalent of the decimal fraction but binary to Hexadecimal change it??

[zaasmi]

@zareen said in decimal fractions:

Aoa sir why we convert binary into decimal by negative signs on digits??

The example which u have studied in lecture, there is a typo mistake. Now, i have explained two examples more to understand this concept. I hope u will understand this idea now. See the attached file.

[zareen]

Aoa sir why we convert binary into decimal by negative signs on digits??

[zaasmi]

@waseem-bai said in decimal fractions:

Dear sir A. O. A I can’t understand millinnum bug digit change

Millennium Bug:

It is a problem in the coding of computerized systems that was projected to create havoc in computers and computer networks around the world at the beginning of the year 2000.

[Waseem Bai]

Dear sir A. O. A I can’t understand millinnum bug digit change

[zaasmi]

@sharooz said in decimal fractions:

AoA, Sir, plz explain why some decimal fractions cannot be finitely represented in binary system.

Dear @sharooz ,

Decimal fractions are either converted to a finit binary or an infinite binary. For example convert 0.375 to binary.

step 1: .375 * 2 = 0.75 (we record 0 and use the leftover .75 in the next step)

step 2: .75 * 2 = 1.5 (we record 1 and use the leftover .5 in the next step)

step 3: .5 * 2 = 1.0 (we record 1 and use the leftover is .0 so this becomes the last step)

So, .375 in binary is .011. This is the case of conversion to a finite binary. Try another example 0.3125.

Now you can look at the example .7625 in the handouts.

In step 5, you record 0 and the left over is .4. You continue but in step 9, things repeat, that is again you record 0 and the left over is .4.

So, this procedure will continue. The digits written in bracket (0011) are those which will repeat continously. This is an example of conversion to infinite binary.

Now the question is when the decimal fraction reduces to finite binary and when to infinite binary. Remove the decimal point, change it to a fraction and start simplifying. When reduced to the simplest form the one in which the denominator is power of base 2 converts to a finit binary. The one in which the denominator cannot be reduced to power of base 2 converts to an infinite binary. Please check this procedure with a) .3125 and b) .7625

Best wishes