Invent Arithmetic complete | ABC of Mathematics

What you noticed

Carrying is breaking up. A place that comes to more than nine holds ten of that place too many, and ten of a place is one of the next. Carrying does that break as you go; writing each total in full and breaking up after does the same thing with nothing held in your head.
Borrowing is carrying run backwards. Taking 7 from 300 works because the ones are there all along, packed inside the hundreds. A trade unpacks one of a place into ten of the place below.
Order does not matter for adding or for times, but it does for taking away. Turning the tray or moving the brackets never counts a pebble, so it works for any numbers. One counterexample was enough to show that taking away keeps neither law.
You never need all 81 facts. Turn the tray and half the times table comes free; from any fact you know, add or take away a row to reach its neighbours.
Times ten moves every pebble up one line. "Add a zero" is the picture of that, not the reason.
A pair of digits lands on the level its places add up to. Tens times tens is hundreds because the zeros add up, and that one rule decides where every piece of a product goes.
Every method gave the same answer, because each adds up the same pieces in a different order.
The best price is in the middle. A higher price brings more from every buyer who stays and loses every buyer who goes; a price cut costs something on every buyer you would have had anyway. Money that is the same whatever you choose, like the bill, never decides the choice.

What you invented

Adding place by place. Break both numbers into places, add each place in full, then break up any place over nine, from the ones up.
Shortcuts for adding and taking away. A running total from the left; making one number round (998 + 467 = 1000 + 465); pairing numbers whose ones make ten; keeping the gap, so 1000 - 467 becomes 999 - 466 with no trade at all.
Multiplying by cutting the tray. Times ten, then times one digit, then the usual way on paper: a big times cut into small ones, glued back together.
Crosswise. Multiplying level by level from the ones, with the answer in one line.
Doubling the Egyptian way. A table of nine doubles that doubles any number, and multiplying with nothing but doubling and adding.

Laws and theorems

Here $a$ , $b$ , $c$ are any whole numbers; in the lines about places, the letters are digits.

	Adding	Multiplying
Commutative	$a + b = b + a$	$a \times b = b \times a$
Associative	$(a + b) + c = a + (b + c)$	$(a \times b) \times c = a \times (b \times c)$
Identity	$a + 0 = a$	$a \times 1 = a$
Zero		$a \times 0 = 0$

Distributive law: $a \times (b + c) = a \times b + a \times c$ .
Taking away is neither commutative nor associative: $5 - 3 = 2$ but $3 - 5$ has no whole-number answer, and $(20 - 7) - 3 = 10$ but $20 - (7 - 3) = 16$ . It undoes adding: $(a + b) - b = a$ , and $a - 0 = a$ .
Times ten moves every digit up one place: $(10 \times d_{1} + d_{0}) \times 10 = 100 \times d_{1} + 10 \times d_{0}$ .
Levels: a digit in place $i$ times a digit in place $j$ lands in place $i + j$ : $(a \times 1 0^{i}) \times (b \times 1 0^{j}) = (a \times b) \times 1 0^{i + j}$ .
Crosswise: $(10 a + b) \times (10 c + d) = 100 \times a c + 10 \times (a d + b c) + b d$ .
Powers of two (seen here, proved in a later module): every whole number $n \geq 1$ is a sum of different powers of two, $n = 2^{k_{1}} + 2^{k_{2}} + \dots$ with $k_{1} > k_{2} > \dots \geq 0$ .

What comes next

Taking away was the one operation that kept breaking: 3 - 5 has no answer among the numbers you have. At the fair it stopped being a puzzle: a stall that took 845 for a bill of 864 is short by 19, and 845 - 864 has no whole-number answer to say so. Invent Negative Numbers starts there.

The mathematics behind it

The laws you found, commutative, associative, the identities (adding 0 or multiplying by 1 changes nothing), and the distributive law that ties times to adding, are almost all there is to calculating with whole numbers. Every method in this module, on paper or in the head, is a rearrangement allowed by these laws, together with place value. Mathematicians call a system with these laws a semiring. Taking away is missing from that list: among whole numbers it does not always have an answer, and it is neither commutative nor associative, which is one way to see why negative numbers were needed.

Crosswise is the distributive law with the pieces sorted by level. Write 23 as 2t + 3 and 64 as 6t + 4, where t is ten, and multiplying them is exactly multiplying polynomials: the pairs whose places add up to the same power of t are collected together. The only thing digits add is the break-up at the end, since a place may not hold more than nine. Fast methods for multiplying very large numbers start from the same view of numbers as polynomials.

Egyptian multiplication, in the Rhind papyrus copied by the scribe Ahmose about 1550 BC, works because every whole number is a sum of different powers of two, each used at most once. Writing down which powers are used, 1 for used and 0 for skipped, is writing the number in binary, and simple computer circuits multiply by shifting and adding in much the same way.

Inspiration

The module draws on Paul Lockhart's Arithmetic (Harvard University Press, 2017), especially its chapters on adding and multiplying with counters and on the page, and its bills of sale and Egyptian doubling.

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