UNIVERSITY CALIFORNIA THE PRINCIPLES AND PRACTICE OF ARITHMETIC. CHAPTER I. DEFINITIONS, PRELIMINARY NOTIONS, NOTATION, NUMERATION, AND FUNDAMENTAL OPERATIONS. ARTICLE I. DEFINITION I. ARITHMETIC is that part of Mathematical Science which treats of the computation of magnitudes, and their relations to one another, with reference to the consideration of how many or how few. 2. DEF. 2. An Unit, or, as it is generally called, Unity, is the representation of any thing considered in its individual capacity, without regard to the parts of which it is made up, and it is the Base or Element of all arithmetical computations and comparisons. 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 bases or elements by means of which several men, several houses, several pounds, &c., may be computed or compared. 3. DEF. 3. Number signifies a multitude or collection of two or more units, or denotes an assemblage of two or more distinct objects of the same kind. Thus, two men, three houses, four pounds, &c., denote more than one individual of the same kind, the single individuals being supposed to be repeated twice, thrice, four times, &c., respectively. Numbers thus viewed are termed Whole Numbers or Integers, and, for the sake of uniformity, the Unit or Unity is considered the first or least integer. 4. DEF. 4. Numbers used to express one or more individuals of specified kinds, as in the instances just given, are called applicate or 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 by means of certain symbols or characters, any proposed number or quantity arithmetically considered. 6. DEF. 2. The Symbol or Representative of unit or unity, is 1; but instead of any other number being expressed by an assemblage or multitude of units placed together, which would soon become embarrassing, other characters or symbols have been invented, by means of which every number, however small or 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: zero: 1, 2, 3, 4, 5, 6, 7, 8, 9, 0: one, two, three, four, five, six, seven, eight, nine, the first nine of which are all defined by their names; and the last, which is variously denominated Nought, Cipher, 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 hereafter to be explained. 7. DEF. 3. Whenever any figure is placed on the right of the same or any other, 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 0, 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 ninetynine, whose symbol will be 99. 8. DEF. 4. Beyond this number, the use of two, either the same or different figures will not enable us to go, but a repetition of the contrivance explained in the last article, will by means of more figures supply the defect. Thus, beginning again with the auxiliary digit 0, and supposing the effect of any figure's being placed on the right of symbols formed as above, to be to increase all their values tenfold, we shall have 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 units' place, the second the place of tens, and the third that of hundreds. Of the auxiliary digit 0, 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. Before we proceed further, we may observe that it is usual, in estimating numerical magnitudes, to proceed in order from hundreds to thousands, tens of thousands, hundreds of thousands, millions, tens of millions, and hundreds of millions, in precisely the same manner as we have done from units to tens, and from tens to hundreds, in the preceding articles. 10. DEF. 6. By a generalization of the principle adopted in article (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 being affected in its own value;" and we are thus enabled to express with facility all numbers whatsoever. Thus, 1000 will represent One thousand. 5493 will represent Five thousand, four hundred and ninety-three. 23456 will represent Twenty-three thousand, four hundred and fifty-six. 729054 will represent Seven hundred and twentynine thousand and fifty-four. 1803205 will represent One million, eight hundred and three thousand, two hundred and five. 32754081 will represent Thirty-two millions, seven hundred and fifty-four thousand and eighty-one. 473025004 will represent Four hundred and seventythree millions, twenty-five thousand and four. And similarly, for larger numbers. 11. DEF. 7. 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; and hence to express 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. 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 will be expressed in figures by 3 5819. Ex. 2. Write down in figures the number; Five millions, 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 0 being affixed to the left of the others without altering their values, to make up the required number of three, and the third is 5; so that the expression required will be 502 5607. Ex. 3. Express by figures the following number; Five hundred and seventy millions, two hundred and six thousand and fifty-four. Here, the first division is 054, the O altering only the values 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 57 0 2 0 6 0 5 4. 12. Examples of the kind just given, might easily be multiplied, but the 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 indeed by a little practice, we shall soon be enabled to write down any proposed number by beginning at the left hand. Ex. 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 000, and that of units 519: and the number expressed by the arithmetical symbols is 6 1 3 0 0 0 5 1 9. 13. A facility in expressing arithmetically, any numerical magnitude that may be presented to his notice, being of the greatest importance to the student, the following additional examples for practice are subjoined. (1) Five hundred and ninety-eight. (2) Seven thousand, eight hundred and four. (4) Six hundred and three thousand, two hundred and forty. (5) Nine millions, forty-three thousand, six hundred and two. (6) Forty-five millions, three hundred and eightyseven thousand and twenty-five. (7) Three hundred and forty-nine millions, four thousand and sixty-five. (8) One hundred millions, ten thousand and one. (9) Eight hundred and forty-two millions, two hundred and forty-eight thousand, four hundred and eighty-four. |