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rejecting or dropping the number 9 as often as the sum amounts to that nuniber, and proceeding with the excess, and finally denote the last excess.
Perform the same operation upon each of the factors; then multiply together the excesses of the factors, and cast out the nines from their product. If the excess of this smaller product be equal to the excess of the larger product first found, the work may be supposed to be right. It is, however, to be observed, that, although this test furnishes satisfactory evidence of the correctness of an operation, it is not an infallible proof; for, if a product chance to contain an error of just 9 units of any degree, the excess of its horizontal sum is not thereby altered.
In order to perceive why the excess above nines found in the horizontal sum of a product, must be equal to the excess found in the product of the excesses of the face tors, observe that, by the law of notation, a figure is increased nine times its value by its removal one place to the left; and hence, however far a figure is removed from the place of units, when its nines are excluded, its remainder can be only itself. Therefore, any number, and the horizontal sum of its figures, must have equal remainders when their nines are excluded. This being understood, observe that, since factors composed of entire nines will give a product consisting of entire nines, it follows, that any excess above nines in a product, must arise from an excess above nines in the factors. Therefore, the product of the excesses of the factors, must contain the same excess that is contained in the product of the whole factors.
Division is the operation by which we find how many times one number is contained in another. It is the converse of multiplication; the product and one factor being given, and the other factor resulting from the operation,
The number which corresponds to the product in multiplication, is the number to be divided, and is called the dividend. The given factor is the number to divide by, and is called the divisor. The factor to be found, that is, the number which shows how many times the dividend contains the divisor, is called the quotient.
As multiplication has been shown to proceed from addition, so division may be shown to proceed from subtraction. If we repeatedly subtract the divisor from the dividend till the latter is exhausted, the number of subtractions performed_will answer to the number of units in the quotent. For example, if the dividend be 24, and the divisor 6, the quotient may be found by subtraction thus, 24-6=18, 18—6=12, 12-6=6, 6—6=0. Here 6 is subtracted four times from 24, and there is nothing remains; therefore, 4 is the number of times that 6 is contained in 24. In division, this operation is denoted thus, 24+=4; or thus, ą4=4.
Division not only investigates the number of times the dividend contains the divisor, but it also serves to divide the dividend into as many equal parts as the divisor contains units; the quotient being one of these parts. This effect of the operation may be understood by considering, that, since the divisor and quotient are factors of the dividend, they must each indicate how many of the other the dividend contains.
It may be observed, that all the preceding operations begin at the place of simple units; division, however, must begin at the highest degree of units; for, the number of times that the divisor is contained in the higher units of the dividend must be taken out first, in order that any remainder, or excess above an exact number of times, may be carried down to the lower degrees of units, and divided therewith.
When the divisor is not contained an exact number of times in the dividend, there will be a remainder end of the operation. This remainder, being a part of the dividend, is to be divided; but its quotient will be smaller than a unit, since a quantity in the dividend just equal to the divisor, gives only a unit in the quotient.
Quantities sinaller than a unit, that is, parts of a unit, are called Fractions. Such quantities are commonly expressed by two numbers, placed one above the other with a line between them, thus . The lower number, called the denominator, shows how many equal parts the unit is divided into; and the upper number, called the numerator, shows how many of the equal parts are embraced in the fraction. When the unit is divided into two equal parts, the parts are called halves; when divided into three equal parts, the parts are called thirds; when divided into four equal parts, the parts are called fourths; and so on; the number of the denominator giving the name. For example, if the unit be divided into five equal parts, one of the parts is denoted thus, }, and called onefifth; two of the parts, thus, }, and called two-fifths; and so on. In this method, the unit may be divided into any number of equal parts, and any number of such parts may be denoted. With this elementary view of fractions, it may be
perceived, that when there is a remainder of 1 unit, it is to be divided into as many equal parts as there are units in the divisor, and one of these parts is to be annexed to the quotient. This is performed by merely writing the 1 as a numerator, and the divisor as the denominator, on the right of the quotient. If the remainder be 2 units, there will be 2 such parts of a unit as the divisor indicates to be annexed to the quotient, and, therefore, the numerator will be 2. If the remainder be 3 units, the numerator will be 3; and so on. Hence, whatever the remainder may be, it becomes, in the quotient, the numerator of a fraction, the divisor being the denominator.
RULE FOR DIVISION. When the divisor does not exceed 9, draw a line under the dividend, find how many times the divisor is contained in the left hand figure, or two left hand figures of the dividend, and write the figure expressing the number of times underneath: if there be a remainder over, conceive it to be prefixed to the next figure of the dividend, and divide the next figure as before. Thus proceed through the dividend.
When the divisor is more than 9, find how many times
it is contained in the fewest figures that will contain it, on the left of the dividend, write the figure expressing the number of times to the right of the dividend, for the first quotient figure ; multiply the divisor by this figure, and subtract the product from the figures of the dividend considered. Place the next figure of the dividend on the right of the remainder, and divide this number as before.' Thus proceed through the dividend. If there be a final remainder, place it as a numerator, and the divisor as a denominator, on the right of the quotient.
PROOF. Multiply the whole numbers of the divisor and quotient together, and to the product add the numerator of any fraction in the quotient: the sum will be equal to the dividend, if the work be right.
Abbreviations of the above rule may frequently be adopted, as follows:
When there are ciphers on the right hand of a divisor, cut them off, and omit them in the operation; alsu cut of and omit the same number of figures from the right hand of the dividend. Finally, place the figures cut of from the dividend, on the right of the remainder.
When the divisor is 10, 100, 1000, fc., cut off as many figures from the right hand of the dividend as there are ciphers in the divisor; the other figures of the dividend will be the quotient, and the figures cut of will be the remainder.
When factors of the divisor are known, divide the dividend by one of these factors, and the quotient thence arising by the other: the last quotient will be the true
To find the true remainder, multiply the last remainder by the first divisor, and to the product add the first remainder.
1. Divide 4062900311 by 9, and prove the operation. 2. How many times is 502 contained in 74260710?
3. Suppose 52076348 to be a dividend, and 8649 the divisor; what is the quotient?
4. If 26537009535 be divided into 27856 equal parts, what will be one of those parts? 5. Divide 16500269842 by 86000 ; abbreviating.
6. Divide 8065743924 by 10000; by abbreviation. 7. Divide 290516 by 63; using factors of the divisor 8. 142375800392 ; 5274=what number?
PROPERTIES OF NUMBERS.
Before proceeding to examine the properties of numbers, a few arithmetical terms, which we shall here collect and define, should be perfectly understood. As an exercise in this article, the learner may give, upon his slate, an example of each term defined, and each property described.
A unit, or UNITY, is any thing considered individually, without regard to the parts of which it is composed.
An Integer is either a unit or an assemblage of units; and a FRACTION is any part or parts of a unit.
One number is said to MEASURE another, when it divides it without leaving any remainder.
A number which divides two or more numbers with out a remainder, is called their comMON MEASURE.
When a number can be measured by another, the former is called the multiple of the latter.
If a number can be measured by two or more numbers, it is called their COMMON MULTIPLE.
A COMPOSITE NUMBER is that which can be measured by some number greater than unity.
The ALIQUOT PARTS of a number, are the parts by which it is measured, or into which it can be divided.
An EVEN NUMBER is that which can be measured, or exactly divided by 2.
An ODD NUMBER is that which cannot be measured by 2; it differs from an even number by 1.
A PRIME NUMBER is that which can only be measure ed by unity, that is, by 1.
One number is PRIME TO ANOTHER, when unity is the only number by which both can be measured.