# What is 1/7 in base 2?

1. Dec 7, 2009

### zeebo17

What is 1/7 in base 2? How would you solve for this?

Thanks!

2. Dec 7, 2009

### Hurkyl

Staff Emeritus
The same as it is in any other base.

I assume you meant to ask how to compute its infinite binary expansion? Use long division.

3. Dec 7, 2009

### sjb-2812

As perhaps a hint, what is 1/2 in binary?

4. Dec 7, 2009

5. Dec 7, 2009

### DoctorBinary

Just multiply your fraction by 2 repeatedly, writing a 0 for products less than 1, and a 1 for products greater than 1 (for the latter, subtract out 1 before continuing with the multiplication). For example:

1/7 * 2 = 2/7
2/7 * 2 = 4/7
4/7 * 2 = 1 1/7
1/7 * 2 = ... (repeats)

The binary value is 0.(001), where () indicates the repeating part.

For a little more detailed explanation, see heading "dec2bin_f()" in my article "http://www.exploringbinary.com/base-conversion-in-php-using-bcmath/". [Broken]

Last edited by a moderator: May 4, 2017
6. Dec 8, 2009

### Steve.pf

wouldn't it be:
Code (Text):
1/111
But in decimal form it would be:
Code (Text):
0.100010111000001001
I'm not sure if it's the correct answer though. My logic was to take the result of 1/7:

Code (Text):
0.142857
remove the decimal (by multiplying it by 1,000,000) then converting that number to binary and then re-placing the decimal.

Is my method correct?

7. Dec 8, 2009

### Integral

Staff Emeritus
sure but not real interesting.

No your method is not correct nor is your result. See the post by DoctorBinary for the correct algorithm. Following his process the integer part of each computation yields a digit of the binary number.

8. Dec 8, 2009

### sjb-2812

Perhaps another way to look at it, what is 0.537 as a decimal fraction? 5/10 + 3/102 + 7/103, so what is 0.111 as a binary fraction? 1/2 + 1/22 + 1/23 = 0.87510

9. Dec 9, 2009

### HallsofIvy

But .111 is NOT the same as 1/111 in decimal or binary.