ARITHMETIC. NUMBER is the abstract ratio of one quantity to another of the same kind, taken for unity. Theoretic Arithmetic is the science of numbers. Practical Arithmetic is the art of numbering. In Arithmetic there are five principal or fundamental rules for its operations, namely, Notation, Addition, Subtraction, Multiplication, and Division. NOTATION.* NOTATION teaches how to read any proposed number, expressed in characters, and to write any proposed number in characters. * As it is absolutely necessary to have a perfect knowledge of our excellent method of notation, in order to understand the reasoning made use of in the following Notes, I shall endeavour to explain it in as clear and concise a manner as possible. 1. It may then be observed, that the characters, by which all numbers are expressed, are these ten; 0, 1, 2, 3, 4, 5, 6, 7, 8, 9; 0 is called a cypher, and the rest, or rather all of them, are called figures or digits. The names and signification of these characters, and the origin or generation of the numbers they stand for, are as follow; O nothing; 1 one, or a single I. To read Numbers. RULE. To the simple value of each figure join the name of its place, beginning at the left and reading toward the right. thing, called an unit; 1 and 1 are 2 two; 2 and 1 are 3 three; 3 and 1 are 4 four; 4 and 1 are 5 five; 5 and 1 are 6 six; 6 and 1 are 7 seven; 7 and 1 are 8 eight; 8 and 1 are 9 nine; and 9 and I are ten, which has no single character; and thus by continual addition of one, all numbers are generated. 2. Besides the simple value of the figures, as above noted, they have each a local value according to the following law, namely, in a combination of figures, reckoning from right to left, the figure in the first place represents its primitive simple value; that in the second place, ten times its simple value; that in the third place, a hundred times its simple value, and so on; the value of the figure in each place being ten times the value of it in that immediately preceding it. 3. The names of the places are denominated according to their order. The first is called the place of units; the second, that of tens; the third, of hundreds; the fourth, of thousands; the fifth, of ten thousands; the sixth, of hundred thousands; the seventh, of millions, and so on. Thus, in the number 3456789; 9 in the first place signifies only nine; 8 in the second place signifies eight tens, or eighty; 7 in the third place is seven hundred; 6 in the fourth place is six thousand; 5 in the fifth place is fifty thousand; 4 in the sixth place is four hundred thousand; and 3 in the seventh place is three million; and the whole number is read thus, three million, four hundred and fifty six thousand, seven hundred and eighty nine. II. To write Numbers. RULE. Write the figures in the same order as their values are expressed in, beginning at the left, and writing toward the right; remembering to supply those places of the natural order with cyphers, which are omitted in the question. 4. A cypher, though it signifies nothing of itself, yet it occupies a place, and, when set on the right of other figures, increases their value like any other in a tenfold proportion; thus, 5 signifies only five; but 50, five tens or fifty; and 500, five hundred, &c. 5. For the more easy reading of large numbers, they are divided into periods, and half periods, each half period consisting of three figures; the name of the first period being units; that of the second, millions; of the third, billions; of the fourth trillions, &c. Also the first part of any period is the part of units; and the latter part, that of thousands. The following Table contains a summary of the whole doctrine. Figures 123,456 789,098 765,432 101,234 567 800| 1=1 A Synopsis of the Roman Notation. 2=II As often as any character is repeated, so many times is its value repeated. 3=III EXAMPLES. Write in figures the following numbers. Eighty one. Two hundred and eleven. One thousand and thirty nine. A million and a half. A hundred and four score and five thousand. Eleven thousand million, eleven hundred thousand and eleven. Thirteen billion, six hundred thousand million, four thousand and one. EXPLANATION OF CHARACTERS. NOTE. It may be proper to explain here certain signs, used in this work. = SIGNIFIES equality; as 20 shillings = 1 pound signifies, that 20 shillings are equal to one pound. + Signifies plus, or addition; as, 4+2=6. Signifies minus, or subtraction; as, 6—2—4. xor., Into, signifies multiplication; as, 3×2 or 3•2=6. →By, or ) ( signifies division; as, 6÷÷2=3, or 2)6(3. 10=X 100 C 500 D or I 1000 M or CIO For every affixed this becomes 10 times as many. For every C and Ɔ, put one at each end, it becomes ten times as much. 5000=1: or V A line over any number increases it Division may also be denoted by placing the dividend over a line, and the divisor under it; thus =6÷÷2=3. --::·· Signifies arithmetical proportion; thus 2 - 4:6 · 8; here the meaning is, 4-2-8-6-2. :::: Signifies geometrical proportion; thus 2:4:: 3:6, which is to be read, as 2 to 4 so is 3 to 6. Signifies continual arithmetical proportion, or arithmetical progression; thus, 2:4:6:8 signifies, that 2, 4, 6, and 8 are in arithmetical progression. Signifies continual geometrical proportion, or geometrical progression; thus, 2:4:8:16 signifies, that 2, 4, 8, 16, are in geometrical progression. ... Signifies therefore. Signifies the second power, or square; thus, x sig nifies the square of x. 73 Signifies the third power, or cube. m Signifies any power. , or, Signifies the square root; thus x, or x signifies the square root of x. 3, or, Signifies the cube root. n, or, Signifies any root. m , Signifies any root of any power. The number, or letter, belonging to the above signs of powers and roots, is called the index, or exponent. A line, called a vinculum, drawn over several numbers, signifies, that the numbers under it are to be considered jointly; thus, 20-7+8=5; but without the vinculum, 20-7+8=21. The same thing is also sometimes expressed by a parenthesis, inclosing two or more numbers or quantities thus, 20—(7+8)=5. Two or more letters, joined together like those of a word, signify, that the numbers, which they represent, are to be multiplied together; thus ab=axb; and abc=axbxc. |