REVIEW III.

Multiplication, strictly speaking, is the operation of finding the sum of two or more equal numbers.

With reference to this operation, this sum is called the product; one of the equal numbers is called the multiplicand; and the number which shows how many times the multiplicand is taken is called the multiplier.

In multiplication we must always have two different kinds of units: So many things taken or repeated so many times.

It is sometimes required to take or repeat the whole of the multiplicand, sometimes a certain part of it, sometimes the whole and a certain part of it; and the meaning of multiplication is extended to cover all of these

cases.

To multiply by a whole number is to take the multiplicand as many times as is indicated by the whole number.

To multiply by a fraction is to take such a part of the multiplicand as is indicated by the fraction.

To multiply by a whole number and a fraction is to take the multiplicand as many times as is indicated by the whole number, and such a part of the multiplicand as is indicated by the fraction.

The product, therefore, will be equal to, greater than, or less than the multiplicand, according as the multiplier is equal to, greater than, or less than one.

The multiplicand and the multiplier are often called factors of the product.

When a product consists of more than two factors, it is called the continued product of the factors.

When a product consists of two or more equal factors, it

is called a power of that factor; and one of the equal factors is called a root of the product.

The index or exponent of a power is a small figure placed at the right of a number to show how many times the number is taken as a factor, and is read the first power, second power, third power, etc., of the number.

The second power of a number is generally called the square of the number, and the third power is called the cube of the number.

The product of two or more powers of the same number may be expressed by writing the number with an exponent equal to the sum of the exponents of the given powers.

The numerical result of multiplying one number by another is the same whichever is taken as the multiplicand, the other being taken as the multiplier; but the product will always denote the same kind of units as the true multiplicand.

A St. Andrew's cross (X) placed between two numbers means that one of the numbers is to be multiplied by the other, and it is read "times," or "multiplied by."

To multiply by 10, 100, 1000, etc., it is necessary only to move the decimal point in the multiplicand as many places to the right, annexing ciphers, if necessary, as there are ciphers in the multiplier,

To multiply by .1, .01, .001, it is necessary only to move the decimal point in the multiplicand as many places to the left, prefixing ciphers, if necessary, as there are places in the multiplier.

The decimal places of a product are equal in number to the decimal places in the multiplicand and multiplier counted together.

To multiply by the product of two or more factors gives the same result as to multiply by one of the given factors, this product by another of the factors, and so on.

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If we have to multiply one number by another, we may separate the multiplicand into parts, multiply each part by the multiplier, and add the results to form the complete product. Or, we may separate the multiplier into parts, find the product of the multiplicand by each part, and add these partial products to form the complete product.

When the multiplier is a single digit, the product is obtained by multiplying each order of the multiplicand, beginning with the lowest order, care being taken to write down the ones of each partial product, and to reserve the tens to be added as ones to the product of the next higher order.

When the multiplier consists of two or more digits, multiply by each digit separately, arrange the partial products so that figures of the same order shall fall in the same column, and add.

To test the accuracy of the work, interchange multiplicand for multiplier. Or, separate the multiplier into parts, multiply the multiplicand by each part separately, and add the results. If the work be correct, their sum will be equal to the original product.

In contracted multiplication of decimals:

Reverse the multiplier, and place the units' figure under the last place of decimals in the multiplicand to be retained.

Multiply by each figure of the multiplier the figure next to the right above it. Do not write the result, but carry the nearest ten to the next product and write this result, multiplying as usual.

Write the first figures of the products in a vertical column.

Add the several products, and point off from the sum as many places for decimals as were retained in the multiplicand.

CHAPTER VII.

DIVISION.

149. Division is the operation by which, when a product and one of its factors are given, the other factor is found.

With reference to this operation, the product is called the dividend; the given factor is called the divisor; and the required factor is called the quotient.

150. When the given factor is the multiplicand, the factor sought is the multiplier.

In this case, the question is: What must we multiply the divisor by to get the dividend?

To answer this question, it is necessary to find how many times the divisor is contained in the dividend, and the answer will be so many times.

Thus, in the question, How many times are 6 cents contained in 30 cents? the factor sought is the multiplier, 5; 5 times 6 cents are 30 cents.

When the given factor is the multiplier, the factor sought is the multiplicand.

In this case, the question is: What must we multiply by the divisor to get the dividend?

To answer this question, it is necessary to divide the dividend into as many equal parts as is indicated by the number in the divisor, and the answer will be so much to each part.

Thus, in the question, How much will each boy receive if 30 cents be divided among 6 boys? the factor sought is the multiplicand, 5 cents; 6 times 5 cents are 30 cents.

The arithmetical process and the numerical result are the same in both cases; but the name to be attached to the result depends upon the nature of the question.

151. Division is indicated by the sign (÷), by the colon (), or by writing the dividend over the divisor, and drawing a line between them; thus, each of the expressions,

15

153, 15:3, means and is read, "Fifteen divided by

three."

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SHORT DIVISION.

152. When the divisor does not exceed 12, the work may be written in the following manner: the divisor is placed to the left of the dividend, the quotient under the dividend, and each remainder is added as so many tens to the next figure of the dividend not divided.

(1) Divide 2736 by 4.

4)2736

30?

Wording. 4 in 27, 6, carry 3; in 33, 8, carry 1;

Answer, 684.

684 in 16, 4.

(2) Divide 2736 by 9.

9)2736 304

Wording. 9 in 27, 3; in 3, 0, carry 3; in 36, 4.

Answer, 304.

(3) Divide 3696 by 12.

12)3696 308

Wording. 12 in 36, 3; in 9, 0, carry 9; in 96, 8.
Answer, 308.

The pupil will observe that, the divisor being a whole number, each quotient figure is of the same order of units as the right-hand figure of the partial dividend used in obtaining it.