Of the nature and formation of equations in general 113 Of the resolution of equations by various methods ALGEBRA. DEFINITIONS. ALGEBRA is the art of computing by symbols. 1. Like quantities are those which consist of the same letters. 2. Unlike quantities are those which consist of different letters. 3. Given quantities are those whose values are known. 4. Unknown quantities are those whose values are unknown. 5. Simple quantities are those which consist of one term only. 6. Compound quantities are those which consist of several terms. 7. Positive or affirmative quantities are those which are to be added. 8. Negative quantities are those which are to be subtracted. 9. Like signs are all affirmative (+), or all negative (一). 10. Unlike signs are when some are affirmative (+) and others negative (-). 11. The co-efficient of any quantity is the number prefixed to it. B 12. A binomial quantity is one consisting of two terms; a trinomal of three terms; a quadrinomial of four, &c. 13. A residual quantity is a binomial where one of the terms is negative. 14. The power of a quantity is its square, cube, biquadrate, &c. 15. The index or exponent of a quantity is the number which denotes its root or power. 16. A surd or irrational quantity is that which has no exact root. 17. A rational quantity is that which has no radical sign (✔) or index annexed to it. 18. The reciprocal of any quantity is that quantity inverted, or unity divided by it. Thus, a + b shows that the number represented by b is to be added to that represented by a. a-b shows that the number represented by b is to be subtracted from that represented by a. as 6 represents the difference of a and b when it is not known which is the greatest. ab, or axb, or a.b denotes the product of the numbers represented by a and 6. a or shows that the number represented by a is to be divided by that represented by b. a:b::c:d denotes that a is in the same proportion to bas cis to d. x = a-b+c is an equation, showing that x is equal to the difference of a and b, added to the quantity c. 1 1 Va, or a2, is the square root of a; Ja, or a3, is the cube root of a; and a" is the ath root of a. L 22 is the square of a; as the cube of a; a4 the fourth power of a; and am the mih power of a. a b b 1 is the reciprocal of -, and - the reciprocal of a. a a a + bx c, or (a + b)c is the product of the compound quantity a + b multiplied by the simple quantity c. a+ba-b, or (a + b)÷(a-b), or quotient of a + b divided by a-b. a+b is the 6 ✓ ab + or (ab + cd) is the square root of the com pound quantity ab + cd. 3 a+b-c3 or (a+bc) is the cube, or third power, of the quantity a + b -c. 5a denotes that the quantity a is to be taken 5 times, and 7(6+ c) is 7 times b + c. 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 known quantities, and the last for those which are unknown. |