ARITHMETIC. ARITHMETIC is the art of computing by numbers. A NUMBER, in Arithmetic, is an expression that denotes how many. Written numbers may be expressed by words, by letters, or by combining the ten Arabic figures invented for this purpose: One, Two, Three, Four, Five, Six, Seven, Eight, Nine, Naught. 1, 2, 3, 4, 5, 6, 7, 8, 9, 0. The first nine of these characters are called digits, from the Latin digitus, a finger: because the numbers they express are often counted on the fingers. The tenth character, 0, expresses no value by itself, and is therefore commonly called Naught. It is also called Zero, or Cipher. The number One is also called a Unit, and serves as a standard for comparing all numbers of a kind. Thus when we speak of ten miles, twenty houses, fourteen men, or sixty bushels, we are supposed already to have a clear idea of one mile, one house, one man, or one bushel, which is assumed as the unit. All numbers may be considered as derived from the Unit; for one and one are two; two and one are three; three and one are four; and so on to any extent we please. An ABSTRACT NUMBER, is a simple number, which has no reference to any thing in particular; as five, seventeen, six. An APPLICATE, or CONCRETE NUMBER, is a collection of particular things; as five books, seventeen days, six men. An INTEGER is a whole number; as ten, twenty-nine, forty-five. A FRACTION is a part or parts of a unit; as one-half, two-thirds, seven-tenths. Arithmetic consists of five fundamental operations, in each of which something is given, and something is required; namely, Numeration, Addition, Subtraction, Multiplication, and Division. In Numeration, numbers are either given in figures to be read in words, or given in words to be written in figures. In Addition, two or more numbers are given, and their sum or amount is required. In Subtraction, two numbers are given, and their difference is required. In Multiplication, two numbers are given, and their product is required. In Division, two numbers are given, and their quotient is required. CHAPTER I. NUMERATION. NUMERATION is the process by which we express numbers in letters or figures, and read them when so expressed. There are two modes of numeration now in use: the ROMAN and the ARABIC. The Roman method employs the following letters: One, Five, Ten, Fifty, One hundred, Five hundred, One thousand. I, V, X, L, C, D, M. When a letter stands before another which denotes a larger number, the value of the first must be taken from that of the second.* In all other cases the values of the different letters must be added together, as in the following table: The Arabic method is much more convenient, and is employed in all the ordinary operations of Arithmetic. By this method, every figure is made to represent a value ten times larger by each removal to the left, and ten times smaller by each removal to the right. Thus if 1 represent one, 10 will represent ten: 100, ten tens, or one hundred; 20, two tens, or twenty; 40, four tens, or forty; &c. A point called the decimal point, or separatrix, is placed at the right of units, to distinguish them from parts of units or decimals. The first figure at the left of the separatrix denotes units, the second tens, the third hundreds. These three are usually embraced under the name of units; the next three figures embrace units, tens, and hundreds, of thousands; the next, units, tens, and hundreds, of millions, &c., as in the following table. * A dash placed over any letter, increases its value a thousand 54 123 006 862 040 000 000 016 500 000 000 479 513 660: The places at the right of units are called tenths, hundredths, thousandths, &c., (denoting that the unit is divided into 10, 100, 1000, &c. parts,) as in the following table. In reading decimals, read first the whole number, (if any,) then read the decimal as if it expressed a whole number, assigning it the value of the right hand decimal place. Thus 29000001.004063, is read, 29 million and 1, and 4 thousand and sixty three millionths. *The English formerly divided numbers into periods of six figures each, calling a million million a billion, a million billion a trillion, &c. The division into periods of three figures is now generally adopted. |