1. A Unit is a single thing; as, 1, 1 apple, 1 cow, 1 peach. ILLUSTRATION.*—Def. 1.—A unit is a single thing. 1 cow, 1 peach represent each a single thing, and, therefore, are units. 2. A Number is an expression that tells how many; as, 1, 8, 5, 19, 4 horses, 3 oxen, 2 cows. 3. An Abstract Number is one used without reference to any object; as, 5, 18, 13, 25. 4. A Concrete, or Denominate Number, is one used with reference to some object or collection of objects; as, 3 pecks, 1 pound, 9 melons, 11 dollars. ILLUSTRATION.-A concrete, or denominate number, is one used with reference to some object, or collection of objects. 1 peck, 9 horses, 11 melons, are each used either with reference to some object or collection of objects, and are, therefore, concrete, or denominate numbers. *NOTE. The teacher should require the pupil to illustrate each of the following definitions and axioms. 5. Arithmetic treats of quantity represented by numbers. 6. Quantity is that which can be measured. 7. A Problem in Arithmetic is a question with conditions, proposed for solution. 8. A Simple Problem is a problem containing but one elementary question. 9. A Complex Problem is a problem involving two or more elementary questions. 10. An Analytical Step is the solution of one of the elementary questions composing a complex problem. 11. An Analysis* is a simplification of a complex problem by resolving it into elementary questions. 12. A Rule is a concise direction for solving problems, and may be deduced from an analysis. 13. A Sign in Arithmetic is a symbol employed to show the relations of numbers, and to denote operations to be performed upon them. 1 14. An Axiom is a self-evident truth. 15. The principal, or fundamental rules in Arithmetic, are Notation and Numeration, Addition, Subtraction, Multiplication, and Division. 16. AXIOMS. (a.) Things can be added to like things only. *NOTE.-The term Analysis is, also, somewhat loosely applied to the solution of an Elementary Question by means of an Arithmetical formula, thus: ELEMENTARY QUESTION.-If 1 pound of sugar costs 11 cents, 'what will 6 pounds cost? ARITHMETICAL FORMULA.-If 1 pound of sugar costs 11 cents, 6 pounds will cost 6 times 11 cents, which are 66 cents. (6.) Things can be subtracted from like things only. (c.) Things can be compared with like things only. (d.) If the same or equal numbers be added to equal numbers, the sums will be equal. (e.) If the same or equal numbers be subtracted from equal numbers, the remainders will be equal. (f) If equal numbers be multiplied by the same or equal numbers, the products will be equal. (g.) If equal numbers be divided by the same or equal numbers, the quotients will be equal. (h.) If the same number be both added to, and subtracted from another number, the value of the latter number will not be changed. (i.) If a number be both multiplied and divided by the same number, its value will not be changed. (j.) If two numbers be equally increased or diminished, their difference will not be changed. (k.) Numbers equal to the same number each other. are equal to (1.) A number is greater than any of its parts. (m.) A number is equal to the sum of all its parts. (n.) If both divisor and dividend be multiplied by the' same number, the quotient will not be changed. (o.) If both divisor and dividend be divided by the same" number, the quotient will not be changed. QUESTIONS.-What is a unit? (1.)* What is a simple problem? (8.) What is an analytical step? (10.) What is a number? (2.) What is an abstract number? (3.) Of what does Arithmetic treat? (5.) *NOTE FOR THE TEACHER.-The numbers at the end of the questions refer to the paragraphs containing the answers. N. B. The Review questions appended to each lesson should by no means be omitted. What is a quantity? (6.) What is a problem? (7.) What is an analysis? (11.) What is a rule? (12.) What is a sign? (13.) What is an axiom? (14.) Which are the fundamental rules? (15.) Illustrate the 1st definition after the model. (1.* See note.) Illustrate the 4th definition. (See note.) Illustrate the 8th definition. Illustrate each of the first fourteen definitions on the blackboard. 17. Notation is a method of expressing numbers by words, letters, or figures. There are two methods in common use, the Roman and the Arabic. 18. The Roman, used in numbering the chapters, sections, and other divisions of books, expresses numbers by letters, thus: (a.) All other numbers are expressed by the use of these letters repeated or combined, as: T one IX nine LXXX eighty II two X ten XC IV four XXX thirty CC VI six L fifty VII seven LX sixty CCC CCCC D VIII eight LXX seventy DC IDCC seven hundred ninety DCCC eight hundred fifteen hundr'd two thousand ten thousand one million (b.) By repeating a letter the value of that letter is multiplied as many times as the letter is repeated; as, XXX, thirty, CC, two hundred. (c.) If a letter of less value be written before a letter of greater value, the value of the less is subtracted from that of the greater; as, X, ten, IX, nine; L, fifty, XL, forty. (d.) If a letter of less value be written after a letter of greater value, the value of the less is added to that of the greater; as, V, five, VI, six; M, one thousand, MD, fifteen hundred. (e.) A dash (—) placed over a letter or combination of letters increases the value a thousand times; as, X, ten, X, ten thousand; XII, twelve, XII, twelve thousand. QUESTIONS.-What is the axiom for addition? (16., a.) What is the axiom for subtraction? (16., b.) What is the axiom for comparison? (16., c.) If you add the same or equal quantities to equal quantities, how will the sums be affected? (16., d.) Illustrate each of the axioms on the blackboard. (16.) What is notation? (17.) What of the Roman notation? (18.) How many letters are used in the Roman notation? (18.) Name them. What number is represented by each? (18.) How are other numbers expressed? (18., a.) What effect does repeating a letter have? (18., b.) What effect has a letter of less value when placed before one of greater value? (18., c.) What effect has a letter of less value when placed after one of greater value? (18., d.) What effect has a dash placed over a letter or combination of letters? (18., e.) LESSON II. 19. The Arabic notation is generally used in arithmetical computations. It consists of ten characters, or figures, nine of which have each a simple and a local value. They are: 1, one; 3, three; 5, five; 7, seven; 9, nine; 2, two; 4, four; 6, six; 8, eight; 0, naught or cipher. |