Ternary Expansion of x ∈ [0,1] - Tips & Tricks

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SUMMARY

The discussion focuses on the challenges of understanding ternary expansions, specifically for values of x in the range [0,1]. Participants explore how to find the ternary expansion of the fraction 5/27, emphasizing the need to identify coefficients a_k in the equation 5/27 = ∑ a_k / 3^k. The conversation highlights the similarity between ternary and decimal expansions while addressing common pitfalls in conversion. A specific example provided is the conversion of 5 in base 3, which is represented as 12_5.

PREREQUISITES
  • Understanding of base conversions, particularly between decimal and ternary systems.
  • Familiarity with fractions and their representation in different bases.
  • Knowledge of series and summation notation, specifically in the context of expansions.
  • Basic mathematical skills to manipulate fractions and powers of 3.
NEXT STEPS
  • Research the method for converting fractions to ternary expansions.
  • Learn about the properties of base 3 and its coefficients in expansions.
  • Study examples of ternary expansions for various fractions to gain practical insights.
  • Explore mathematical series and their applications in base conversions.
USEFUL FOR

Students, mathematicians, and educators interested in number theory, particularly those seeking to understand ternary expansions and their applications in mathematical analysis.

jakey
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Hi guys, I'd like to ask about ternary expansions. they seem easy but I'm having a hard time doing this as well as searching for tips online, specifically for x \in [0,1].

I know that ternary expansions are similar to decimal expansions but for example, how do you find the ternary expansion of \frac{5}{3^3} ?
 
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In base 3, 5= 3+2 is 12_5. Of course, 9= 3^2 is 200_3 so the fraction 5/9, in base 3, would be the fraction 12/200. Can you write that in "trinary"?
 
Hi hallsofivy, sorry I didnt quite understand your reply.

And oh, it's 5/27 not 5/9. Perhaps a better way of explaining it to me would be how to identify the coefficients a_k such that \frac{5}{27} = \sum \frac{a_k}{3^k} ?
 

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