| hundreds | tens | units |
Addition.We can have bags containing 10 buttons, 100 buttons, and then we can get change. Sonnenschein’s box makes carrying very clear. Suppose I want to put down 5 thousands, 9 tens and 3 units or ones. I should write it thus, and if I wanted to add to this 2 thousands, 9 hundreds and 9, I should write that below.
| th. | hun. | tens | units |
|---|---|---|---|
| 5 | 9 | 3 | |
| 2 | 9 | 9 | |
| 8 | 2 |
Then I should say 9 units and 3 units make 12 units. But this is equal to 1 ten and 2 units, so I should carry on 10 to the second row, and write down 2 in the unit row. Then I add the 1 to the 9, that makes 10, but 10 in the second row is the same as 1 in the third, so I carry that on; 9 and 1 make 10, but 10 in the third row makes 1 in the fourth, so I carry again, and get 5 + 2 + 1 = 8 thousands, and we should read it 8 thousands and 2.
Subtraction.Then after a while people said, “Why need we have all the chequers? suppose we put a nought when there is no number, just to mark that there is a row, and all will come right;” so they wrote thus:—
| th. | h. | t. | u. |
|---|---|---|---|
| 5 | 0 | 9 | 3 |
| 2 | 9 | 0 | 9 |
| 2 | 1 | 8 | 4 |
And a little later they left off writing anything at the top of the line, because every one knew. Here is a subtraction sum. We cannot take 9 units from 3 units, so we get change from the next row, that gives 13 units, from which we take 9, and have 4 left. We have nothing to take from our 8 remaining tens, so we write 8. We have no hundreds, so we cannot take away 9, but we change one of our thousands into 10 hundreds, and take away 9, leaving 1; lastly we take away 2 from our 4 thousands, and get 2—altogether 2184.
Decimal fractions.Now would come in naturally the extension of this system of notation to decimal fractions, marking the unit by a full stop. If numbers decrease as we go from left to right, they might get smaller than one; the next row to the right would be one-tenth of a penny or of an inch, and the next one-hundredth, and so on. Sums in addition and subtraction might be worked at this stage with decimal fractions. Then it should be pointed out that to push the number a row farther from the point which marks the unit row increases it tenfold, and pushing to the right diminishes tenfold.
| hun. | tens | units | tenths | hundth. | thousandths | |
|---|---|---|---|---|---|---|
| 1 | 3 | 2 | . | 7 | 9 | |
| 2 | 5 | . | 8 | 9 | 7 |
It is good practice and interests young children to work in different scales of notation—one may suggest that Goliath would prefer the 6 or 12 scale.