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COMMON-PLACE BOOK,

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CHAPTER I.

ARITHMETIC.

SECTION I.-Definitions and Notation.

ARITHMETIC is the science of numbers.

We give the name of number to the assemblage of many units, or of many parts of an assumed unit; unit being the quantity which, among all those of the same kind, forms a whole, which may be regarded as the base or element. Thus, when I speak of one house, one guinea, I speak of units, of which the first is the thing called a house, the second that called a guinea. But when I say four houses, ten guineas, three quarters of a guinea, I speak of numbers, of which the first is the unit house repeated four times; the second is the unit guinea repeated ten times; the third is the fourth part of the unit guinea repeated three times.

In every particular classification of numbers, the unit is a measure taken arbitrarily, or established by usage and convention.

Numbers formed by the repetition of an unbroken unit are called whole numbers, or integers, as seven miles, thirty shillings: those which are formed by the assemblage of many parts of a unit are called fractional numbers, or simply fractions; as two-thirds of a yard, three-eighths of a mile.

When the unit is restricted to a certain thing in particular, as one man, one horse, one pound, the collection of many of those units is called a concrete number, as ten men, twenty horses, fifty pounds. But if the unit does not denote any particular thing, and is expressed simply by one, numbers which are constituted of such units are denominated discrete or abstract, as five, ten, thirty. Hence, it is evident that abstract numbers

can only be compared with their unit, as concrete numbers are compared with, or measured by, theirs; but that it is not possible to compare an abstract with a concrete number, or a concrete number of one kind with a concrete number of another; for there can exist no measurable relations but between quantities of the same kind.

The series of numbers is indefinite; but only the first nine of them are expressed by different characters, called figures: thus,

Names. one, two, three, four, five, six, seven, eight, nine. Figures. 1, 2, 3, 4, 5, 6, 7, 8, 9. Besides these, another character is employed, namely 0, called the cipher or zero; which has no particular value of itself, but by its position is made to change the value of any significant figures with which it is connected.

In the system of numeration now generally adopted, and borrowed from the Indians, an infinitude of words and characters is avoided, by a simple yet most ingenious expedient, which is this-every figure placed to the left of another assumes ten times the value that it would have if it occupied the place of the latter.

Thus, to express the number that is the sum of 9 and 1, or ten units, called ten, we place a 1 to the left of a 0, thus 10. So again the sum of 10 and 1, or eleven, is represented by 11; the sum of 11 and 1, or of 10 and 2, called twelve, is represented by 12; and so on for thirteen, fourteen, fifteen, &c. denoted respectively by 13, 14, 15, &c., the figure 1 being all along equivalent to ten, because it occupies the second rank.

In like manner, twenty, twenty-one, twenty-two, &c. are represented by 20, 21, 22, because the 2 in the second rank is equivalent to twice ten, or twenty. And thus we may proceed with respect to the numbers that fall between twenty and three tens or thirty 30, four tens or forty 40, five tens or fifty 50, six tens or sixty 60, seven tens or seventy 70, eight tens or eighty 80, nine tens or ninety 90. After 9 are added to the 90 (ninety) numbers can no longer be expressed by two figures, but require a third rank to the left hand of the second.

The figure that occupies the third rank, or of hundredths, is expressed by the word hundred. Thus 369, is read three hundred and sixty-nine; 428, is read four hundred and twentyeight; 837, eight hundred and thirty-seven and so on for all numbers that can be represented by three figures.

But if the number be so large that more than three figures are required to express it, then it is customary to divide it into periods of three figures each, reckoning from the right hand towards the left, and to distinguish each by a peculiar name.

The second period is called that of thousands, the third that of millions, the fourth that of milliards or billions,* the fifth that of trillions, and so on; the terms units, tens, and hundreds, being successively applied to the first, second, and third ranks of figures from the right towards the left, in each of these periods.

Thus, 1111, is read one thousand one hundred and eleven. 23456, twenty-three thousands, four hundred and fifty-six. 421835, four hundred and twenty-one thousands, eight hundred and thirty-five.

732846915, seven hundred and thirty-two millions, eight hundred and forty-six thousands, nine hundred and fifteen.

The manner of estimating and expressing numbers we have here described is conformable to what is denominated the decimal notation. But, besides this, there are other kinds invented by philosophers, and others indeed in common use; as the duodecimal, in which every superior name contains twelve units of its next inferior name; and the sexagesimal, in which sixty of an inferior name are equivalent to one of its next superior. The former of these is employed in the measurement and computation of artificer's work; the latter in the division of a circle, and of an hour in time.

To the head of notation we may also refer the explanation. of the principal symbols or characters employed to express operations or results in computation. Thus,

The sign + (plus) belongs to addition, and indicates that the numbers between which it is placed are to be added together, Thus, 5+7 expresses the sum of 5 and 7, or that 5 and 7 are to be added together.

The sign (minus) indicates that the number which is placed after it is to be subtracted from that which precedes it. So, 9. 3 denotes that 3 is to be taken from 9.

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The sign denotes difference, and is placed between two quantities when it is not immediately evident which of them is the greater.

The sign x (into), for multiplication, indicates the product. of two numbers between which it is placed. Thus 8 × 5 denotes 8 times 5, or 40.

The sign (by), for division, indicates that the number which precedes it is to be divided by that which follows it; and the quotient that results from this operation is often

It has been customary in England to give the name of billions to millions of millions, of trillions to millions of millions of millions, and so on: but the method here given of dividing numbers into periods of three figures instead of six, is universal on the continent; and, as it seems more simple and uniform than the other, I have adopted it.

represented by placing the first number over the second with a small bar between them. Thus, 15÷8 denotes that 15 is to be divided by 8, and the quotient is expressed thus .

The sign, two equal and parallel lines placed horizontally, is that of equality. Thus, 2+3+4=9, means that the sum of 2, 3, and 4, is equal to 9.

Inequality is represented by two lines so drawn as to form an angle, and placed between two numbers, so that the angular point turns towards the least. Thus, 7> 4, and A > B, indicate that 7 is greater than 4, and the quantity represented by A greater than the quantity represented by B: on the other hand, 35 and C <D indicate that 3 is less than 5, and C less than D.

Colons and double colons are placed between quantities to denote their proportionality. So, 3 : 5 :: 9: 15, signifies that 3 are to 5 as 9 to 15, or = %.

The extraction of roots is indicated by the sign ✔, with a figure occasionally placed over it to express the degree of the root. Thus 4 signifies the square root of 4, 27 the cube root of 27, ✔ 16 the fourth or biquadrate root of 16; and so on. These characters find their most frequent use in algebra and the higher departments of mathematics; but may, without hesitation, be employed whenever they secure brevity without a sacrifice of perspicuity.

SECTION II.-Addition of Whole Numbers.

ADDITION is the rule by which two or more numbers are collected into one aggregate or sum.

Suppose it were required to find the sum of the numbers 3731, 349, 12487, and 54. It is evident that if we computed separately the sums of the units, of the tens, of the hundreds, of the thousands, &c. their combined results would still amount to the same. We should thus have 15 thousands + 14 hundreds + 20 tens + 21 units, or 15000+1400+200+21; operating again upon these, in like manner, rank by rank, we should have 10 thousands + 6 thousands + 6 hundreds + 2 tens + 1, or 16621, which is the sum required.

But the calculation is more commodiously effected by this

RULE.

Place the given numbers under each other, so that units stand under units, tens under tens, hundreds under hundreds, &c.

Add up all the figures in the column of units, and observe for every ten in its amount to carry one to the place of tens in

the second column, putting the overplus figure in the first column.

Proceed in the same manner with the second column, then with the third, and so on till all the columns be added up; the figures thus obtained in the several amounts indicate, according to the rules of notation, the sum required.

Note. Whether the addition be conducted upwards or downwards, the result will be the same; but the operation is most frequently conducted by adding upwards.

3731

349

12487

54

Example. Taking the same numbers as before, and disposing them as the rule directs, we have 4+7+9+1=21, of which we put down the 1 in the place of units, and carry the 2 to the tens : then 2+5+8+ 4 + 3=22, of which we put down the left hand 2 in the place of tens, and carry the other to the hundreds then 2+4+3+7= 16, of which the 6 is put in the place of hundreds, and the 1 carried to the thousands. This progress continued will give the same sum as before.

Other Examples.

16621

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SECTION III.—Subtraction of Whole Numbers.

SUBTRACTION is the rule by which one number is taken from another, so as to show the difference or excess.

The number to be subtracted or taken away is called the subtrahend; the number from which it is to be taken, the minuend: the quantity resulting, the remainder.

RULE.

Write the minuend and the subtrahend in two separate lines, units under units, tens under tens, and so on.

Beginning at the place of units, take each figure in the subtrahend from its corresponding figure in the minuend, and write the difference under those figures in the same rank or place.

But if the figure in the subtrahend be greater than its cor

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