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Number theory bases problem

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683
4
avatar+55 

For each part of this problem, convert the fraction to a "decimal" in the specified base.

(a) Convert the fraction 1/2 to base 3

(b) Convert the fraction 1/7 to base 8

(c) Convert the fraction 1/(b-1) to base b

 

Thank you in advance!  :)

 Jun 19, 2020

4+0 Answers

 #1
avatar+9675 
+2

Because 13+132+133+⋯=131−13=12,

 

12=0.¯13

 

By a similar approach,

 

17=0.¯18

and 

1b−1=0.¯1b

 Jun 19, 2020
 #2
avatar+55 
+2

Thank you for replying!  I do not really know how you got the first equation, I might be missing something really simple, but if you have the time, could you explain it to me? Thanks!

ConfuzzledKitten  Jun 19, 2020
 #3
avatar+9675 
+2

It is the sum of geometric series.

 

It looks like this:

For -1 < r < 1, and an arbitrary a,

a+ar+ar2+ar3+⋯+arn+⋯=a1−r

MaxWong  Jun 19, 2020
 #4
avatar+55 
+2

Thank you!  I searched up a proof on Google and I think I understand now! :-)

 Jun 19, 2020

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