Fractions arise from Division, 16 Miscellaneous Questions, involving the Principles of the preceding Rules, 40 Different Denominations, Federal Money, Reduction, COMPOUND NUMBERS. to find the Value of Articles sold by the 100, or 1000, Tables of Money, Weight, Measure, &c., Addition of Compound Numbers, Subtraction, Multiplication and Division, FRACTIONS. COMMON OF VULGAR. Their Notation, Proper, Improper, &c. To change an Improper Fraction to a Whole or Mixed Number, a Mixed Number to an Improper Fraction, To reduce a Fraction to its lowest Terms, Greatest Common Divisor, how found, To divide a Fraction by a Whole Number; two ways, To multiply a Fraction by a Whole Number; two ways, a Whole Number by a Fraction, . one Fraction by another, . General Rule for the Multiplication of Fractions, General Rule for the Division of Fractions, Common Denominator, how found, Rule for the Addition and Subtraction of Fractions, Reduction of Fractions, . DECIMAL Their Notation, Division of Decimal Fractions, Multiplication of Decimal Fractions, Addition and Subtraction of Decimal Fractions, Pago To reduce Vulgar to Decimal Fractions, Reduction of Decimal Fractions, To reduce Shillings, &c., to the Decimal of a Pound, by Inspection, Reduction of Currencies, To reduce English, &c. Currencies to Federal Money, ..... Time, Rate per cent., and Amount given, to find the Principal, Time, Rate per cent., and Interest given, to find the Principal, Principal, Interest, and Time given, to find the Rate per cent., . Principal, Rate per cent., and Interest given, to find the Time, . To find the Interest on Notes, Bonds, &c., when partial Payments have 124 Application and Use of the Square Root, see Supplement, Annuities at Compound Interest, Having the Diameter of a Circle, to find the Circumference; or, having the Circumference, to find the Diameter, ex. 171-175. ARITHMETIC. NUMERATION. T1. A SINGLE or individual thing is called a unit, unity, or one; one and one more are called two; two and one more are called three; three and one more are called four; four and one more are called five; five and one more are called six; six and one more are called seven; seven and one more are called eight; eight and one more are called nine; nine and one more are called ten, &c. These terms, which are expressions for quantities, are called numbers. There are two methods of expressing numbers shorter than writing them out in words; one called the Roman method by letters, and the other the Arabic method by figures. The latter is that in general use. In the Arabic method, the nine first numbers have each an appropriate character to represent them. Thus, In the Roman method by letters, I represents one; V, five; X, ten; L, fifty; C, one hundred; D, five hundred; and M, one thousand. As often as any letter is repeated, so many times its value is repeated, un. Less it be a letter representing a less number placed before one representing a greater; then the less number is taken from the greater; thus IV represents four, IX nine, &c., as will be seen by the following • I is used instead of D to represent five hundred, and for every additional annexed at the right hand, the number is increased ten times. †CIO is used to represent one thousand, and for every C and end, the number is increased ten times. put at each A line over any number Increases its value one thousand times. A unit, unity, or one, is represented by this character, 123456789 Seven Eight Nine Ten has no appropriate character to represent it; but is consi- One ten and one unit are called Three tens are called Four tens are called Five tens are called Eight tens are called Nine tens are called Ten 10 Eleven 11 Twelve 12 Thirteen 13 Fourteen 14 Fifteen 15 Sixteen 16 Seventeen 17 Eighteen 18 Nineteen 19 Twenty 20 Thirty 30 Forty 40 Fifty 50 Sixty 60 Seventy 70 Eighty 80 Ninety 90 Ten tens are called a hundred, which forms a unit of a still higher order, consisting of hundreds, represented by the same character (1) as a unit of each of the foregoing orders, but is written one place further toward the left hand, that is, on the left hand side of tens; thus, One hundred 100 One hundred, one ten, and one unit, are called One hundred and eleven 111 T2. There are three hundred sixty-five days in a year. In this number are contained all the orders now described, viz. units, tens, and hundreds. Let it be recollected, units occupy the first place on the right hand; tens the second place from the right hand; hundreds the third place. This number may now he decomposed, that is, separated into parts, exhibiting each order by itself, as follows:-The highest order, or hundreds, are three, represented by this character, 3; but, that it may be made to occupy the third place, counting from the right hand, it must be followed by two ciphers, thus, 300, (three hundred.) The next lower order, or tens, are six, (six tens are sixty,) represented by this character; ; but, that it may bccupy the second place, which is the place of tens, it must be followed by one cipher, thus, 60, (sixty.) The lowest order, or units, are five, represented by a single character, thus, 5, (five.) We may now combine all these parts together, first writing down the five units for the right hand figure, thus, 5; then the six tens (60) on the left hand of the units, thus, 65; then the three hundreds (300) on the left hand of the six tens, thus, 365, which number, so written, may be read three hundred, six tens, and five units; or, as is more usual, three hundred and sixty-five. T3. Hence it appears that figures have a different value according to the PLACE they occupy, counting from the right hand towards the left. Hund. Take for example the number 3 3 3, made by the same figure three times repeated. The 3 on the right hand, or in the first place, signifies 3 units; the same figure, in the second place, signifies 3 tens, or thirty; its value is now increased ten times. Again, the same figure, in the third place, signifies neither 3 units, nor 3 tens, but 3 hundreds, which is ten times the value of the same figure in the place immediately preceding, that is, in the place of tens; and this is a fundamental law in notation, that a removal of one place towards the left increases the value of a figure TEN TIMES. Ten hundred make a thousand, or a unit of the fourth order. Then follow tens and hundreds of thousands, in the same manner as tens and hundreds of units. To thousands succeed millions, billions, &c., to each of which, as to units and to thousands, are appointed three places,* as exhibited in the following examples : |