1. MATHEMATICS is the science of quantity. 2. QUANTITY is that which can be measured; as distance, time, weight. 3. ARITHMETIC is the science of numbers. In Arithmetic quantities are represented by figures. 4. ALGEBRA is Universal Arithmetic. In Algebra quantities are represented by either letters or figures, and their relations by signs. ΝΟΤΑΤΙΟΝ. 5. ADDITION is denoted by the sign +, called plus; thus, 32, i. e. 3 plus 2, signifies that 2 is to be added to 3. 6. SUBTRACTION is denoted by the sign, called minus; thus, 74, i. e. 7 minus 4, signifies that 4 is to be subtracted from 7. 7. MULTIPLICATION is denoted by the sign X; thus, 6 × 5 signifies that 6 and 5 are to be multiplied together. Between a figure and a letter, or between letters, the sign X is generally omitted; thus, 6 ab is the same as 6 × a × b. Multiplication is sometimes denoted by the period; thus, 8.6.4 is the same as 8 × 6 × 4. 8. DIVISION is denoted by the sign; thus, 9 ÷ 3 sig nifies that 9 is to be divided by 3. cated by the fractional form; thus, Division is also indi is the same as 9 ÷ 3. An ex 9. EQUALITY is denoted by the sign; thus, $1 = 100 cents, signifies that 1 dollar is equal to 100 cents. pression in which the sign occurs is called an equation, and that portion which precedes the sign is called the first member, and that which follows, the second member. 10. INEQUALITY is denoted by the sign> or <, the smaller quantity always standing at the vertex; thus, 8 > 6 or 6 < 8 signifies that 8 is greater than 6. 11. THREE DOTS .. are sometimes used, meaning hence, therefore. 12. A PARENTHESIS (), or a Vinculum indicates that all the quantities included, or connected, are to be considered as a single quantity, or to be subjected to the same operation; thus, (84) × 3 = 12 X 3, or = 24 +1236; 21 6315 ÷ 3, or = 7 2 = 5. Without the parenthesis, these examples would stand thus: 84 X 3 = 8 + 12 = 20; 21 6 ÷ 3 -219; the sign X, in the former, not affecting 8; nor the sign, in the latter, 21. : 21 44 + 14 > 144 + 13 75. 9. Prove that 216 10. Place the proper sign (=, >, or <) between these two expressions, (247 +104) and (546 195). 11. Place the proper sign (=, >, or <) between these two expressions, (11947+ 16) and (317 104). < 12. Place the proper sign (=, >, or <) between these two expressions, (417+31) — (18772) and (127 + > 179). AXIOMS. 13. All operations in Algebra are based upon certain self-evident truths called AXIOMS, of which the following are the most common : 1. If equals are added to equals the sums are equal. 2. If equals are subtracted from equals the remainders are equal. 3. If equals are multiplied by equals the products are equal. 4. If equals are divided by equals the quotients are equal. 5. Like powers and like roots of equals are equal. 6. The whole of a quantity is greater than any of its parts. 7. The whole of a quantity is equal to the sum of all its parts. 8. Quantities respectively equal to the same quantity are equal to each other. 1* SECTION II. ALGEBRAIC OPERATIONS. 14. A THEOREM is something to be proved. 15. A PROBLEM is something to be done. 16. The Solution of a Problem in Algebra consists, 1st. In reducing the statement to the form of an equation; 2d. In reducing the equation so as to find the value of the unknown quantities. EXAMPLES FOR PRACTICE. 1. The sum of the ages of a father and his son is 60 years, and the age of the father is double that of the son; what is the age of each? It is evident that if we knew the age of the son, by doubling it we should know the age of the father. Suppose we let x equal the age of the son; then 2x equals the age of the father; and then, by the conditions of the problem, x, the son's age, plus 2x, the father's age, equals 60 years; or 3x equals 60, and (Axiom 4) x, the son's age, is of 60, or 20, and 2x, the father's age, is 40. Expressed algebraically, the process is as follows: |