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posite number is equal to the sum of as many products by one number as there are units in another, and may therefore be found by multiplying one product by the number of products. That is, to multiply by a number composed of two factors, we may multiply by one first, and this product by the other.
If the multiplier be composed of more than two factors, they may be reduced to two, and the product formed by multiplying by each of these. But the products by each of these two may be formed in the same manner by multiplying in succession by each of two factors of which it may be composed; and the same may be said of every multiplier, which is composite. So that generally to multiply by any composite number, we may multiply by each factor in succession.
Prop. 10.- To prove the Rule for Multiplication by
10, 100, &c. Since 10 times 6 (or any other number) are equal to 6 times 10 (Prop. 6), therefore the multiplication of units, tens, &c. by 10, changes them respectively to tens, hundreds, &c. Now if we place a () on the right of a number, it has the effect of increasing by 1 the distance of each figure from the right, and therefore of increasing tenfold its signification, i. e. it has the same effect as multiplication by 10. Therefore to multiply by 10 we have only to place a 0 on the right of the number. Similarly to multiply by 100, 10000, &c. we have only to place on the right of the number as many ciphers as there are in the multiplier.
Cor. Hence, and by the previous Proposition, the multiplication by any number, having ciphers on the right, is effected by adding on the right as many ciphers as there are in the multiplier, and multiplying the result by the figures of the multiplier, when the ciphers have been removed. Prop. 11.-T'o explain the Rule for Multiplication by any
If the multiplier be not greater than 12, then (Prop. 7) we may find the product by multiplying each part of units, &c. in succession by the multiplier (with the aid of the Multiplication Table) and adding the products, which in practice is done as they are formed.
If the multiplier be greater than 12, since the product of one number by another is equal to the sum of the products by any parts into which the multiplier is divided; therefore we are at liberty to divide the multiplier into any parts we please, to multiply by each part separately, and add the products; and doing so we shall obtain the whole product. Now by the Rule for Mul. tiplication, the multiplier is divided into units, tens, &c. and the successive products formed are the products by these several parts, the first being that
by the units, the second that by the tens, where the 0, which should be in
5765 The same as usually written 5765
40355 = 7 times the multiplicand
40355 51885 46120 23060
28231205 = 4897
Prop. B.-To explain the Rule for Compound Multiplication. If the multiplier be less than 12, the Multiplication is performed precisely on the same principle as Addition, the sums of the several denominations being found by the short process of Multiplication.
If the multiplier be greater than 12, and a composite number, the true product is found by multiplying by the factors in succession. This is evidently true, when the multiplicand is only of one denomination (Props. 6, 7, 8, 9); and when it is of several, it may be reduced to one, and the Multiplication then performed. But since the product of a whole is equal to the sum of the products of the several parts, therefore we may multiply the parts of the multiplicand as they stand, without reduction.
If the multiplier be greater than 12, and not composite, then by Prop. 8 we may separate it into any parts we please, and, multiplying by these separately,
, may add the products. And this is done by the rule, according to which the multiplier is separated either into units, tens, &c.; or into the sum of a composite number and another less than 12.
Again since the result of the repetition of a quantity a certain number of times may be obtained by repeating it a greater number of times, and by subtracting the result of its repetition a number of times equal to the difference, therefore the product by any number may be found by multiplying by a greater number, and subtracting from the result the product by the difference.
Prop. 12.—To prove that the quotient of one number by
another is equal to the sum of the quotients of parts of
the dividend divided by the divisor. Since the quotient expresses the number of times which the divisor is contained in the dividend ; or, in the case of a concrete dividend, is the part
of it denoted by the divisor, therefore the dividend is equal to the product of divisor and quotient. Hence
Divisor X quotient = dividend = sum of parts of dividend but each part of dividend = divisor X its quotient 1. Divisor X quotient = sum of products of divisor into each partial quotient
(Prop. 8) = divisor X sum of partial quotients
.. Quotient = sum of partial quotients. The same Prop. exhibited algebraically:—Let D be the dividend, d the divisor; and let D=a+b+c+ &c.; let Q be the whole quotient; 9, r, s, &c. the quotients of a, b, c, &c. divided by d; Then
=dx9+ d xgtdxs + &c.
q+r+s + &c.
or more numbers may be obtained by dividing one of them,
and multiplying the quotient by the rest. Divisor X quotient = dividend = product of the factors of dividend. Putting therefore in place of one factor the product of the divisor into its quotient, which is equal to the factor, we have Divisor X quotient=(divisor X partial quotient) X the other factors
(Prop, 10) =divisor X (partial quotient X the other factors)
.. Quotient = partial quotient X the other factors. The same Prop. exhibited algebraically:-Let D be the dividend, d the divisor, Q the quotient; and let D=a XbXcX &c.; let the quotient of a by d be q: then
d XQ=D= axbXCX &c.
= (d X 9) X 6 XCX &c.
= dx (X 6 XCX &c.)
.. Q=qXlXcX &c. Cor. Hence if we divide an exact number of tens, hundreds, &c. by any number, the quotient obtained by throwing away the ciphers and dividing will be number of tens, hundreds, &c. in the whole quotient: in other words a number of tens, hundreds, &c. when divided, give a number of tens, hundreds, &c. as the quotient. Prop. 14.—To explain the Rule for Division of Numbers.
The quotient of one number divided by another is equal to the sum of the quotients of any parts into which the dividend is separated (Prop. 12.) Hence we are at liberty to separate the dividend into any parts we please,
to divide each of them, and add the quotients. It remains to be seen what method of separation is most convenient.
To determine this, it is to be considered that our object is to find the number of units, tens, &c. in the quotient. Now no denomination of figures can arise from the division of a lower denomination, but may arise from the division of one higher. Therefore separating the dividend into units, tens, &c. and commencing the division with the highest denomination in the dividend, we shall certainly determine the true number of that denomination in the quotient (Cor. Prop. 13.) Let then the division be commenced by dividing the number of the highest denomination in the dividend ; if this number contain the divisor, i.e. be not less than the divisor, the number of exact times the divisor is contained may be written as the number of this denomination in the quotient. If there be any remainder, (which is ascertained by subtracting the product of divisor and the quotient from the figures divided,) or if the number do not contain the divisor, then there cannot arise from the division of this remainder, or number, any number of this denomination in the quotient, but there may arise a number of the next denomination, and also from the division of the next number in the dividend ; therefore let the remainder be converted into the next lower denomination, and let the number of this denomination in the dividend be added to it, which is done by affixing this number to the remainder. Now if this number be divided, the exact part of the quotient will be the number of this denomination in the whole quotient. In the same manner the numbers of the other denominations may be found.
Thus the dividend will at length have been separated into several parts, each of which contains the divisor a number of single times, tens of times, &c. The quotients of these parts being written after one another as they occur are in fact added (Cor. Prop. 1), and are equal to the whole quotient.
It being evident that there can be no denomination of figures in the quotient, of which there is not in the dividend a number greater than the divisor, therefore in commencing the division, we may at once take as our first dividend the least number of figures on the left, which compose a number not less than the divisor, and the denomination of the last of them will be the highest denomination in the quotient.
The foregoing explanation will be better understood by an example.Divide 87444 by 347.
Here we see at once that the highest possible denomination of figures in the quotient is hundreds. Therefore let the dividend be separated as below;
874 hundreds + 4 tens + 4 units. We may now commence the division with the 874 hundreds; and we find that 347 is contained twice in 874 with 180 over. Therefore there are 2 hundreds in the quotient. We have next to ascertain the number of tens, which may arise from the division of the 180 hundreds in the remainder just obtained, and also from the 4 tens in the dividend. As the quotients of parts of a dividend are equal to the quotient of the whole, therefore the quotients of 180 hundreds, and 4 tens are equal to the quotient of their sum, i.e. of 1804 tens. Dividing this number by 347, we find 5 as the number of tens in the quotient with 69 tens over. In the same way the number of units in the quotie is found to be 2. Therefore the whole quotient contains 2 hundreds, 5 tens, 2 units, or is equal to 252. The process of Division may be written thus:
divisor. hundreds. tens, units. hundreds.
4 ( 2 2 X 347 = 694
180 = 1800 tens
add 4 tens
347)1804(5 tens 5 X 347 = 1735
69 = 690 units
347)694(2 units 2 X 347 = 694
The ordinary method will be seen to be merely an abbreviation of this.
Cor. In precisely the same manner the Rule for Compound Division may be explained by merely substituting the denominations of the quantities for the units, tens, &c. in the Proposition.
Prop. 15.- To prove that the quotient obtained by successive
division by several numbers is the same as from the division by their product; and to prove the Rule for the formation of the total remainder.
Let the dividend be represented by counters, equal in number to the units in the dividend. Let them be divided into equal heaps (which call A), the number of heaps being equal to the Ist divisor. The number of counters in each heap will represent the 1st quotient, and the number of those that are over will be the 1st remainder. Now let each of the heaps (A) be divided into equal heaps (which call B), the nnmber of heaps being equal to the 2nd divisor. Then there will altogether be a number of heaps (B) equal to the product of the 1st and 2nd divisors. And the number of counters in each heap will represent the quotient of the 1st quotient by