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

In summary, To convert (0.333...)4 to base 10, the geometric formula can be used with a=3 and r=1/4, resulting in an answer of 4. However, this does not match the expected answer of 10 in base 4. For converting (0.333...)10 to base 4, the formula a*(b + b2 + b3 + ...) = a/(1 - b) - a can be used, resulting in an answer of 1. Another method is to recognize that 0.33333 = 1 - 0.00000..., meaning it can't get any closer to 1.
  • #1
srfriggen
306
5

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


 
Physics news on Phys.org
  • #2
hi srfriggen! :smile:
srfriggen said:
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 + … ? :wink:

(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 :smile:​
 
  • #3
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?
 
  • #4
3*1 + 3*10-1+3*10-2+3*10-3...= a(1/1-r) :wink:
 
  • #5
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?
 
  • #6
a*(1 + b + b2 + b3) + … = a/(1 - b)

you left out the a*1 (= 3), which if you subtract from 4 is …? :wink:
 
  • #7
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.
 
  • #8
your formula does not have an a*1 …

it starts with a*b :smile:​
 
  • #9
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.
 
  • #10
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!
 
  • #11
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
 
  • #12
ahhh, now I see what you meant by "your formula" !
 
  • #13
ok, alternative method: can you see that 0.33333 = 1 - 0.00000… ?

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

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

1. What are change of base calculations?

Change of base calculations are mathematical calculations used to convert numbers from one given base to another. This is commonly seen in logarithmic calculations, where the base of a logarithm needs to be changed in order to simplify the calculation.

2. Why do we need to change the base in calculations?

Changing the base in calculations can make the problem easier to solve. In logarithmic calculations, changing the base can eliminate the use of logarithms altogether and make the calculation simpler.

3. How do you perform a change of base calculation?

To perform a change of base calculation, you can use the change of base formula, which is logb(x) = loga(x) / loga(b). This formula allows you to convert a logarithm with a different base into a logarithm with a base that is easier to work with.

4. What are some common bases used in change of base calculations?

The most commonly used bases in change of base calculations are 10, e (or Euler's number), and 2. These bases are often used because they have special properties that make calculations easier.

5. Can change of base calculations be used in other types of math problems?

Yes, change of base calculations can be used in a variety of mathematical problems, not just logarithmic calculations. They can also be used in trigonometry, algebra, and even calculus to simplify calculations and make them more manageable.

Similar threads

How does one convert a decimal float to binary using the multiply by 2 method?
    • Computing and Technology
Replies
4
Views
971
How can I convert directly between two non-power-of-ten bases?
    • General Math
Replies
2
Views
2K
How do I convert a number with a decimal point to binary?
    • Precalculus Mathematics Homework Help
Replies
3
Views
4K
B How to convert a fraction into a repeating decimal
    • General Math
Replies
9
Views
2K
How Do You Convert 212 from Decimal to Base-9?
    • Precalculus Mathematics Homework Help
Replies
4
Views
7K
Convert repeating decimal to improper fraction
    • Precalculus Mathematics Homework Help
Replies
10
Views
7K
B Calculating Odds of Getting at Least One Pair in Pai Gow
    • Set Theory, Logic, Probability, Statistics
Replies
18
Views
2K
Thought experiment: How would ocean levels change?
    • Advanced Physics Homework Help
Replies
32
Views
2K
Rational Representation of a Repeating Decimal
    • Precalculus Mathematics Homework Help
Replies
6
Views
9K
How Do I Design an EMI Filter for a Converter with Only Voltage and Power Specs?
    • Electrical Engineering
Replies
8
Views
1K

Hot Threads

Recent Insights

Back
Top