The beads on this third wire then are counted one by one: one hundred, two hundreds, three hundreds, four hundreds, five hundreds, six hundreds, seven hundreds, eight hundreds, nine hundreds, ten hundreds. These ten hundred-beads are equal to one of the thousand-beads.
There also are ten thousand-beads: one thousand, two thousands, three thousands, four thousands, five thousands, six thousands, seven thousands, eight thousands, nine thousands, ten thousands. The child can picture ten separate "thousand chains"; this symbol is in direct relation, therefore, to a tangible idea of quantity.
Now we must transcribe all these acts by which we have in succession counted, ten units, ten tens, ten hundreds, and ten thousands. On the first vertical line to the extreme right (the green line) we write the units, one beneath the other; on the second line (blue) we write the tens; on the third line (red) the hundreds; and, finally, on the line beyond the dots we write the thousands. There are sufficient horizontal lines for all the numbers, including one thousand.
Having reached 9, we must leave the line of the units and pass over to that of the tens; in fact, ten units make one ten. And, similarly, when we have written 9 in the tens line we must of necessity pass to the hundreds line, because ten tens equal one hundred. Finally, when 9 in the hundreds line has been written, we must pass to the thousands line for the same reason.
The units from 1 to 9 are written on the line farthest to the right; on the next line to the left are written the tens (from 1 to 9); and on the third line, the hundreds (from 1 to 9). Thus always we have the numbers 1 to 9; and it cannot be otherwise, for any more would cause the figure itself to change position. It is this fact that the child must quietly ponder over and allow to ripen in his mind.
It is the nine numbers that change position in order to form all the numbers that are possible. Therefore, it is not the number in itself but its position in respect to the other numbers which gives it the value now of one, now of ten, now of one hundred or one thousand. Thus we have the symbolic translation of those real values which increase in so prodigious a way and which are almost impossible for us to conceive. One line of ten thousand beads is seventy meters long! Ten such lines would be the length of a long street! Therefore we are forced to have recourse to symbols. How very important this position occupied by the number becomes!
How do we indicate the position and hence the value of a certain number with reference to other numbers? As there are not always vertical lines to indicate the relative position of the figure, the requisite number of zeros are placed to the right of the figure!
The children already know, from the "Children's House," that zero has no value and that it can give no value to the figure with which it is used. It serves merely to show the position and the value of the figure written at its left. Zero does not give value to 1 and so make it become 10: the zero of the number 10 indicates that the figure 1 is not a unit but is in the next preceding position—that of the tens—and means therefore one ten and not one unit. If, for instance, 4 units followed the 1 in the tens position, then the figure 4 would be in the units place and the 1 would be in the tens position.
The bead material used for addition and subtraction. Each of the nine numbers is of different colored beads.