In countries where the metric system has long been in use, it is not looked upon as just another collection of units but as a way of "bundling" objects in a systematic manner. Indeed, the whole way arithmetic is presented in the U. S. is a bit mysterious to a metric native: What do, for instance, terms like "carry over" and "borrow" allude to? We suggest that the algorithmic aspect of arithmetic is well worth taking into consideration.
Example: Suppose that Jack and Jill go into a partnership. Say Jack holds, on the one hand, 8 Hamiltons, 9 Washingtons, 5 Dimes, 4 Pennies and that, on the other hand, Jill holds 2 Franklins, 7 Washingtons, 8 Dimes, 6 Pennies. The situation is:
We shall join the holdings step by step and we should note that the children should not be able to count beyond ten (the number of their fingers). Thus, the numbers written in italicized English are solely for the reader of this paper. In order to change, the children only count up to Ten and change that for 1 of the next unit up and then count the remainder to see if further change is needed. Putting the Pennies together results in ten Pennies:
But the ten Pennies must be changed for 1 Dime:
Putting the Dimes together results in fourteen Dimes:
But ten of the fourteen Dimes must be changed for 1 Washington leaving 4 Dimes:
Putting the Washingtons together results in seventeen Washingtons:
But ten of the seventeen Washingtons must be changed for 1 Hamilton leaving 7 Washingtons.
Putting the Hamiltons together results in 9 Hamiltons:
Altogether then, the Jack and Jill partnership has:
After the children have had time to familiarize themselves with the above, they readily understand the following set up:
Example: Say Jill holds, on the one hand, 3 Hamiltons, 0 Washingtons, 5 Dimes and that she must pay 7 Dimes which is more dimes than she has even though she does have enough money to pay. Normally, to get dimes, she would change one Washington for ten dimes but since she does not have any Washington, she must begin by changing one Hamilton for ten Washingtons. Then she can change one Washington for ten dimes so that she now has fifteen Dimes from which she can now pay the seven Dimes that she owes leaving her with eight dimes. Altogether, she is left with: 2 Hamiltons, 9 Washingtons, 8 Dimes. Absolutely no "borrowing" is involved!
Example: Say three robbers stick up the bank of a one-bank town and run away (not too far) with the following: 2 Clevelands, 9 Franklins, 0 Hamilton, 5 Washingtons. Now they want to divide the loot. Clearly, each will only get 0 Cleveland, 3 Franklins, 0 Hamilton, 1 Washington as they can hardly return to the bank to exchange the remaining 2 Cleveland, 0 Franklin, 0 Hamilton, 2 Washington for further division. But this is not the result which the division algorithm would give because it is based on the usually unstated assumption that changing facilities are available. Here is how it goes: They would first have to go to a faraway bank to change the 2 Clevelands for twenty Franklins which they would divide, 6 Franklins each; then they would return to the bank to change the remaining 2 Franklins for twenty Hamiltons which they would divide, 6 Hamiltons each; then, they would return again to the bank to change the remaining 2 Hamiltons for twenty Washingtons; this would give them, with the already remaining 2 Washingtons, twenty two Washingtons which they would divide, 7 Washingtons each, with 1 Washington left over which they may or may not want to change it for dimes. Etc. But, by that time, either one of the robbers should have killed by the other two in order to avoid the whole division process or, less likely, the police should have arrived and arrested them all.
| Kilo-$ | Hecto-$ | Deka-$ | $ | Deci-$ | Centi-$ | Milli-$ |
|---|
This allows for a much more compact notation and a great deal of flexibility: Given a holding, we can want to see it in terms of a particular unit.
Example: If we hold 4 Hecto-$, 0 Deka-$, 5 Deci-$ and if we want to think, say in terms of Deka-$, we write 40.65 Deka-$ where the decimal point indicates that the digit immediately to its left refers to Deka-$. Of course, the metric "heading" is understood.
Another advantage of this notation is that changing the particular unit in terms of which we want to think of the holding is absolutely instantaneous: all we have to do is to "point" at the digit corresponding to the desired unit.
Example: Say that we want to look at 34.58 Hecto-$ from the Deci-$ viewpoint. Since the Deci-$ unit is three places to right of the Hecto-$ unit, all we have to do is to move the decimal point three places to the right! Of course, the metric "heading" is understood.
While this process is usually called unit conversion, observe that it requires no changing facility!
There are several other possible headings which work, of course, in exactly the same manner and which have taken a new everyday importance with the advent of cheap scientific calculators:
| Thousand-$ | Hundred-$ | Ten-$ | $ | Tenth-$ | Hundredth-$ | Thousandth-$ |
|---|---|---|---|---|---|---|
| 1000/1-$ | 100/1-$ | 10/1-$ | 1/1-$ | 1/10-$ | 1/100-$ | 1/1000-$ |
| 103 $ | 102 $ | 101 $ | 100 $ | 10-1 $ | 10-2 $ | 10-3 $ |
| 1000.-$ | 100.-$ | 10.-$ | 1.-$ | 0.1-$ | 0.01-$ | 0.001-$ |
A Final Example: Suppose we want to multiply 30,000 by 0.004. The idea is to move the two decimal points in such a way as to turn the given numbers into 3. and 4. respectively because these are easy to multiply. So, we move the decimal point in 30,000. four places to the left and we must move the decimal point in 0.004 three places to the right. Multiply 3. by 4. to get 12. and now undo what we just did: move the decimal point in 12. four places to the left and three places to the right to get 120. as a result.
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