Base introduction

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

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A base is a way of representing a number. A number written in another base has a subscript indicating the base: for instance, $1011_3$ 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.


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 $\overline{abcd...z}$ where $a,b,c,d,...,z$ are digits of the number. This is not multiplication; this is referring to the digits of the number.


Formally, let $N$ be some base 10 number and let $b$ be some base. Let $(d_0,d_1,...,d_n)$ be a list of numbers fulfilling:

  • $0\leq d_i<b$ for any $i$ between 0 and $n$, inclusive.
  • $d_0\cdot b^0 + d_1 \cdot b^1 + d_2 \cdot b^2 + d_3 \cdot b^3 + \cdots + d_n \cdot b^n=N$

Then, the number representing $N$ in base $b$ is just the number $\overline{d_n\cdots{}d_3d_2d_1d_0}_b$. In other words, the number in base $b$ has digits $d_0,d_1,d_2,...,d_n$ from right to left.

For instance, $1011_3=31$. The list (1,1,0,1) fulfills the two conditions for $b=3$ and $N=31$; thus the number representing $N$ in base $b$ is $1011_3$.

How do we convert into a base?

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

  • Put $N$ as the only member of the list: (N)
  • Now, if any of the members of the list are greater than or equal to $b$, subtract $b$ 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 $\overline{d_n\cdots{}d_2d_1d_0}_b$ is called 'converting $N$ into base $b$.'

How would we go about finding $N$ from a list?

Note that we know that $d_0\cdot b^0+d_1\cdot b^1+\cdots+d_n\cdot b^n=N$. 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 $\overline{d_n\cdots{}d_2d_1d_0}_b$ 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:

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

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