Base introduction

Bases. These confuse a lot of us. Hopefully this will make it better.

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Overview

A base is a way of representing a number. A number written in another base has a subscript indicating the base: for instance, is the number whose digits are 1011 in base 3. In this article, every number in another base has a subscript. Every number in base 10 (our normal numbers) does not have a subscript.

Convention

A list is, obviously, a list of numbers. It may contain duplicates, and it has an order: the first element, the second element, etc. for instance (1,3,5) is a list.

A number may be represented by where are digits of the number. This is not multiplication; this is referring to the digits of the number.

Definition

Formally, let be some base 10 number and let be some base. Let be a list of numbers fulfilling:

• for any between 0 and , inclusive.
• 

Then, the number representing in base is just the number . In other words, the number in base has digits from right to left.

For instance, . The list (1,1,0,1) fulfills the two conditions for and ; thus the number representing in base is .

How do we convert into a base?

One way to generate a list of numbers fulfilling those two conditions is this:

• Put as the only member of the list: (N)
• Now, if any of the members of the list are greater than or equal to , subtract from that member and add 1 to the next member of the list on the right.
• Repeat while possible.

Your list now satisfies the two conditions.

For instance, 31 into base 3: (31) (28,1) (25,2) ...skipping some steps... (1,10) (1,7,1) (1,4,2) (1,1,3) (1,1,0,1)

Now we're done and our list satisfies the two conditions.

The process of creating this list, then putting these as digits in a number is called 'converting into base .'

How would we go about finding from a list?

Note that we know that . Thus, we can just perform these multiplications and additions.

The process of taking the digits of the number to the list, and then performing these actions, is called 'converting to base 10'

Math in a base that isn't our normal base

This is what confuses the most people.

First, it's helpful to make a multiplication table. For instance, base 4:

\[ \begin{tabular}{r|c|c|c|c|} &0_4&1_4&2_4&3_4\\\hline 0_4&0_4&0_4&0_4&0_4\\ 1_4&0_4&1_4&2_4&3_4\\ 2_4&0_4&2_4&10_4&12_4\\ 3_4&0_4&3_4&12_4&21_4 \end{tabular} \]