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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, <math>1011_3</math> 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 <math>\overline{abcd...z}</math> where <math>a,b,c,d,...,z</math> are digits of the number. This is not multiplication; this is referring to the digits of the number.

===Definition===
Formally, let <math>N</math> be some base 10 number and let <math>b</math> be some base. Let <math>(d_0,d_1,...,d_n)</math> be a list of numbers fulfilling:
* <math>0\leq d_i<b</math> for any <math>i</math> between 0 and <math>n</math>, inclusive.
* <math>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</math>

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

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

==How do we convert into a base?==

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

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

==How would we go about finding <math>N</math> from a list?==

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

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