# Convert decimal base to binary

1. Oct 2, 2010

### Trentonx

1. The problem statement, all variables and given/known data
Convert 33.9 to binary.

2. Relevant equations
Division by two and remainders

3. The attempt at a solution
I'm unsure since it is .9. Using the dividing by two method with the remainders wouldn't account for that. Could I do something like this?
33.9 = 339 *10^-1, convert to binary (101010011) and then account for the scientific notation in some way?

2. Oct 2, 2010

### NobodySpecial

think about the 'divide by two' method with 1/2, 1/4,1/8 etc

3. Oct 2, 2010

### vela

Staff Emeritus
It's easiest if you just do the integer part and fractional part separately. Convert the integer part as usual. For the fractional part, multiply by 2, pick off the integer part, and repeat the process with the new fractional part. For example, for 13/16=0.8125=0.11012, you'd do

0.8125 x 2 = 1.625
0.625 x 2 = 1.25
0.25 x 2 = 0.5
0.5 x 2 = 1.0

4. Oct 3, 2010

### DoctorBinary

You could proceed this way, though it would be cumbersome; it requires long division in binary. 339 = 101010011, and 10 = 1010 (you know how to convert to integers so those two conversions are straightforward). Then do the division: 101010011/1010 (Sorry, I tried to put the long division here but the formatting didn't work.) In any case, you'd find that 33.9 has an infinitely repeating binary equivalent, which is 100001.1(1100) (the part in parentheses is the repeating part).

But @vela's way is the way I recommend doing it.

This article, with more details on the algorithm: http://www.exploringbinary.com/base-conversion-in-php-using-bcmath/ (see section labeled "dec2bin_f())"

5. Oct 3, 2010

### Trentonx

Except for the bit where the binary is infinite, that's not too bad. Thanks for the help. On a related note does vela's way work to convert to octal and hexadecimal the same way?

6. Oct 3, 2010

### vela

Staff Emeritus
That's just like the fact that the decimal representation of 1/3=0.33333... requires an infinite number of 3s.
Yes, as long as you realize you're multiplying by the base you're using. Say a number has the base-b representation $(0.d_1d_2d_3d_4\cdots)_b$. When you multiply by b, you shift each digit to the left, so you get $(d_1.d_2d_3d_4\cdots)_b$, so the integer part is the first digit in the representation. You can get the subsequent digits by discarding the integer part and repeating the process.

7. Oct 4, 2010

### DoctorBinary

Actually, you're multiplying by the base you're converting to. In this case, we're multiplying a decimal number by 2 to convert to binary. To convert to hex, you would multiply the decimal number by 16; to convert to octal, you would multiply the decimal number by 8.