DIVISION

, is one of the four principal Rules of Arithmetic, being that by which we find out how often one quantity is contained in another, so that Division is in reality only a compendious method of Subtraction; its effect being to take one number from another as often as possible; that is, as often as it is contained in it. There are therefore three numbers concerned in Division: 1st, That which is given to be divided, called the dividend; 2d, That by which the dividend is to be divided, called the divisor; 3d, That which expresses how often the divisor is contained in the dividend; or the number resulting from the division of the dividend by the divisor, called the quotient.

There are various ways of performing Division, one called the English, another the Flemish, another the Italian, another the Spanish, another the German, and another the Indian way, all equally just, as finding the quotient with the same certainty, and only differing in the manner of arranging and disposing the numbers.

There is also division in integers, division in fractions, and division in species, or algebra.

Division is performed by seeking how often the divisor is contained in the dividend; and when the latter consists of a greater number of figures than the former, the dividend must be taken into parts, beginning on the left, and proceeding to the right, and seeking how often the divisor is found in each of those parts.

For ex. If it be required to divide 6758 by 3. First seek how often 3 is contained in 6, which is 2 times; then how often in 7, which is likewise 2 times, with 1 remaining; which joined to the next figure 5 makes 15, then the 3's in 15 are 5 times; and lastly | the 3's in 8 are 2 times, and 2 remaining. All the numbers expressing how often 3 is contained in each of those parts, are to be written down according to the order of the parts of the dividend, or from left to right, and separated from the dividend itself by a crooked line, thus:

It appears therefore, that 3 is contained 2252 times in 6758, with 2 remaining over; or that 6758 being divided into 3 parts, each part will be 2252 2/3, viz, the figures of the quotient before found, together with the fraction 2/3 formed of the remainder and the divisor.

When the divisor is a single digit, or even as large as the number 12, the division is easily performed by setting down only the quotient as above. But when the divisor is a larger number, it is necessary to set down the several remainders and products &c. This process may be seen at large in most books of arithmetic, as well as various contractions adapted to particular cases: such as, 1st, when the divisor has any number of ciphers at the end of it, they are cut off, as well as the same number of figures from the end of the dividend, and then the work is performed without them both, annexing only the figures last cut off, to the last remainder; 2d, when the divisor is equal to the product of several single digits, it is easier to divide successively by those digits, instead of the divisor at once; 3d, when it is required to continue a quotient to a great many places of figures, as in decimals, a very expeditious method of performing it, is as follows: Suppose it were required to divide 1 by 29, to a great many places of decimals. Adding ciphers to the 1, first divide 10000 by 29 in the common way, till the remainder become a single figure, and annex the fractional supplement to complete the quotient, which gives 1/29 = 0.03448 8/29: next multiply each of these by the numerator 8, so shall 8/29 = 0.27584 64/29 or rather 0.27586 6/29; which figures substituted instead of the fraction 8/29 in the first value of 1/29, it becomes 1/29 = 0.0344827586 6/29: again, multiply both of these by the last numerator 6, and it will be 6/29 = 0.2068965517 7/29; which figures substituted for 6/29 in the last-found value of 1/29, it becomes 1/29 = 0.03448275862068965517 7/29: and again, multiplying these by the numerator 7, gives 7/29 = 0.24137931034482758620 10/29; which figures substituted instead of 7/29 in the last-found value of 1/29, it becomes 1/29 = 0.0344827586206896551724137931034482758620 10/29 and so on; where every operation will at least double the number of figures before found by the last one.

Proof of Division. In every example of division, unity is always in the same proportion to the divisor, as the quotient is to the dividend; and therefore the product of the divisor and quotient is equal to the product of 1 and the dividend, that is, the dividend itself. Hence, to prove division, multiply the divisor by the quotient, to the product add the remainder, and the sum will be equal to the dividend when the work is right; if not, there is a mistake.

, in Decimal Fractions, is performed the same way as in integers, regard being had to the number of decimals, viz, making as many in the quotient as those of the dividend exceed those in the divisor.

, in Vulgar Fractions, is performed by dividing the numerators by each other, and the denominators by each other, if they will exactly divide; but if not, then the dividend is multiplied by the reciprocal of the divisor, that is, having its terms inverted; for, taking the reciprocal of any quantity, converts it from a divisor to a multiplier, and from a multiplier to a divisor. For ex. (15/16)÷by 5/8 gives 3/2, by dividing the numerators and denominators; but 15/16÷by 4/3 is the same as 15/16 X 3/4, which is = 45/64. Where X is the sign of multiplication, and the character ÷ is the mark of division. Or division is also denoted like a vulgar fraction; so 3 divided by 2, is 3/2.

, or Species, is performed like that of common numbers, either making a fraction of the dividend and divisor, and cancelling or dividing by the terms or parts that are common to both; or else dividing after the manner of long division, when the quantities are compound ones. Thus, ab divided by a, gives b for the quotient: and 12ab divided by 4b, gives 3a for the quotient: and 16abc² divided by 8ac, gives 2bc: and a divided by 3b, gives a/3b: and 15abc³ divided by 12bc², gives

In some cases, the quotient will run out to an infinite series; and then, after continuing it to any certain number of terms, it is usual to annex, by way of a fraction, the remainder with the divisor set under it.

It is to be noted that, in dividing any terms by one another, if the signs be both alike, either both plus, or both minus, the sign of the quotient will be plus; but when the signs are different, the one plus and the other minus, the sign of the quotient will be minus.

Division by Logarithms. See Logarithms.

Division of Mathematical Instruments. See GRADUATION, and Mural Arc or Quadrant.

Division in Music, is the dividing the interval of an octave into a number of lesser intervals.

Division by Napier's Bones. See Napier's Bones.

Division of Powers, is performed by subtracting their exponents. Thus, ; and . |

Division of Proportion, is comparing the difference between the antecedent and consequent, with either of them. Thus,

, is the dividing number; or that which shews how many parts the dividend is to be divided into.

Common Divisor. See Common Divisor.