Convert 0.1001(repeating) to Decimal

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Homework Help Overview

The discussion revolves around converting the repeating binary fraction 0.1001 (repeating) into its decimal equivalent. Participants are exploring the methods for handling recurring patterns in binary numbers and how these relate to mathematical conversions.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • One participant expresses uncertainty about converting a repeating binary fraction to decimal and questions whether this is a computing or mathematical problem. Another participant suggests a method similar to converting repeating decimal fractions to rational numbers, involving multiplication by powers of 2. There is also a discussion about the implications of base differences in the calculations.

Discussion Status

The discussion is active, with participants offering insights and methods for approaching the problem. There is a recognition of the complexity introduced by different bases, and some participants are clarifying their understanding of the conversion process. No explicit consensus has been reached, but guidance has been provided regarding the approach to take.

Contextual Notes

Participants are navigating the challenge of converting a repeating binary fraction, with some noting confusion regarding the relationship between binary and decimal representations. There is mention of notes that provide an answer, which may not align with the calculations being discussed.

SherlockOhms
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Homework Statement


Not so much a problem as a general wondering. I know how to convert from decimal to binary and binary to decimal apart from when there's a recurring pattern involved. One which I really can't figure out it 0.1001 (repeating). How do you go about converting that to decimal?

Homework Equations


None really.

The Attempt at a Solution


I really don't have any attempt as I can't find much about it on the internet. It features in a worked example in our notes and it's never explained. The answer is 0.6 apparently. (Not sure if this would be considered a computing problem or a mathematical one)
 
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SherlockOhms said:

Homework Statement


Not so much a problem as a general wondering. I know how to convert from decimal to binary and binary to decimal apart from when there's a recurring pattern involved. One which I really can't figure out it 0.1001 (repeating). How do you go about converting that to decimal?

Homework Equations


None really.

The Attempt at a Solution


I really don't have any attempt as I can't find much about it on the internet. It features in a worked example in our notes and it's never explained. The answer is 0.6 apparently. (Not sure if this would be considered a computing problem or a mathematical one)

The trick here is similar to what you do to convert a repeating decimal fraction to a rational number. For example, if you need to convert .35353535... to a fraction, you multiply by a power of 10 large enough to move the part that repeats to the other side of the decimal point, which in this case would be 102. If we write x = .353535..., then 100x = 35.353535...

Subtracting the equation with x from the equation with 100x, we get
99x = 35 ==> x = 35/99

For your binary fraction, write an equation x = .10011001...2. Get a new equation that you can subtract this one from by multiplying by 24 (or 16).
 
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Won't you then end up with 15x = 1001, and that equals 66.733... and not 3/5 as it says in my notes. Is there a mistake in the notes or am I misunderstanding you?
 
No, you won't. In my work I was careful to note that the fraction was in base-2 (something you neglected to do). You should have gotten 15x = 10012.

This is a bit ungainly, having a decimal number on one side, and a binary number on the other, but at least it's marked to indicate that different bases are being used.
 
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Got it! Thanks a million.
 

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