# Base introduction

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

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## Contents

## 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:

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