Generalising place-value numbers to polynomials

“Place-value” numbers, also known as Hindu-Arabic numbers, are the ‘normal’ way we write down numbers, e.g. $123$ to represent “a hundred and twenty three” (AKA ‘CXXIII’ in roman numerals).

Place-value notation is usually taught very early on, since it’s such a fundamental part of how we do mathematics. That’s good, but it also means we’re not very mathematically sophisticated when learning it: we might just rote-learn some simple rules (and hopefully gain some intuition over time); once we are more sophisticated, e.g. at high school, we may be so used to place-value numbers that we never think to re-visit those rules, and the underlying theory.

Simple rules

We may have learned place-value notation via rules like the following:

The right-most digit counts how many “ones” (or “units”) we have
The left neighbour of a digit counts ten times more

Hence the digits, from right-to-left, count “ones”, “tens”, “hundreds”, “thousands”, etc.

TODO: LINK TO LOGARITHMIC NAMING

Our example of $123$ contains $3$ ones, $2$ tens and $1$ hundred. The overall number is the sum of these parts, so $3 + 20 + 100 = 123$ .

Note that the ‘ten times more’ rule ensures we get zeros on the right, which makes the final sum pretty easy.

More sophisticated rules

Once we’ve spent a few years becoming comfortable with arithmetic, we can re-visit the rules of place-value notation. In particular, we can make the above rules more precise as powers of a base.

The “base” is the multiplier between adjacent digits; decimal numbers are “base ten”, but other bases are possible (and some are actually preferable! TODO: LINK). Note that I’ll be using the word “digit” regardless of base (rather than “bit”, “octet”, “dozit”, etc.).

The next key insight is that multiplying the base over and over (for each extra digit going left) is the same as the power operation in arithmetic:

$10^{0} = 1$

$10^{1} = 10 \times 1 = 10$

$10^{2} = 10 \times 10 \times 1 = 100$

For the above example:

123 = 1 \times 10^{2} + 2 \times 10^{1} + 3 \times 10^{0} = 1 \times 100 + 2 \times 10 + 3 \times 1

We will keep things general by referring to our base using a variable $b$ .

Columns raise the base to increasing powers ** We tend to use base 10 *** Link to units post section on bases ** Power of 1 is just the number as-is ** Power of 0 is just the number 1 (except for 0, which doesn’t work as a base anyway) ** Decimals raise the base to decreasing powers
We can extend columns to the left, e.g. ‘123’ is normally interpreted as ‘1 * 10^2 + 2 * 10^1 + 3 * 10^0’, but can also be interpreted as ‘1 * 10^2 + 23 & 10^0’ or ‘123 * 10^0’
We can extend this interpretation to negative numbers too ** Link to negatives with bar notation