How do I convert a repeating decimal from one base to another?

[srfriggen]

Homework Statement

1. Convert (0.333...)₄to base 10.

2. Convert (0.333...)₁₀to base 4.

Homework Equations

For question 1:


I see this can be written as (3/4+3/4²+3/4³...)₁₀

Can I just use the geometric formula and arrive at the answer 12/3=4 ?

But that doesn't match, because 4 in base 10, written in base 4 is just 10.




For problem 2 I'm not sure how to start. There was a trick regarding division and keeping the remainder, but I'm not sure that applies to decimals.

The Attempt at a Solution

 

[tiny-tim]

hi srfriggen!

I see this can be written as (3/4+3/4²+3/4³...)₁₀

Can I just use the geometric formula and arrive at the answer 12/3=4 ?

yes, but doesn't your formula apply to 1 + 3/4 + … ?

(btw, doesn't that look a lot to you like 0.9999… ?)

For problem 2 I'm not sure how to start. There was a trick regarding division and keeping the remainder, but I'm not sure that applies to decimals.

you mean to quaternary? (quaternals?)

yes, long division works in any system ​

 

[srfriggen]

the series can be written as 3*10^-1+3*10^-2+3*10^-3...

And isn't the formula for a geometric series: a(1/1-r)?

So here, a=3 and r=1/4, right?

When I plug in and multiply out I get an answer or 4, not .9...

What am I missing?

 

[tiny-tim]

3*1 + 3*10^-1+3*10^-2+3*10^-3...= a(1/1-r)

 

[srfriggen]

Wait, I wrote that out wrong, it's not:

he series can be written as 3*10-1+3*10-2+3*10-3...

those 10s should be replaced with 4s, right?


I'm still getting a=3 and r=1/4. Am I right with that?

 

[tiny-tim]

a*(1 + b + b² + b³) + … = a/(1 - b)

you left out the a*1 (= 3), which if you subtract from 4 is …?

 

[srfriggen]

so isn't the formula a=3 and b=1/4? If so that still give the answer or 4. I don't see where subtracting 3 comes into play.

 

[tiny-tim]

your formula does not have an a*1 …

it starts with a*b ​

 

[srfriggen]

I'm looking at my book now and it says the formula is a(1/1-r). If a=3 and r=1/4, I get the answer to 4. I'm not sure what is wrong of the previous statement? Either my formula is wrong or my a and r choices are wrong.

 

[srfriggen]

I'm sorry tiny tim I'm just not getting it today. I'm going to take a break, so some studying later, and come back to this one. Thanks for your patience so far!

 

[tiny-tim]

a*(1 + b + b² + b³ + …) = a/(1 - b) = 3/(1 - 1/4) = 4

a*(b + b² + b³ + …) = a/(1 - b) - a = ab/(1 - b) = 4 - 3 = 1

 

[srfriggen]

ahhh, now I see what you meant by "your formula" !

 

[tiny-tim]

ok, alternative method: can you see that 0.33333 = 1 - 0.00000… ?

(ie you can't get any closer to 1 ! )