Comparing Numbers in Different Bases

Date: 07/28/99 at 00:40:19
From: Agnes Morrison
Subject: Bases other than 10

If I am given a number in base 4 and asked whether it is bigger or 
smaller than another number in another base, should I always convert 
both to base 10 to compare, or is there an easier way?

Thank you.

Date: 07/28/99 at 12:41:18
From: Doctor Rick
Subject: Re: Bases other than 10

Hi, Agnes. 

If you know how to convert a number in the other base to base 4 
without converting it first to base 10, that _might_ be faster; you 
can compare any two numbers in the _same_ base the same way you 
compare numbers in base 10. But you'd have to know your base-4 
multiplication tables really well for that to be easy. I would stick 
with converting both numbers to base 10.

In some cases you might not have to do the whole conversion. Let's say 
I need to compare 331 (base 4) with 304 (base 5). I know that

     331 (base 4) < 1000 (base 4) = 64 (base 10)

I also know that

     304 (base 5) > 300 (base 5) = 3*25 = 75 (base 10)

Putting these facts together, I know that

     331 (base 4) < 64 (base 10) < 75 (base 10) < 304 (base 5)
so
     331 (base 4) < 304 (base 5)

Also, I don't have to do _any_ conversion to know that

     331 (base 5) > 304 (base 4)

Why? Because I know that

     331 (base 5) > 331 (base 4)

Any numeral in one base is always greater than the same numeral in a 
smaller base. If this isn't clear to you, write out the numbers in 
expanded form:

     331 (base 5) = 3*25 + 3*5 + 1

     331 (base 4) = 3*16 + 3*4 + 1

Once you see this, it's obvious that

     331 (base 4) > 304 (base 4)

Putting the facts together, I know that

     331 (base 5) > 331 (base 4) > 304 (base 4)
so
     331 (base 5) > 304 (base 4)

Does that help? These "shortcut" observations will help sometimes, but 
converting both numbers to base 10 will always give you the answer.

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/