How To Do Direct Conversion From One Number Base To A New Number Base

Question

I want to understand direct conversion from one number base to a different number base. What are the steps that work for any number base to another number base? What I am looking for is not a short cut method which works for say, binary to octal or hexa-decimal number system. It can be anything like converting number base-22 to base-37.

I don't want any explanation that converts to base-10 first & then do the conversion to the destination number base. I already understand it. The direct conversion steps (if they exist) with a short explanation would be really helpful. I am really trying to get my head around it.

Answer 1

It is pretty heavy topic. The steps do exist. You can do conversion between any two number systems. But you will be doing arithmetic calculation in a different number base. That itself is pretty hard to get your head around it.

I can quote the general concept behind direct base conversion from a W3 Spot article:

1. Divide the number by the value of other base.

2. Get the remainder. That is the least significant digit.

3. Divide the quotient with the value of other base.

4. Add the remainder as next least significant digit.

We need to keep doing steps 3 & 4 repeatedly till quotient becomes zero.

But that's not all. There are subtle nuances when you convert from lower to higher base than the opposite way. Let's say, you are doing base-3 to base-6 conversion. You will have to do some arithmetic operation on the base-3 number. And all the calculation will be done in base-3 number system. How do we convert to base-6 number then? base-3 system doesn't have 3, 4 & 5 digits that are valid in base-6 system. You need to have a mapping first between base-3 digits & base-6 digits.

Also doing division in non-decimal number system is not easy. If you do it using pen & paper, you should create multiplication table in that number system.

All these things are not easy. That's why we convert to decimal system first. Anyways just go through the article I mentioned above. It explains the steps in details with examples for both lower to higher base & the opposite.