. Definitions, Notation, and Numeration Miscellaneous Questions and Examples Subtraction of Vulgar Fractions Multiplication of Vulgar Fractions Miscellaneous Examples, worked out Miscellaneous Questions and Examples Vulgar Fractions expressed as Decimals Miscellaneous Questions and Examples Concrete Numbers (Tables). Money Miscellaneous Questions and Examples Miscellaneous Questions and Examples Square and Cubic Measure. "Cross Multiplication or Duodec 4II Miscellaneous Questions and Examples Addition and Subtraction of Fractions Proof of the Rules for the multiplication and division of Decimals 332 Miscellaneous Questions and Examples Problems involving Quadratic Equations Simultaneous Quadratic Equations involving two unknown quantities 416 Problems which produce Simultaneous quadratic Equations, involving Miscellaneous Questions and Examples Division into Proportional Parts Miscellaneous Questions and Examples from Cambridge Examination ARITHMETIC. DEFINITIONS, NOTATION, AND NUMERATION. ARTICLE 1. By a Unit is meant a single object or thing, considered as one and undivided. 2. NUMBER is the name by which we signify how many objects or things are considered, whether one or more. When, for instance, we speak of one horse, two apples, three yards, or four hours, the number of the things referred to will be one, two, three, or four, according to the case ; and so one, two, three, four, and the rest, are called numbers. 3. NUMBERS are considered either as ABSTRACT or CONCRETE. Abstract numbers are those which have no reference to any particular kind of unit; thus, five, as an abstract number, signifies five units only, without any regard to particular objects. Concrete numbers are those which have reference to some particular kind of unit; thus, when we speak of five hours, six yards, seven horses, the numbers five, six, seven, are said to be concrete numbers, having reference to the particular units one hour, one yard, one horse, respectively. 4. ARITHMETIC is the science of Numbers. 5. All numbers in common Arithmetic are expressed by means of the figure 0, commonly called zero or a cypher, which has no value in itself, and nine significant figures, 1, 2, 3, 4, 5, 6, 7, 8, 9, which denote respectively the numbers one, two, three, four, five, six, seven, eight, nine. These ten figures are sometimes called Digits; but this name is often improperly limited to the nine significant figures above mentioned, which are then called the nine digits. The number one, which is represented by the figure 1, is called UNITY. 6. When any of these figures stands by itself, it expresses its simple or intrinsic value ; thus, 9 expresses nine abstract units, or nine particular things: but when it is followed by another figure, it then expresses ten times its simple value ; thus, 94 expresses ten times nine units, together with four units more: when it is followed by two figures, it then expresses one hundred times its simple value ; thus, 943 expresses one hundred times nine units, together with ten times four units, and also three units more: and so on by a tenfold increase for each additional figure that follows it. The value, which thus belongs to a figure in consequence of its position or place, is called its LOCAL VALUE. Therefore all numbers have a simple or intrinsic value, and also a local value. 7. It appears then, that in common Arithmetic we proceed towards the left from units to tens of units; from tens of units to tens of tens of units, or hundreds of units ; from hundreds of units to tens of hundreds of units, or thousands of units ; from thousands of units to tens of thousands of units ; from tens of thousands of units to tens of tens of thousands of units, that is, to hundreds of thousands of units; thence to tens of hundreds of thousands of units, or millions of units; thence to tens of millions of units, hundreds of millions of units, &c., till we come to millions of millions of units, which are called billions of units, and so on to trillions, quadrillions, &c. Thus, 10 represents one ten of units, together with no units; or, as it is briefly read, ten. 11 represents one ten of units, together with one unit; or, as it is briefly read, eleven. Similarly 12, 13, 14, 15, 16, 17, 18, 19, respectively represent one ten of units together with two, three, four, five, six, seven, eight, nine units; they are respectively read twelve, thirteen, fourteen, fifteen, sixteen, seventeen, eighteen, nineteen. The next ten numbers are expressed by 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, which respectively represent two tens of units together with no, one, two, three, four, five, six, seven, eight, nine units; they are briefly read twenty, twenty-one, twenty-two, twenty-three, twenty-four, twentyfive, twenty-six, twenty-seven, twenty-eight, twenty-nine. The next ten numbers are expressed by 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, which are respectively read thirty, thirty-one, thirty-two, thirtythree, thirty-four, thirty-five, thirty-six, thirty-seven, thirty-eight, thirtynine: we thus arrive at 40 (forty), 50 (fifty), 60 (sixty), 70 (seventy), 80 (eighty), 90 (ninety). 99 is the largest number which can be expressed by two figures, since it represents nine tens of units together with nine units; the next number |