EXPLANATION OF SIGNS USED. The following signs are made use of both in arithmetic and algebra :SIGNS OF OPERATION. 1. The sign+, called plus (which is the Latin for more), signifies additive, or to be added, and shows that the number before which it stands is to be added; thus 3 + 4 (read three plus four) means that 4 is to be added to 3, making 7. 2. The sign —, called minus (which is the Latin for less), signifies subtractive, or to be subtracted, and shows that the number before which it stands is to be subtracted; thus 13 5 (read as thirteen minus five) means that 5 is to be subtracted from 13, leaving 8. 3. The sign x (into), signifies multiplied by, and shows that the numbers between which it stands are to be multiplied; thus 3 X 4 (read three into four, or three multiplied by four), means that 3 is to be multiplied into 4, making 12. Sometimes a full stop at the bottom of the figure is used for this; thus, 2 × 7, or 2.7, are both used to express twice seven. 4. The sign, signifies divided by, and means that the number which stands before it is to be divided by the one which follows it; thus 14÷2 (read fourteen by two), means that 14 is to be divided by 2, making 7. The operation of division is also indicated by writing the divisor under the dividend with a line between them; thus 14 by 2 is also frequently denoted thus . 5. The sign, signifying equal to (or amounting to), means that the numbers between which it stands are equal to each other; that is, have the same arithmetical value, each taken as a whole. Examples of the preceding, with the results in each case, will stand thus :I. 14 and 3 equals 17, or 14 + 3 = 17. 3. 7 X 5 = 35. 2. 10- -3=7. 4. 1427, or 4* = 7. 6. The signs is to, so is, are the signs of proportion: as 2:4:: 8:16; that is, as 2 is to 4 so is 8 to 16. 7. The signs 14 4+ 10 = 20, show that the difference between 4 and 14 added to 10 is equal to 20. The line drawn over 14 and 4 is called a vinculum. 8. The signs 10 − 2 + 5 = 3 signify that the sum of 2 and 5 taken from 10 is equal to 3. 9. 82 is read 8 squared, and means that the 8 is to be multiplied by itself; thus, 8 x 8 64; hence 64 is called the square of 8. 10. The sign V, prefixed to any number, signifies that the Square Root of that number is required; thus, ✔ 64 is read the square root of 64, and means that number which when multiplied by itself gives 64; hence 64 is called the square of 8. THE PRINCIPLES AND PRACTICE OF ARITHMETIC. I. CHAPTER I. DEFINITIONS, PRELIMINARY NOTIONS, NUMERATION, ARTICLE I.-DEFINITION I. Arithmetic is the science which treats of numbers-of the mode of expressing them-of the manner of computing by them-and of the various uses to which they are applied in the practical business of life. 2. A Unit or Unity is the name given to that quantity which is to be reckoned as one when other quantities of the same kind are to be measured. Thus, each of the terms, a man, a house, a pound, &c., denotes one individual of its kind, being the same as one man, one house, one pound, &c., respectively: and these are the basis or elements by means of which several men, several houses, several pounds, &c., may be computed. 3. Number is the relation of a quantity to its unit, the notion of number being suggested by successive repititions of the individual unit; or number is the name by which we signify how many objects or things are considered whether one or more. When, for instance, we speak of one ship, two steamers, three masts, or four yards, the number of things referred to will be one, two, three, or four, according to the case; and so one, two, three, four, and the rest, are called numbers. Numbers thus viewed are termed whole numbers or integers; and for the sake of uniformity, the unit is considered the first or least integer. 4. Numbers used to express one or more individuals of specified kinds are called concrete numbers; whereas two, three, four, &c., by themselves, not particularizing the kinds of individuals, are termed abstract numbers. NOTATION. 5. Def. 1.-Notation is the method of expressing any proposed number by certain symbols or characters appropriated for that purpose. 6. Def. 2.-The symbol or representative of unit or unity is 1; but instead of other numbers being expressed by assemblages or multitudes of units placed together, which would soon become embarrassing, other characters or symbols have been invented, by means of which every number, however great, may be expressed; and instead of a different symbol being adopted for every different number, which would soon become equally inconvenient, all numbers are expressed by means of the following ten symbols, or as they are usually termed, figures, and sometimes digits, which have their names respectively annexed: the first nine of which are called by their names; and the last, which is variously denominated nought, cypher, or zero, when standing by itself has no signification, or at most, denotes the absence of number, and is to be regarded merely as an auxiliary digit, for the purposes to be hereafter explained. 7. Def. 3.—Whenever a figure is placed on the right of the same or any figure, it has, by universal agreement, the effect of increasing the value of the last-mentioned figure tenfold, at the same time that it retains its own value. Thus, beginning with the auxiliary digit o, we have the following numbers and their representations: and it is obvious that by means of two figures, this kind of notation may be continued till we arrive at ninety-nine, whose symbol is 99. 8. Def. 4.-Beyond this number, the use of two, either of the same or different figures, will not enable us to go, but a repitition of the contrivance in the last Article will, by means of more figures, supply the defect. Thus, supposing the effect of any figures being placed on the right of symbols formed as above, to be to increase all their values tenfold, we shall have one hundred one hundred and one one hundred and two so likewise of succeeding numbers; thus, we have &c. &c. five hundred and eighty-seven and again 999 will be nine hundred and ninety-nine, which is the largest number capable of being expressed by three figures. Here, the first figure on the right hand is said to occupy the unit's place, the second the place of tens, and the third that of hundreds. Of the auxiliary digit o, the sole use is in the effect specified in the last two articles; and all figures to the right of it will therefore be unaffected by it.* 9. Def. 5.-In estimating numerical magnitudes, we proceed in order from hundreds, to thousands, tens of thousands, and hundreds of thousands, millions. At the end or in The word cypher is from the Arabic word Tsaphara, blank or void. the middle of any number the cypher is of use to keep the significant digits in their proper rank, when the units or the hundreds or any other denomination may be wanting, e.g., 60 means 6 tens followed by no units, 606 means 6 hundreds with no tens, but 6 units. At the beginning of a number cyphers would be useless; if so placed they could only indicate the absence of any higher class, e.g., 096 means only 9 tens and 6 units, the cypher showing that there are no hundreds, which is equally intelligible if the cypher be omitted. tens of millions, and hundreds of millions, in precisely the same manner as we have done above from units to tens, and from tens to hundreds. 10. Agreeable to the principle of Art. 7, it is assumed that "any figure placed on the right of one or more others, has the effect of increasing every one of them tenfold, without altering its own value," and this enables us to express with facility any number whatever. Thus I. 1000 will represent one thousand. 2. 5493 will represent five thousands, four hundreds, and ninety-three. 3. 23456 will represent twenty-three thousands, four hundreds, and fifty-six. 4. 729054 will represent seven hundred and twenty-nine thousands and fifty-four. 5. 1803205 will represent one million, cight hundred and three thousands, two hundreds, and five. 6. 32754081 will represent thirty-two millions, seven hundred and fifty-four thousands, and eighty-one. 7. 473025004 will represent four hundred and seventy-three millions, twenty-five thousands, and four. II. If the first three figures, beginning from the right hand, be denominated so many units, tens of units, and hundreds of units, it follows that the next three figures taken the same way will be thousands, tens of thousands, and hundreds of thousands; the next three in order will be millions, tens of millions, and hundreds of millions, and so on. Whence to express in figures any number proposed, we have only to consider in which of these divisions each part of it ought to be found, observing that three figures from the right must be taken to make each division complete, before we proceed to the next. EXAMPLES. Ex. 1. Express, by means of figures, thirty-five thousand eight hundred and nineteen. Here, eight hundred and nineteen belongs to the first division on the right, and is written 819; also, thirty-five thousand must be found in the second division from the right, and is 35, whence the proposed number may be expressed by 35819. Ex. 2. Express in figures the number five million twenty-five thousand six hundred and seven. In this case the first division on the right will be 607; the second will be 025, the digit o being affixed to the left of the others without altering their value, to make up the required number of three; and the third is 5, so that the expression will be 5025607. Ex. 3. Express by figures the following number, five hundred and seventy million two hundred and six thousand and fifty-four. Here, the first division is 054, the o altering only the value of the figures in the subsequent divisions; the second division is 206; and the third is 570, whence the number proposed is correctly expressed by 570206054. EXAMPLES FOR PRACTICE. 1. One hundred. 2. One hundred and one. 3. One hundred and ten. 4. Nine thousand and nine. 5. Nine thousand and ninety. 6. Nine thousand nine hundred and nine. 7. Five thousand and seventy-four. 8. Ten thousand seven hundred. 9. Ninety thousand and ninety. 10. Three hundred and five thousand. II. Nine hundred thousand nine hundred. 12. Five hundred and five thousand five hundred and fifty. 13. One million three thousand and eight. 14. Five million thirty thousand and forty-nine. 15. Nine million nine hundred thousand and six. 16. Fifty-eight million and nine. 17. Seventy million three hundred and two thousand four hundred and forty-one. 18. Two hundred and twenty-two millions and thirty-five. 19. Six hundred and four million sixty thousand and five. 20. Eight hundred million three thousand and thirty-three. 21. Nine hundred million nine hundred thousand and nine hundred. 22. Seven hundred million and seven. 23. One hundred and eighty million. 24. Five hundred million. 25. Five hundred and eighty million two hundred and forty-five thousand one hundred and ninety-two. 26. Seven hundred and seven million seven thousand and seventy-seven. 12. This method of notation can never present any difficulty, provided it be carefully remembered that every division of figures as we proceed from the right hand towards the left must be completed, as far as it is possible, and, by a little practice, we shall be enabled to write down any number by beginning at the left hand. EXAMPLE. Ex. 1. To write down six hundred and thirteen millions five hundred and nineteen, we observe that the division of millions will be 613; that of thousands will be ooo, and that of units 519; so that the number is expressed in arithmetical symbols by 613000519. 13. It will be observed, from what has been said, that each of the nine figures or digits, 1, 2, 3, 4, 5, 6, 7, 8, 9, has an absolute value of itself, whereas the auxiliary digit o has no such value; and on this account the former are termed significant figures, in contra-distinction to the last. It will, moreover, have occurred to the reader that every one of these significant digits, in addition to its absolute value, which is fixed and certain, possesses also a local value dependent upon the situation in which it is placed; thus, in the expression of the number four thousand three hundred and twenty-one, which will be 4321, the in the first place on the right hand retains its absolute value; the second figure 2, in its situation, denotes two tens, or twenty; the third is three hundred, and the fourth is four thousand; so that the local values of 2, 3, and 4, are respectively, ten times, a hundred times, and a thousand times, as great as their absolute values; and it is the circumstance of assigning to each of the significant figures a local as well as an absolute value, which confers upon the system the immense power which it possesses. I NUMERATION. 14. Def. 6.-Numeration is the art of reading or estimating the value of a number expressed by figures, and is, therefore, the reverse of Notation. 15. From the circumstance of every figure possessing a local as well as an absolute value, it follows that the value of each figure must be estimated by the place which it occupies; hence, a figure standing by itself expresses so many units; a figure in the second place so many tens; a figure in the third place so many hundreds, and so on; consequently, if we suppose any numerical expression to be divided into periods or portions, each consisting of three figures as far as they go, the figures of the period on the right will be D |