0.101010... in base 3?

Number Theory Level pending

In base ten, ( 0. 10 ‾ ) = 10 99 (0.\overline{10}) =\frac{10}{99} .

What about in base three?

What is ( 0. 10 ‾ ) 3 (0.\overline{10})_3 ?

Give your answer to 3 decimal places in base 10.

Notation:

  • 0. 10 ‾ = 0.10101010101010 … 0.\overline{10} = 0.10101010101010\ldots .
  • The subscripts indicate the base.


The answer is 0.375.

Geoff Pilling by Geoff Pilling , 53, USA

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1 solution

Geoff Pilling
Jan 12, 2017

Here are two ways to approach this one:

1) ( 0.101010... ) 3 = 1 3 1 + 1 3 3 + 1 3 5 + . . . = 3 8 = 0.375 (0.101010...)_3 = \frac{1}{3^1} + \frac{1}{3^3} + \frac{1}{3^5} + ... = \frac{3}{8} = \boxed{0.375}

2) Let x = ( 0.101010... ) 3 x = (0.101010...)_3

  • 10 0 3 x = ( 10.101010... ) 3 100_3 x = (10.101010...)_3
  • 10 0 3 x − x 3 = 1 0 3 100_3 x - x_3 = 10_3
  • 2 2 3 x = 1 0 3 22_3 x = 10_3
  • x = 1 0 3 2 2 3 = 3 10 8 10 = 0.375 x = \frac{10_3}{22_3} = \frac{3_{10}}{8_{10}} = \boxed{0.375}
2 Helpful 0 Interesting 0 Brilliant 0 Confused

Why u place 3 in the place of 10 in the infinite G.P. series.???

My approach was first convert the 10 in to base 3 i.e. 101 and then apply this infinite G.P. series.

Can u please clear my doubts.

Kushal Bose - 4 years, 4 months ago

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The subscript indicates the base.

Geoff Pilling - 4 years, 4 months ago

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Ohh I missed out that

Kushal Bose - 4 years, 4 months ago

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@Kushal Bose – Good point... I'll add it to the notation section.

Geoff Pilling - 4 years, 4 months ago

Totally misread the question,Almost failed by trying to find base 3 representation of ( 0. 10 ‾ ) 10 (0.\overline{10})_{10} that seems harder though.

Anirudh Sreekumar - 4 years, 4 months ago

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