older (262).—Annuities on Joint Lives (263).—An Annuity on the Survivor of two or more Lives (264). LIFE INSURANCE-The Premium for Life Insurance (265).—Rate of In- terest in Life Insurance (266).—Present Value of a Sum Insured (267).—Annual Premium to be paid (268).—Insurance on Joint Lives (269).—Temporary Life Insurance (270).—Annual Premiums to be paid for a Temporary Life Insurance (271).—Mutual Insurance ARITHMETIC. CHAPTER I. PRELIMINARY DEFINITIONS. -NUMERATION.-NOTATION. (1.) ARITHMETIC is the science of numbers: when practically applied, it becomes the art of calculation. Science is knowledge reduced to a system, so as to be conveniently taught, and readily applied: Art is knowledge · applied to practical purposes. The Rules of art are founded on the principles of science. Unity and Number.-Quantity. (2.) A Unit is any thing regarded simply as one: Number consists, properly, in a repetition of units, as two, three, &c.; but a unit is also called the number one. Can you recite the names of numbers, from one to a hundred? (3.) Quantity is any thing which admits of being measured, so as to be expressed in units of that quantity. Thus a line is a quantity, since a line may be measured, so as to be expressed in inches, or feet, &c.; as when we say that a line is ten inches long. Numbers are quantities; for every number is necessarily measured by a unit. Is weight a quantity? Is time a quantity? Is industry a quantity ? Abstract and Concrete Numbers. (4.) An abstract number is a number without any kind of units expressed; as the numbers one, five, ten, a hundred. (5.) A concrete number is a number of some kind of units expressed; as the numbers one book, five men, a hundred dollars. Is twenty an abstract or a concrete number? Is nine pounds an abstract or concrete number? Is two hundred miles an abstract or a concrete number? Is one thousand an abstract or a concrete number? Give two other examples of abstract, and two of concrete numbers. Similar and Dissimilar Numbers. (6.) Similar concrete numbers are such as express the same kind of units; as three dollars and five dollars. (7.) Dissimilar concrete numbers are such as express different kinds of units; as two dollars and five miles. Are four inches and seven inches similar or dissimilar concrete numbers? Are nine pounds and twelve yards similar or dissimilar? Are one cent and ten dollars similar or dissimilar? Twenty men and five hundred men? Give another example of similar concrete numbers; and another of dissimilar concrete numbers: The groundwork of a thorough knowledge of Arithmetic must be laid in the principles of Numeration and Notation; for on these principles depend the four fundamental operations in Arithmetic-Addition, Subtraction, Multiplication and Division. NUMERATION. (8.) NUMERATION is the method of naming numbers by units, tens, hundreds, &c. Thus the name Eleven denotes ten and one ; A Hundred is ten tens; A Thousand is ten hundred; A Million is a thousand thousands, A Trillion is a thousand billions; and so on, through quadrillions, quintillions, sextillions, septillions, octillions, nonillions, decillions, undecillions, duodecillions, &c. What two numbers are implied in the name fifteen? In the name sixteen? In the name seventeen? In eighteen? In nineteen? What is implied in the name forty? In fifty? Twenty-three? Thirty-one? Forty-five? Fifty-four? Sixty? Sixty-seven? Seventy ? A Quadrillion is how many? A Quintillion? A Sextillion? &c. Different Orders of Units. (9.) The naming of numbers by units, tens, hundreds, &c., introduces different orders of units in Numeration. The numbers, two, three, four, &c., are repetitions of the simple unit one, which is a unit of the first order. Twenty, thirty, forty, &c., are respectively two tens, three tens, four tens, &c.; and in these repetitions of ten, ten is regarded as a unit of the second order. In repetitions of a hundred, as two hundred, three hundred, &c., one hundred is a unit of the third order. In like manner, one thousand is made a unit of the fourth order; and so on. How many units of the first and second orders, respectively, are contained in the number thirteen? How many in the number twenty-five? In the number thirty-four? In the number forty-nine? In the number seventy-seven? How many units of distinct orders make up the number five hundred and twenty-one? How many units of distinct orders make up the number nine hundred and fifty-two? Scale of Numeration. (10.) Ten of any lower order of units make one of the next higher order; or one of a higher order makes ten of the next lower order. Thus ten units (of the first order) make one ten. How many tens make one hundred? How many hundreds make one thousand? How many hundreds make two thousand? How many hundreds make five thousand? One million is how many hundred thousand? One billion is how many hundred millions? One trillion is how many hundred billions? Numeration Table. (11.) The ascending orders of units are given in the fol⚫lowing Table (to be recited from right to left). Billions, &c., Hund. of mill., Millions, Hund. of thous., Thousands, Hundreds, Tens, Units, |