# 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 | + 15a^{2}bc |

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 |

a^{2} + ab | a^{2} - ab | a^{2} + ab |

+ ab + b^{2} | - ab + b^{2} | - ab - b^{2} |

a^{2} + 2ab + b^{2} | a^{2} - 2ab + b^{2} | a^{2} - b^{2} |

2a - 3b | 2a + 4x | a^{2} - ax |

4a + 5b | 2a - 4x | 2a + 2x |

8a^{2} - 12ab | 4a^{2} + 8ax | 2a^{3} - 2a^{2}x |

+ 10ab - 15b^{2} | - 8ax - 16x^{2} | + 2a^{2}x - 2ax^{2} |

8a^{2} - 2ab - 15b^{2} | 4a^{2} - 16x^{2} | 2a^{3} - 2ax^{3} |

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√(acx^{2}) | √35 | √^{3}(28a^{2}bc) | √(36a^{2}) = 6a | 12ab√(6acx^{2}) |

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.