.9. Thus, a + b shews that the number represented by & is to be added to that represented by a. 20. a -b shews that the number represented by bis to be subtracted from that represented by a.' 21. ab represents the difference of a and b when it is not known which is the greatest. 22. ab, or a x b, or a.b, denotes the product of the numbers represented by a and b. α 23. ab, or, shews that the number represented by a is to b be divided by that represented by b 24. a b 25. x c d denotes that a is in the same proportion to b as c is to d. bc is an equation, shewing that x is equal to the difference of a and b, added to the quantity c. 26. √a, or aa1 is the square root of a; Va, or a is the cube 29. a + b x c, or (a + b) c is the product of the compound quantity a + b multiplied by the simple quantity c. quotient of a + b divided by a — b. 31. √ ab + cd or (a + cat) is the square root of the compound quantity ab + cd. 32. a + b • c3 or a + b c3) is the cube, or third power, of the quantity a + b — c. 33. 5a denotes that the quantity a is to be taken 5 times, and (7b+c) is 7 times b+c.. N. B. It is also to be remarked that the sign is generally expressed by the word plus, or more, and the sign, by minus, or less. And in the computation of problems, it must be observed, that the first letters of the alphabet are usually put for nown quantities, and the last for those which are unknown. EXPLANATION OF THE ALGEBRAIC METHOD 1. Known or determined quantities are expressed by a, b, c, d, &c. 2. Unknown, or undetermined quantities are expressed by x, y, z, &c. 3. The multipliers of quantities, such as 2a, 3b, 5x, &c. of a, b, c, &c. The sign (plus) between two or more quantities, means that these quantities are to be added together; as, a+b+x+ &c. means the sum of the quantities a, b, x, &c. 5. The sign (minus) before any quantity, means that the said quantity should be subtracted from the quantity which is placed before it. Thus, a b means the difference between a and b. Viz. And, b is to be subtracted from a. a + b -c, means the difference between a +b&c. c is to be subtracted from a and b, after a and b have been added together. 6. + placed before or between quantities, signifies that the said quantities are positive or affirmative quantities. prefixed, the quantities are negative. A sign not being placed before a quantity, means that the quantity is +. Thus, a 2b 4x + 3y — 5z, a general expression of quantities. Then, a, +26, + 3y, aré positive quantities. And-4x, - 5≈, are negative quantities. 7. The general sign of multiplication is x; but the product of two or more quantities is varied. Thus the product of single letters is expressed by placing such letters in alphabetical order. Thus the product of a and b is expressed by ab. of a, b, and x, of 3a, x, and y, of a + b and c +d, by ubx. by a + bx c +d." or by a+b.c+d. or by (a+b) (c+d). N. B. The line over a+b is a mark to shew that they are distinet quantities, and the line is called a vinculum. . The sign between two quantities, means that the formOT quantity is to be divided by the latter quantity. Thus. ab means that a is to be divided by 5. The powers of quantities are known by a small figure at the right hand of a letter. This figure is called the index or exponent of the power. 10. The roots of quantities are expressed by the sign, with the proper index above the sign; as, Va, expresses the square root of a, expresses the cube root of b. 11. Like quantities have the same letter, or same combination of letters. Thus, 5a and 7a; 4ab and 9 ab; 2b.ro and 6b.ra. 9. Unlike quantities have different letters, or different combina tion of letters. Thus, 4a, 3b, 7ax, 5bx2. Exemplification of the Algebraic Signs and Symbols. The use of these several signs, symbols, and abbreviations, may be exemplified in the following manner. EXAMPLE 1. In the algebraic expression a+b—e, let a=9, bi?, and o=3; then a+b-c=9+7-3 =16-3=13. Then ay2 + b2=3×6×6+5 × 5=108+25=133, There is a certain algebraic expression consisting of 6 termo connected together by the sign plus. The first term of it arises from multiplying three times the square of a by the quantity b. The second term is the sum of the squares of a and b divided by the quantity c. The third is the product of a, b, and c. The fourth is two thirds of the product of a and b. The fifth arises from dividing the square of a by the cube of d. The sixth is a fraction, whose binomial numerator is the difference between a and b, and whose trinomial denominator is the sum of the cubes of a and b and the fourth power of c. All this is expressed in one line of algebraic writing, thus ; |