ARITHMETIC. ARITHMETIC teaches to compute numbers by means of certain signs and characters. The word Arithmetic is derived from the Greek arithmos, which means number. The fundamental rules of Arithmetic are Notation, Enumeration, Addition, Subtraction, Multiplication, and Division. The scholar should become thoroughly acquainted with these rules before leaving them. I. Notation. Notation is the method of expressing numbers by characters or figures. Different kinds of notation have been employed by different nations. The Greeks and Romans employed the letters of their alphabets to express numbers. The Arabic method of notation employs these ten characters,-1 one, 2 two, 3 three, 4 four, 5 five, 6 six, 7 seven, 8 eight, 9 nine, 0 cipher. It will be necessary to consider, in this work, only the Roman and Arabic methods of expressing numbers. 1. ROMAN METHOD OF NOTATION. In this method, seven letters are used to express numbers, viz. I, V, X, L, C, D, M; I denoting 1; V, 5; X, 10; L, 50; C, 100; D, 500; and M, 1000. I are sometimes used for D, and CI for M. 2 I ...1/X .10|C II...2 XX....20 CC ...... TABLE. ...100 M........1000........10000 .....200 MM or II.2000 XX ..20000 III ..3 XXX ..30 CCC ...300 III ....3000 XXX ..... ..4000 XXXX....40000 ...30000 ..5000 ...50000 ...6000 C.. ...100000 .1000000 ...2000000 VI..6 LX ...60 DC ...600 VI. .....7000 M These characters pos- In this method of notation, the following characters are employed, 1, 2, 3, 4, 5, 6, 7, 8, 9,0. sess two values, primitive and local. of a figure is that which it has when compared with another of the same denomination. Thus, 1, 2, 3, 4, being units, 2 has twice the value of 1; 3 has three times the value of 1, &c. If 1, 2, 3, 4, are considered any other denomination than units, but are all of the same kind, then 2 has twice the value of 1; 3 has three times the value of 1, as before. The local value of a figure is that which it has, when compared with another figure of a different denomination. Thus, 1, 1, 1, taken as different denominations, have three different values, according to their locality. The local value of figures increases from the right hand to the left, in a ten-fold proportion. If we wish to express the number of days in a year, in Arabic characters, thus, 365, the first right-hand figure, 5, represents five days; the next left-hand figure, 6, represents sixty days; and the figure 3 represents three hundred days. If we wish to express in figures, one dollar, twenty-five cents, and five mills, thus, 1,255, the right-hand figure 5 represents five mills; the next left-hand figure, 5, represents five cents; the next left-hand figure, 2, represents twenty cents; and the figure 1, one hundred cents, or one dollar. On the other hand, the local value of figures diminishes in a ten-fold proportion, from the left hand to the right. II. Enumeration. Enumeration teaches to express in language the value of figures, by ascertaining the local value of each figure in the given sum. The principles of Enumeration may be best learned from the following Should the learner find it necessary to employ a larger number of figures than are given in the preceding table, he can proceed with trillions, as he did with billions, making use of quadrillion, quintillion, sextillion, septillion, octillion, nonillion, decillion, undecillion, duodecillion, &c. III. Addition. ADDITION IS UNITING TWO OR MORE NUMBERS INTO ONE SUM. RULE. Write the numbers to be added under one another, so that units, tens, hundreds, tenths, hundredths, and thousandths, &c., may be respectively under one another; that is, so that those of the same local value may be under each other. Draw a line under the whole; then, beginning at the right hand column, add them, one after another. If the sum be less than 10, write it under the column added; if it be 10 or more, set down the right-hand character or figure, and add the left-hand figure to the next column. Observe the same rule with each column, and at the last column write the whole amount. PROOF.-Begin at the top of the right-hand column, and reckon all the figures downward; if the amount agree with the answer, the work may be supposed to be right. SIGNS. The sign of addition is a short, horizontal line, crossed by a perpendicular, thus, +, and shows that a number placed before it, is to be added to a number placed after it. Two parallel horizontal lines, thus, sign of equality; as, 8+4=12. Ex. 1.-47,5 150,75 are the In this example, three numbers are given to be united into one sum. Begin at the right-hand figure, and say, five are five, and place the 5 directly under Ans. 235,625 the figure 5. Then say, seven and five are twelve, and place 2 directly under 7, and add 1 to the next left-hand column. Proceed in this manner to the left-hand column, where you must write the whole amount. 4. A man sold a horse for seventy-five dollars; a saddle for nine dollars and seventy-five cents; a bridle for one dollar, thirty-seven and a half cents. How much money did he receive? Ans. $96,125. 5. A man bought a lot of land for ninety-seven dollars, eighty-seven cents and five mills; a yoke of oxen for one hundred and twelve dollars; a horse for seventy-nine dollars and ninety cents; a wagon for thirty-two dollars and twentyfive cents; and a harness for nineteen dollars, sixty-two and a half cents. How much money did he pay? Ans. $337,65. IV. Subtraction. Subtraction is taking a less number from a greater. The greater number is called the minuend, the less number is called the subtrahend, and the result is called difference, or remainder. RULE. Write the numbers according to their local value, units under units, tens under tens, hundreds under hundreds, tenths under tenths, hundredths under hundredths, &c. Draw a line under them, and beginning at the right hand, subtract the figure of the subtrahend from the figure above it in the minuend, and write the difference below. If there be no figure in the minuend, from which to subtract the figure in the subtrahend, or, if the figure in the minuend be less than the figure below it in the subtrahend, suppose 10 to be added to the minuend, and from that amount subtract, observing to add 1 to the next left-hand figure in the subtrahend before subtracting. PROOF. Add the remainder to the subtrahend, and if the amount agree with the minuend, the work may be supposed to be right. SIGN. A short, horizontal line, thus, -, placed between |