Question 894634: Express 0.312 in base 10 as a binary number.
2.)convert the following in base ten to decimal
a.) (57/64)ten
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! .312 = 312/1000
312 = 100111000 in binary.
1000 = 1111101000
.312 = 312/1000 = 100111000 / 1111101000
dividing in binary, you get:
100111000 ÷ 11111010 = 1.001111110111110011
i did not do this by hand.
i used a binary calculator.
the process is not very complex but very cumbersome and highly prone to errors and is not recommended to attempt by hand for anything but the simplest numbers.
i'll try a simpler number that should show you that the process is correct.
.25 = 25/100 = 11001 / 1100100
we'll use the calculator again to get:
11001 ÷ 1100100 = 0.01
.01 in binary is 0 / 2^1 + 1 / 2^2 which is equal to 0 + 1/4 which is equal to 1/4 which is equal to .25
binary number go like this:
going left from the decimal point, you have:
2^0
2^1
2^2
2^3
2^4
....
going right from the decimal point you have:
1/2^1
1/2^2
1/2^3
1/2^4
....
the binary division calculator can be found at:
http://www.miniwebtool.com/binary-calculator/?number1=11001&operate=4&number2=1100100
if you want to learn how to do the calculations manually, then there's some links that will show you how to do it.
some of them are:
http://academic.evergreen.edu/projects/biophysics/technotes/misc/bin_math.htm
http://www.exploringbinary.com/binary-division/
http://www.binarymath.info/multiplication-division.php
it is obvious that the number you asked for in binary was very complicated and certainly not desirable to perform by hand.
the calculator came in quite handy for that even though we had to convert .312 into 312 / 1000 and then convert that to binary to get 100111000 / 1111101000 and then plugging that into the calculator.
the second part of your question appears to be easier.
i used my calculator to get 57/64 = .890625
that could also have been done by hand in a similar manner that the binary division is done by hand, but why go through all that if you don't have to?
bottom line is that binary division is not much different then decimal division.
the process if the same, but the base of the number system you are dealing with is different and you have to account for that.
i couldn't find a reference that talked about converting directly from decimal to binary when the decimal number was a fraction that was converted to a decimal.
perhaps there's a way.
i just couldn't find it.
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