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Converting from Base 10 to Base 3


Date: 02/12/2002 at 10:27:50
From: Ian Mahoney
Subject: Base 3

How do you convert a base 10 number (example 2315) to base 3?


Date: 02/12/2002 at 12:43:14
From: Doctor Ian
Subject: Re: Base 3

Hi Ian,

I'll show you how to convert it to base 5, and then you can follow the 
same steps to convert to base 3, okay?

In base five, we're going to end up with something like 

  abcd (base 5)

which _means_

  a*(5^3) + b*(5^2) + c*(5^1) + d*(5^0)

in exactly the same way that 

  abcd (base 10)

means

  a*(10^3) + b*(10^2) + c*(10^1) + d*(10^0)

I don't know that there will be 4 digits. There might be more. In 
fact, to find out how many there will be, I need to start cranking out 
powers of 5:

  5^0 =    1
  5^2 =   25
  5^3 =  125
  5^4 =  625
  5^5 = 3125

Now, 3125 is greater than the number I'm converting, so the 
coefficient of 5^5 and all larger powers is going to be zero.  
Therefore, I don't have to worry about them. 

So I need to know:  How many times does 5^4 go into 2315?

  2315 = 3*625 + 440

So we know that the answer is going to be

  3---- (base 5)

Now we do the same thing with the remainder, 440:

  440 = 3*125 + 65

So now we have

  33--- (base 5)

And so on:

  65 = 2*25 + 15         ->     332-- (base 5)

  15 = 3*5 + 0           ->     3323- (base 5)
  
   0 = 0*1               ->     33230

It would be nice if there were an easier way to do this kind of thing, 
but there really isn't.  

Can you do the same thing in base 3 instead of base 5? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Number Theory

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