Module Complete

Invent Place Value

What you noticed

  • More marks is not more. 236 and 263 have the same number of marks, and 198 has more marks than 205. What a mark is worth depends on what kind of mark it is, or where it sits.
  • No kind ever needs ten. Whenever there are ten of one kind, they make one of the next, so no kind ever holds more than nine.
  • On a board, where matters. You can shake a pile of signs and it says the same amount. Roll a pebble to another line and the amount changes.
  • An empty place is easy to see and easy to lose. On the board an empty line is plain. Written in a row, it vanishes, and 36, 306 and 360 look alike.

What you invented

  • A sign for a group, and for a group of groups. Ten strokes became a hobble, and ten hobbles a coil, so a few hundred fit in a handful of signs.
  • Adding by pooling and tying. Pour two amounts together, then tie ten of a kind into one of the next, smallest kind first.
  • Worth by place. Plain pebbles on ruled lines, where the line says what a pebble is worth.
  • One sign for each count. Nine signs, one for each count from one to nine, with the line or the place in the row saying of what.
  • A mark for an empty place. A sign that holds a place open, so that every other sign keeps its place. We write it 0.
  • Adding in writing. Column by column from the right, with a carried 1 where a column comes to ten or more: the board's lift-ten, in ink.

What comes next

You can now write any number and add in writing. Taking away is still to come. There, a place can run short, and a group has to be untied into ten of the next kind down.

The mathematics behind it

Writing a number as a row of digits, each worth ten times the one to its right, is called positional notation, here in base ten. A row like 407 means 4 hundreds, 0 tens and 7 ones. Because you tie whenever there are ten, every amount ends up with exactly one way of being written, with no place holding more than nine. That is why tying always gave the same answer, whatever order you tied in.

Nothing about ten is special. Any group size from two up works the same way, with that many signs, 0 included. Groups of five or six, from the last module, would do; the Babylonians grouped in sixties, and computers group in twos, which is called binary.

Egyptian signs, and Roman letters written the plain additive way used here (IIII for four, not IV), are sign-value systems: each sign says its own worth, so their order and the empty places do not matter, but large numbers need many signs. Positional notation needs only one sign per place, so a number ten times bigger takes just one more sign. The price is the 0, and a fixed order.

Carrying is the tie in disguise, and that is why adding column by column is right: it regroups ten of one place into one of the next without changing the amount. Here 0 has only been a placeholder. Treating zero as a number in its own right, one you can add and multiply, is a further step.

Inspiration

The module draws on Paul Lockhart's Arithmetic (Harvard University Press, 2017), especially its account of marked-value systems, the Egyptian numerals and reckoning with counters.

Further reading