Page:Collier's Cyclopedia of Commercial and Social Information.djvu/249

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lars, and cents under cents, in such order that the points stand in a vertical line.

The sign $, and the point ( . ) should never be omitted.

MULTIPLICATION is a short method of addition under certain circumstances. If we wish to ascertain the amount of twelve times the number 57, instead of setting down twelve rows of 57, and adding them together, we adopt a shorter plan by which we come to the same conclusion. For ascertaining the amount of all simple numbers as far as 12 times 12, young persons commit to memory the following Multiplication Table, a knowledge of which is of great value, and saves much trouble in after-life:—

1	2	3	4	5	6	7	8	9	10	11	12
2	4	6	8	10	12	14	16	18	20	22	24
3	6	9	12	15	18	21	24	27	30	33	36
4	8	12	16	20	24	28	32	36	40	44	48
5	10	15	20	25	30	35	40	45	50	55	60
6	12	18	24	30	36	42	48	54	60	66	72
7	14	21	28	35	42	49	56	63	70	77	84
8	16	24	32	40	48	56	64	72	80	88	96
9	18	27	36	45	54	63	72	81	80	99	108
10	20	30	40	50	60	70	80	90	100	110	120
11	22	33	44	55	66	77	88	99	110	121	132
12	24	36	48	60	72	84	96	108	120	132	144

This table is so well known, that it is almost superfluous to explain that, when any number in the top row is multiplied by any number in the left-hand side row, the amount is found in the compartment or square beneath the one and opposite the other. Thus, 2 times 2 are 4; 5 times 6 are 30; 12 times 12 are 144.

The multiplying of numbers beyond 12 times 12 is usually effected by a process of calculation in written figures. The rule is to write down the number to be multiplied, called the multiplicand; then place under it, on the right-hand side, the number which is to be the multiplier, and draw a line under them. For example, to find the amount of 9 times 27, we set down the figures thus—

27	(Multiplicand.)
9	(Multiplier.)
——
243	(Product.)

Beginning with the right-hand figure, we say 9 times 7 are 63; and putting down 3 we carry 6, and say g times 2 are 18, and 6, which was carried, makes 24; and writing down these figures next the 3, the product is found to be 243.

5463

34

————

21852

16389

————

185742

76843

4563

————

230,529

4,610,58 

38,421,5  

307,372   

——————

350,634,609

When the multiplier consists of two or more figures place it so that its right-hand figure comes exactly under the right-hand figure of the multiplicand; for instance, to multiply 5463 by 34, we proceed as here shown. Here the number is multiplied, first by the 4, the product of which being written down, we proceed to multiply by 3, and the amount produced is placed below the other, but one place farther to the left. A line is then drawn, and the two products added together, bringing out the result of 185742, We may, in this manner, multiply by three, four, five, or any number of figures, always placing the product of one figure below the other, but shifting a place farther to the left in each line. An example 1s here given in the multiplying of 76843 by 4563.

Multiplication is denoted by a cross of this shape ×: thus 3 × 8 = 24, signifies, that by multiplying 8 by 3, the product is 24. A number which is produced by the multiplication of two other numbers, as 30 by 5 and 6, leaving nothing over, is called a composite number. The 5 and 6, called the factors (that is, workers or agents), are said to be the component parts of 30, and 30 is also said to be a multiple of either of these numbers. The equal parts into which a number can be reduced, as the twos in thirty, are called the aliquot parts. A number which cannot be produced by the multiplication of two other numbers, is called a prime number. When the multiplicand and multiplier are the same, that is, when a number is multiplied by itself once, the product is called the square of that number: 144 is the square of 12.

SUBTRACTION is the deducting of a smaller number from a greater, to find what remains, or the difference between them.

The Sign of Subtraction is−. minus, and signifies less.

when placed between two numbers, it indicates that the one after it is to be subtracted from the one before it. Thus, 12 − 7 is read 12 minus 7, and means that 7 is to be subtracted from 12.

A Parenthesis ( ) is used to include within it such numbers as are to be considered together. A Vinculum      has the same signification, Thus, 25 — (12 + 7), or 25 — 12 + 7, signifies that from 25 the sum of 12 and 7 is to be subtracted.

Principles.—1. Only like numbers and units of the same order can be subtracted.

2. The minuend must be equal to the sum of the subtrahend and remainder.