·ARITHMETICAL DEVELOPMENTS. ARITHMETIC* is the SCIENCE and ART for comprehending multitude, and also magnitude, in some degree, by certain characters called figures. The characters most used to express numbers, are 1 (unit or one), 2(two), 3 (three), 4 (four), 5(five), 6 (six), 7 (seven), 8 (eight), 9 (nine); and will be those introduced into this work. Either of these characters, by itself, or if the right hand one of other figures, describes as many units as practice has allowed it to represent. Hence these figures or representatives have a value, which we will call primitive, it being a first value. Thus, practice describes the figure 2 as having twice the value of 1, or otherwise, to be equal in value to 1 and 1. Also, 3, as having the value of 2 and 1; 4, to be 3 and 1; 5, to be 4 and 1; 6, to be 5 and 1; 7, to be 6 and 1; 8, to be 7 and 1; and 9, to be 8 and 1. This may be seen best by the following: Comparative View of the Value of Figures. Arithmetic consists of theory and practice. Its theory is the Science, which considers the nature and quality of numbers, demonstrating the reason of practical operations. The practice shows the most useful and expeditious method of applying numbers to business, and is termed the Art, which admits of but three distinct fundamental divisions, viz.: NUMERATION, ADDITION, and SUBTRACTION. Each of the two last are severally * Arithmetic is from the Greek arithmos, signifying number. + Numbers are represented differently in different parts of the world. divided into Simple and Compound. Besides the two last named divisions of each, addition or subtraction, these emanate the rules of multiplication and division, with the simples and also the compounds of each, respectively. Questions. What is Arithmetic?-From what is it derived? Of what does it consist? As a science, what does it consider?-What, as an art?-What are the names of the distinct divisions and their attributes?What does either figure used in this work describe?-What name is given to their value?-Why? NUMERATION. It teaches the different values of figures by the different places occupied. Besides the primitive value, to figures are assigned a value termed local.* Any figure situated at the left of one or more figures, has a local value. By common usage, the local value of figures increase from the right hand to the left-but enumerating from right to left, except for convenience, is quite as proper. Every left hand place, in numbering from the right to the left hand, is as many times larger than its right hand place, as the ratio used in such enumeration implies. Ratio, as here applied, is the number used by adoption, to signify the respective values of the different places, according to their relative position; and, more particularly, ratio describes the number of times any place is increased or diminished in value, to occupy the next, or any other place in the same enumeration. The custom of using the tenfold ratio, in arithmetical computations, although the most convenient of any, is entirely arbitrary, as any ratio might be adopted, by giving it a corresponding number of characters, one of which should be 0 (zero), signifying nothing. For if we have no cipher, two or more of the characters would express the same value, and thereby cause confusion. For instance, if, by the common ratio, seven tens are to be represented the character signifying seven units, viz. 7, occupies the left hand place, receiving a local value of ten times its primitive number, and 1, 2, 3, 4, 5, 6, 8 or 9, if we have no cipher, occupies the right hand place, representing more than is required. * Local is from the Latin locus, signifying place, or value, according to the place occupied. The values of the different places may be seen by the following: Although each piece of the diagram is but 1 of the place it occupies, yet it will be seen that the tens' place is ten small pieces as large as the piece in the units' place. Also, the 1 hundred is a hundred of the unit's piece, or ten of the tens'. The tenfold ratio being adopted, it follows, that 1 in a right hand place is only 1 of the 10, making but 1 in its left hand place, ad infinitum, which may be seen by the preceding diagram. Its value is 111, viz. one hundred and eleven. And it is thus by this ratio, respecting the value of the places, without any respect to the point, which is generally used for the purpose of showing that all places to the left hand are whole numbers. Consequently, figures situated at the right hand of this distinguishing point, are but parts of a whole, and are termed Decimal Fractions; and the addition or subtraction of which is the same as whole numbers. Figures are divided into periods of six figures each, by the English method of enumeration, and the first part of any period is so many units of the period, and the latter part so many thousands, from the principle that the places of small value are but parts of the larger. The French method of enumeration admits of but three figures to the period, and although much in use, whether it is best is very doubtful. *Ad infinitum signifies to endless extent. + Decimal is from the Latin decem, signifying ten. Fraction is from the Latin frango, signifying to break. See Decimal Fractions, which follow, if it is thought necessary. The manner of dividing numbers into periods of three or six figures each, with the names, order, and several values of the same, may be seen by the following: The names of periods, which follow in succession, are Nonil lions, Decillions, Undecillions, Duodecillions and Tridecillions. Pupils should become thoroughly acquainted, in every respect, with the above table, it being the basis of arithmetic. Questions.-Numeration teaches what?-What value have figures, besides primitive? How situated, when having a local value?-How does the local value of figures increase?-How much?-How much by custom? How many figures are necessary?-How many necessary by the common ratio?-What else is necessary?-Why?-What do you understand by the diagram?-What are figures situated at the right hand of the distinguishing point?-What are these parts termed?-Why?-How do you add or subtract such?-How do you divide periods of figures, by the English method?-By the French method, how?-Repeat the table and remaining periods. SIMPLE ADDITION. It is the uniting of two or more numbers, of the same kind.* RULE.-Having placed those numbers of the same local value, in columns, under each other, draw a line underneath. Add together the units' column; if it be 10 or more, write the unit figure down, and add the left hand figure to the next column; but if the sum be less than 10, write it under the column added. Should the sum of any column added, be more than units to the next, write the right hand figure under such column, and add the remaining figures to the next column of greater value, as units and tens to that next column. Observe the same method with each column, and at the last column, write down the whole amount. Demonstration.-This rule is founded on the known axiom"The whole is equal to the sum of all its parts." It has been shown that enumeration reduces every number to a certain number of places or orders, each of which is tenfold less in value, than its preceding place. Consequently one is "carried" for every ten in addition of numbers, because common consent adopts the tenfold ratio in arithmetical computations, allowing 10 of a right hand place to be but one of a left hand place. EXAMPLE. To further illustrate the rule and axiom, let the numbers, 437, 258, 345, and 653 be decomposed, and the several parts added, as in the margin. The first is The fourth is OPERATION. 400+30 + 7 = 437 200 + 50 + 8 = 258 300+40 +5 = 345 600+ 50+ 3 = 653 † Ans. 1500+170+231693 It may be seen that each number wrote under each column, has one place more than the column added. And it is obvious, from what precedes the operation, that each left hand figure of these sums is, in value, the same as 1 in a next left hand place. But to save the trouble of setting down and adding up so many separate amounts, a method has been adopted of adding the left hand figure, or figures, to the next column of greater value. Hence we may also perceive that tens are always the numbers added to the next left hand place, when adding, which are not always tens of units, however. *Thus, 5 dollars, 3 dollars, and 4 dollars are of the same kind, viz. dollars-but, 5 hats, 3 slates, and 4 books are not all hats, slates, nor books. The sign+(plus) signifies added to. The sign = signifies equal to. |