### Overview

The numbers that we use every day for things like prices, math, banking, how
many pizzas to get for a party, etc are in a system called **base10**, or
**decimal**. We call it that because each 'column', or place, is associated
with a power of 10. The system also has a **radix** of 10, meaning that each
column can hold 1 of 10 possible digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9.

ex:
`$20{3}_{10}$`

```
$203=(2\ast 1{0}^{2})+(0\ast 1{0}^{1})+(3\ast 1{0}^{0})$
$203=200+0+3$
```

The digit in each *place* is multiplied by a power of 10, and then when the
products are added together, the result is the number written in base10. The
`1's`

column is multiplied by
`$1{0}^{0}$`

, which is just
`1`

, the
`10's`

column is multiplied by
`$1{0}^{1}=10$`

, the
`100's`

column is multiplied
by
`$1{0}^{2}=100$`

, and so on. It's a different way of looking at the numbers
we use every day, but it hopefully gives you some insight into how and why they
are constructed the way that they are.

**Note:** *The subscript 10 is used to denote the base of the number.
We don't include this in every day math because it's assumed that unless
specified, numbers are already written in base10.*

### Numbers smaller than 1

It's a pretty straight-forward process for getting bigger and bigger numbers- just increase the power of 10 of the multiplier. But what about numbers smaller than 1?

Let's look at
`$0.3{1}_{10}$`

as an example.
Thinking in the context of number bases, we can see that the decimal point
actually specifies where our
`$1{0}^{0}$`

column is. When moving to the right
of the decimal point, the powers of 10 simply continue to decrease:

**ex.**
`$0.3{1}_{10}$`

```
$0.31=(0\ast 1{0}^{0})+(3\ast 1{0}^{-1})+(1\ast 1{0}^{-2})$
$0.31=0+0.3+.01$
```

Recall that:

```
$1{0}^{-1}=\frac{1}{1{0}^{1}}=\frac{1}{10}=0.1$
$1{0}^{-2}=\frac{1}{1{0}^{2}}=\frac{1}{100}=0.01$
$1{0}^{-3}=\frac{1}{1{0}^{3}}=\frac{1}{1000}=0.001$
```

If you've never experienced numbers in different bases before, hopefully this is some interesting insight into the number system we use every day. Now that we've figured out how our decimal system works as written, it should be an easy task to apply this to other number bases.