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As already indicated in Chapter X, decimals may be regarded as a natural and legitimate extension of the notation with which the pupils are already familiar. Taking this view of decimals as a basis for teaching the subject, we shall see how easily and naturally all the ordinary processes are established, and, further, how this mode of treatment recalls and confirms all that was said in building up the simple rules.
Notation and Numeration.—Consider the number 111: the first 1, starting at the right, denotes one unit; the second, one ten, or ten units; the third, one hundred, or ten tens, or one hundred units. The third 1 is equivalent to one hundred times the first 1, and to ten times the second 1; the second 1 is equivalent to ten times the first 1, and to one tenth of the third 1; the first 1 is equivalent to one tenth of the second 1, and to one hundredth of the third 1. Let us now rewrite the number already taken, place a point after the first 1 to indicate that that 1 is to be regarded as the unit, and then place after the point three l’s, so that we have
111.111. We may ask what each of these 1's should mean, if the same relation is to hold among successive digits that we have supposed hitherto to hold. The 1 after the point, standing next to the 1 which the point tells us is to be looked upon as a unit, would naturally mean one tenth of that 1—that is, one tenth of a unit, or, as we shall say, one tenth. The next 1, passing to the right, standing two places to the right of the unit, is one hundredth of the unit, or one hundredth; it is one tenth of the preceding 1-that is, one tenth of one tenth. Similarly, the next 1 would signify one thousandth, and would equal one hundredth of the one tenth or one tenth of the one hundredth. Thus the number above written may be read as follows: One hundred, one ten, one unit, one tenth, one hundredth, and one thousandth. But just as in ordinary numbers it is convenient, for the purpose of reading, to combine the elements into groups, here also it will be well to adopt a similar method. The 1 to the extreme right is 1 thousandth; the next 1 is, from its position, equivalent to 10 thousandths; and the next 1 is 100 thousandths; so that to the right of the point we have 111 thousandths. The whole number may now be read, one hundred and eleven, and one hundred and eleven thousandths. Very little practice will suffice to acquaint the pupil with the extended notation and numeration. A few questions, such as the following, will prove useful:
(1) Read 539:7423, and show that the reading properly expresses the number.
(2) Explain how it is that the insertion of a zero between the point and the 5 in the decimal 5 changes the value of the decimal, but that the addition of zeros to the right of the 5 does not change the value.
(3) Name the decimal consisting of three digits which lies nearest in value to :573245.
These will serve to bring out in a new relation some of the essential features of the decimal system, and throw light on some facts that at an earlier stage in the pupil's progress were necessarily somewhat dimly seen.
I. SIMPLE RULES. Multiplication.—When once the notation is understood, addition and subtraction of decimals can offer no difficulties, and we pass them by to consider multiplication. In this connection the most striking application is the multiplication by 10, 100, etc. The pupil will be asked to compare 7 and 7, :3 and .03, .009 and .0009, and he will see at once that the first number in each case is, in virtue of the position of the point, 10 times the second number. Next, when asked to compare the numbers 37 and 3.7, he will see that the 3 in the first number is 10 times as great as the 3 in the second, that the 7 in the first number is 10 times as great as the 7 in the second, and that therefore the first number is 10 times as great as the second number. He has thus been led to discover that by moving the points one place to the right we get a number 10 times as great as the original number. Similarly, a corresponding conclusion may be reached for multiplication by 100, 1000, etc., and the conclusions in each case should be arrived at and stated by the pupil. It will at once follow that to divide a number by 10, 100, 1000, etc., we have only to move the point one place, two places, three places, etc., to the left. We pass next to the multiplication by any integral number.
251.94 The multiplication in each of the foregoing cases is based on the same considerations as the multiplication of integers by integers. Thus, in the second case, 7 times 2 hundredths are 14 hundredths—that is, 1 tenth and 4 hundredths, and the 4 must be in the hundredths place; 7 times 4 tenths are 28 tenths, which with the former 1 tenth make up 29 tenths or 2 units and 9 tenths, and the 9 must be in the tenths place; thus, the 4 and the 9 will be properly placed if the point is introduced before the 9, etc.; next, multiplying by 5, we must write the results one place to the left, for reasons explained in an earlier chapter. The pupil will now understand multiplication by an integer, and is ready to proceed with multiplication by a decimal. 3:12
7.1 76 He will be asked to multiply some number, say 3.12, by some number, say 23; the result is 71.76. If, then, we propose to multiply 3:12 by 2:3, it will be seen that this differs from the former only in that the multiplier is 10 times as small; the product then will be 10 times as small, and may at once be written down 7.176. A further example or two, in which a different number of decimal places are taken, will suffice to show that to multiply two decimals we proceed as in the multiplication of integers, and mark off in the resulting product as many places as there are in both multiplier and multiplicand.
Division.—To teach division, it is well to begin with the division by an integer, as this will connect the process with what is already known. Consider the following examples:
5).001 5( 0 0 0 3 2 3)1 4 568 1(5:47
1 3 5
1.6 1 The pupil who can explain the first division can at once explain the second, the third, and the fourth ; and he will see how to divide whenever the divisor is a whole number. Then he may be asked to explain why, in the following divisions, we have the same quotient: 3)15
He will be led to recognise a principle that he already knows, namely, that the multiplication of divisor and dividend by the same number does not change the quotient. He may then be asked to state a quotient equiva