Tens and Ones, Up to 120

Position gives each digit its worth. In 25, the 2 works the tens job and earns twenty; in 52, the same 2 works the ones job and earns two. And zero is a real worker: in 40, the 0 holds the ones place open so the 4 keeps its tens job. Without it, 40 would collapse into 4.

Bundles can pile up too. Ten bundles of ten make one hundred. The count keeps walking: 99, 100, 101, 102, on through 109, then 110 — the tens tick up by one — and the walk carries on steadily to 120.

Expanded form spells out the jobs: 52 = 5 × 10 + 2. Each place is worth ten of the place to its right — ten ones trade for one ten, and ten tens trade for one hundred. Past 99 the pattern simply grows a third place: 120 = 1 hundred + 2 tens + 0 ones.

The hundred chart is this system drawn as a map. Every row holds one full ten. Step down a column and you add exactly ten; step right and you add one. Reading the chart, you are reading the writing system itself.

The general rule: a numeral's value is the sum of digit × place value, the places worth 10⁰, 10¹, 10², and so on, reading right to left. Compare Roman numerals, where XXXIV is a tally to be added and subtracted — column arithmetic is miserable in it. Positional notation earns its keep in calculation: columns line up like with like, and the zero placeholder, which Rome never had, keeps every digit in its lane. That zero is why 40, 104, and 140 can all be told apart.

Look at 34. The 4 on the right counts loose ones. The 3 on the left counts bundles of ten. Three bundles make thirty. Thirty and four is thirty-four.

Now read 43. Four bundles, three loose sticks. Forty-three. Same digits, different jobs — the spot a digit sits in tells you its job.

The representation theorem: for any base b ≥ 2, every natural number n has exactly one expansion n = aₖbᵏ + … + a₁b + a₀ with 0 ≤ aᵢ < b and the leading digit nonzero. Existence falls out of repeated division with remainder; uniqueness from a size bound — if two expansions first differ at place k, the difference there is at least bᵏ, while all lower places together can supply at most bᵏ − 1.

Base ten is anatomy, not mathematics: the same theorem serves base 2, 12, or 60 equally well. A child bundling sticks with rubber bands is executing the division algorithm by hand — quotient in bundles, remainder loose on the table.

What positional notation buys is compression: n occupies roughly log₁₀ n symbols, and every schoolroom column algorithm runs in time polynomial in that length — the quiet reason digit-wise arithmetic scales. Canonical base-b strings under shortlex order realize an order-isomorphism with (ℕ, ≤), and variant systems — bijective base ten with digits 1 through 10 and no zero, balanced ternary with a signed digit set — satisfy parallel existence-and-uniqueness theorems, each trading zero-handling for other conveniences.

Historically the full package — positional digits plus a zero placeholder — consolidated in India by the seventh century, reached Europe through al-Khwārizmī's arithmetic and Fibonacci's Liber Abaci, and displaced counting-board practice only after centuries of resistance. The bundling a six-year-old performs is a faithful physical model of quotient-remainder recursion; the notation is that recursion, frozen onto paper.