Converting 1/7 to Base 2: A Scientific Approach

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Discussion Overview

The discussion revolves around converting the fraction 1/7 into its binary representation. Participants explore various methods for calculating its infinite binary expansion, including long division and repeated multiplication, while also addressing potential misconceptions about binary fractions.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested, Mathematical reasoning

Main Points Raised

  • One participant asks how to compute the binary representation of 1/7, seeking clarity on the process.
  • Another suggests that the computation involves finding its infinite binary expansion using long division.
  • A participant provides a method involving repeated multiplication by 2, explaining how to determine the binary digits based on whether the product is less than or greater than 1.
  • One participant proposes an alternative method involving converting the decimal representation of 1/7 to binary after manipulating it by removing the decimal point, though they express uncertainty about its correctness.
  • Another participant challenges the validity of the proposed method and result, referencing a previous post that outlines the correct algorithm for determining the binary representation.
  • There is a discussion about the representation of binary fractions, with one participant questioning the equivalence of certain binary and decimal representations.

Areas of Agreement / Disagreement

Participants express differing views on the methods for converting 1/7 to binary, with some supporting traditional approaches while others propose alternative methods. There is no consensus on the correctness of all proposed methods, and some participants challenge the validity of others' claims.

Contextual Notes

Some methods discussed may depend on specific interpretations of binary fractions and the handling of decimal representations, which could lead to confusion. The discussion includes unresolved mathematical steps and assumptions regarding the conversion process.

zeebo17
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What is 1/7 in base 2? How would you solve for this?

Thanks!
 
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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.
 
As perhaps a hint, what is 1/2 in binary?
 
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/".
 
Last edited by a moderator:
zeebo17 said:
What is 1/7 in base 2? How would you solve for this?

Thanks!


wouldn't it be:
Code: 
1/111

But in decimal form it would be:
Code: 
0.100010111000001001

I'm not sure if it's the correct answer though. My logic was to take the result of 1/7:

Code: 
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?
 
Steve.pf said:
wouldn't it be:
Code: 
1/111
sure but not real interesting.

But in decimal form it would be:
Code: 
0.100010111000001001

I'm not sure if it's the correct answer though. My logic was to take the result of 1/7:

Code: 
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?

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.
 
Steve.pf said:
wouldn't it be:
Code: 
1/111

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

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