MULTIPLICATION
, is, in general, the taking or repeating of one number or quantity, called the Multiplicand, as often as there are units in another number, called the Multiplier, or Multiplicator; and the number or quantity resulting from the Multiplication, is called the Product of the two foregoing numbers or factors.
Multiplication is a compendious addition; performing at once, what in the usual way of addition would require many operations: for the multiplicand is only added to itself, or repeated, as often as is expressed by the units in the multiplier. Thus, if 6 were to be multiplied by 5, the product is 30, which is the sum arising from the addition of the number 6 five times to itself.
In every Multiplication, 1 is in proportion to the mulplier, as the multiplicand is to the product.
Multiplication is of various kinds, in whole numbers, in fractions, decimals, algebra, &c.
1. Multiplication of Whole Numbers, is performed by the following rules: When the multiplier consists of only one figure, set it under the first, or righthand figure, of the multiplicand; then, drawing a line underneath, and beginning at the said first figure, multiply every figure of the multiplicand by the multiplier; setting down the several products below the line, proceeding orderly from right to left. But if any of these products amount to 10, or several 10's, either with or without some overplus, then set down only the overplus, or set down 0 if there be no overplus; and carry, to the next product, as many units as the former contained of tens. Thus, to multiply 35092 by 4.
| Multiplicand | 35092 |
| Multiplier | 4 |
| Product | 140368 |
When the multiplier consists of several figures; multiply the multiplicand by each figure of it, as before, and place the several lines of products underneath each other in such order, that the first figure or cipher of each line may fall straight under its respective multiplier, or multiplying figure; then add these several lines of products together, as they stand, and the sum of them all will be the product of the whole multiplication. Thus, to multiply 63017 by 236:
| Multiplicand | 63017 |
| Multiplier | 236 |
| Product of 63017 by 6 | 378102 |
| Product of 63017 by 30 | 189051 |
| Product of 63017 by 200 | 126034 |
| Whole product | 14872012 |
The several lines of products may be set down in any order, or any of them first, and any other of them second, &c; for the order of placing them can make no difference in the sum total. There are many abbreviations, and peculiar cases, according to circumstances, which may be seen in most books of arithmetic.
The mark or character now used for Multiplication, is either the × cross or a single point .; the former being introduced by Oughtred, and the latter I think by Leibnitz.
To Prove Multiplication. This may be done various ways; either by dividing the product by the multiplier, then the quotient will be equal to the multiplicand; or divide the same product by the multiplicand, and the quotient will come out equal to the multiplier; or in general divide the product by either of the two factors, and the quotient will come out equal to the other factor, when the operations are all right. But the more usual, and compendious way of proving Multiplication, is by what is called casting out the nines; which is thus performed: Add the sigures of the multiplicand all together, and as often as the sum amounts to 9, reject it always, and set down the last overplus as in the margin; this in the foregoing example is 8. Then do the same by the multiplier, setting down the last overplus, which is 2, on the right of the former remainder 8. Next multiply these two remainders, 2 and 8, together, and from their product 16, cast out the 9, and there remains 7, which set down over the two former. Lastly, add up, in the same manner, all the figures of the whole product of the multiplication, viz 14872012, casting out the 9's, and then there remains 7, to be set down under the two first remains. Then when the figure at top, is the same as that at bottom, as they are here both 7's, the work it may be presumed is right; but if these two figures should not be the same, it is certainly wrong.
2. To Multiply Money, or any other thing, consisting of different Denominations together, by any number, usually called Compound Multiplication. Beginning at the lowest, multiply the number of each denomination separately by the multiplier, setting down the products below them. But if any of these products amount to as much as 1 or more of the next higher denominations, carry so many to the next product, and set down only the overplus. For Ex. To find the amount of 9 things at 1l 12s 4 1/2d. each; or to multiply 1l 12s 4 1/2d by 9.
| l | s | d |
| 1 | 12 | 4 1/2 |
| 9 | ||
| 14 | 11 | 4 1/2 |
3. To Multiply Vulgar Fractions.—Multiply all the given numerators together for the numerator of the product, and all the denominators together for the denominator of the product sought.
Thus, 2/3 multiplied by 4/5, or .
And .
And here it may be noted that, when there are any common numbers in the numerators and denominators, these may be omitted from both, which will make the operation shorter, and bring out the whole product in a fraction much simpler and in lower terms. Thus, , by leaving out the two 3's, become
Also, when any numerators and denominators will both abbreviate or divide by one and the same number, let them be divided, and the quotients used instead of them. So, in the above example, after omitting the two 3's, let the 2 and 6 be both divided by 2, and use the quotients 1 and 3 instead of them, so shall the expression become , as before.
4. To Multiply Decimals.—Multiply the given numbers together the same as if they were whole numbers, and point off as many decimals in the whole product as there are in both factors together;
| 2.305 |
| 21.86 |
| 13830 |
| 18440 |
| 2305 |
| 4610 |
| 50.38730 |
5. Gross Multiplication, otherwise called Duodecimal Arithmetic, is the multiplying of numbers together whose subdivisions proceed by 12's; as feet, inches, and parts, that is 12th parts, &c; a thing of very srequent use in squaring, or multiplying toge-| ther the dimensions of the works of bricklayers, carpenters, and other artificers. For Example. To multiply 5 feet 3 inches by 2 feet 4 inches. Set them down as in the
| F | I |
| 5 | 3 |
| 2 | 4 |
| 10 | 6 |
| 1 | 9 |
| 12 | 3 |
6. Multiplication in Aigebra. This is performed, 1. When the quantities are simple, by only joining the letters together like a word; and if the simple quantities have any coefsicients or numbers joined with them, multiply the numbers together, and prefix the product of them to the letters so joined together. But, in algebra, we have not only to attend to the quantities themselves, but also to the signs of them; and the general rule for the signs is this: When the signs are alike, or the same, either both + or both -, then the sign of the product will always be + ; but when the signs are different, or unlike, the one +, and the other -, then the sign of the product will be -. Hence these
| Mult. | + a | - 2a | + 6x | - 8x | - 3ab |
| By | + b | - 4b | - 3a | + 5a | - 5ac |
| Products | + ab | + 8ab | - 18ax | - 40ax | + 15a2bc |
2. In Compound quantities, multiply every term or part of the multiplicand by each term separately of the multiplier, and set down all the products with their signs, collecting always into one sum as many terms as are similar or like to one another.
| a + b | a - b | a + b |
| a + b | a - b | a - b |
| a2 + ab | a2 - ab | a2 + ab |
| + ab + b2 | - ab + b2 | - ab - b2 |
| a2 + 2ab + b2 | a2 - 2ab + b2 | a2 - b2 |
| 2a - 3b | 2a + 4x | a2 - ax |
| 4a + 5b | 2a - 4x | 2a + 2x |
| 8a2 - 12ab | 4a2 + 8ax | 2a3 - 2a2x |
| + 10ab - 15b2 | - 8ax - 16x2 | + 2a2x - 2ax2 |
| 8a2 - 2ab - 15b2 | 4a2 - 16x2 | 2a3 - 2ax3 |
3. In Surd quantities, if the terms can be reduced to a common surd, the quantities under each may be multiplied together, and the mark of the same surd prefixed to the product; but if not, then the different surds may be set down with some mark of multiplication between then, to denote their product.
| 7√(ax) | √7 | √3(7ab) | √(12a) | 6a√(2cx) |
| 5√(cx) | √5 | √3(4ac) | √(3a) | 2b√(3ax) |
| 35√(acx2) | √35 | √3(28a2bc) | √(36a2) = 6a | 12ab√(6acx2) |
4. Powers or Roots of the same quantity are multiplied together, by adding their exponents. Thus, ; and : also ; and
To Multiply Numbers together by Logarithms.—This is performed by adding together the logarithms of the given numbers, and taking the number answering to that sum, which will be the product sought.
Des Cartes, at the beginning of his Geometry, performs Multiplication (and indeed all the other common arithmetical rules) in geometry, or by lines; but this is no more than taking a 4th proportional to three given lines, of which the first represents unity, and the 2d and 3d the two factors or terms to be multiplied, the product being expressed by the 4th proportional; because, in every multiplication, unity or 1 is to either of the two factors, as the other factor is to the product.
