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. 123123 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:

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

TODO: LINK TO LOGARITHMIC NAMING

Our example of 123123 contains 33 ones, 22 tens and 11 hundred. The overall number is the sum of these parts, so 3+20+100=123 plus320100123.

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:

100=1 100 1

101=10×1=10 101 times10110

102=10×10×1=100 102 times10101100

For the above example:

123=1×102+2×101+3×100=1×100+2×10+3×1 123plustimes1 102times2 101times3 100plustimes1100times210times31

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