REVIEW IV.

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, the divisor; and the required factor, the quotient.

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

In this case, it is required to find what we must multiply the divisor by to get the dividend.

It is, therefore, necessary to find how many times the divisor is contained in the dividend; and the answer will be, so many times.

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

In this case, it is required to find what we must multiply by the divisor to get the dividend.

It is, therefore, 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.

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

Division is indicated by the sign, the colon, :, or by writing the dividend over the divisor and drawing a line between them. Whatever symbol of division is employed, it is read, "divided by."

The dividend may be separated into parts, and each part divided by the given divisor; the sum of the results will then be the entire quotient.

Hence, in dividing,

Find by trial the first figure of the quotient.

Subtract from the partial dividend the product of the divisor by this quotient figure.

Consider the remainder with the next figure of the dividend annexed as a new partial dividend, and proceed as before.

When the divisor is so small that the necessary multiplications and subtractions can be carried on mentally, the operation is called Short Division.

When the work is written in full, the operation is called Long Division.

The decimal point may be suppressed in any divisor, if the point in the dividend be moved as many places to the right as there are decimal places in the divisor. And if the first quotient figure be written over the right-hand figure of the first partial dividend, the position of the decimal point in the quotient will be directly over that in the dividend.

To divide by 10, 100, 1000, etc., it is necessary only to move the decimal point in the dividend as many places to the left as there are ciphers in the divisor.

To divide by .1, .01, .001, etc., it is necessary only to move the decimal point in the dividend as many places to the right as there are decimal places in the divisor.

To divide by the product of two or more divisors gives the same result as to divide first by one divisor, the quotient by another divisor, and so on. That is,

Equal factors of the dividend and divisor may be suppressed without altering the value of the quotient.

Suppressing equal factors in dividend and divisor is called cancelling.

When the dividend is 1, the divisor and quotient are reciprocals of each other. Hence,

Multiplying by the reciprocal of a number is equivalent to dividing by the number.

Dividing by the reciprocal of a number is equivalent to multiplying by the number.

The quotient of a power by another power of the same number may be expressed by writing the number with an exponent equal to the exponent of the dividend diminished by the exponent of the divisor.

Any number with a minus exponent is the reciprocal of the number with an equal plus exponent.

The quotient of two equal powers of the same number may be expressed by unity, or by the number with zero for an exponent. That is,

The zeroth power of any number is equal to 1.

To test the accuracy of the work in Division, multiply the divisor by the quotient. The product should be equal to the dividend.

When a parenthesis includes two or more numbers, the included numbers must first be reduced to a single number, and the result put in place of the parenthesis.

In contracted division of decimals:

First, determine the number of significant figures required in the quotient.

Secondly, begin to cut off figures from the right of the divisor, when the number of figures still required in the quotient is two less than the number of digits in the divisor.

Thirdly, in multiplying the divisor by each quotient figure, multiply the figure of the divisor cut off, and carry the nearest ten.

CHAPTER VIII.

MISCELLANEOUS EXERCISES.

BEFORE giving any more examples, we shall give a method of testing the accuracy of results in the four operations by what is called

CASTING OUT NINES.

170. A product of two integral factors is called a multiple of either of its factors.

Every power of 10 is one more than some multiple of 9. Thus, 10 9+1; 102=11x9+1; 103-111 x 9+1, etc. Every multiple of a power of 10 by a single digit is, therefore, some multiple of 9, plus that digit. For example, 500 5 x 11 x 9+5; 7000 777 x 9+7, etc.

=

=

But as every number consists of the sum of such multiples of powers of 10, every number is a multiple of 9, plus the sum of its own digits.

Thus, 24,573 is a multiple of 9 plus 2+4+5+7+3. If a number, therefore, be divided by 9, the remainder will be the same as that arising from dividing the sum of its digits by 9.

In finding the remainder from dividing the sum of the digits by 9, we may, of course, omit the nines, or any two or three digits which we see at a glance will make 9. Thus, to find the remainder on dividing 1,926,754 by 9, we see at once that 1, 2, 6, and 5, 4, make nines, and the single 7 will be the remainder. So in 254,786, we reject 5, 4, and 2, 7, and only add 8+6, from the sum of which reject 9, and there is left 5.

171. This truth may be applied to test the accuracy of results obtained in addition, subtraction, multiplication, and division.

From the several numbers to be added, the nines are cast out; then, from the sum of the several remainders, a final remainder of 6 is obtained, which corresponds with remainder obtained by casting out the nines from the sum. Therefore, the work may be presumed to be correct.

SUBTRACTION.

(2) 176,543 = a number of nines + 8 }

85,674 =
90,869

+3

Subtract

5

The nines are cast out from the minuend and subtrahend, and the remainder of the subtrahend is subtracted from that of the minuend, giving a final remainder, 5, which corresponds with the remainder obtained by casting out the nines from the difference.

It is obvious, that a number of nines + a remainder, subtracted from a number of nines + a remainder, will give an exact number of nines the difference of the remainders.

(3) 51,786,531 = a number of nines +0

Subtract.