ALGEBRA. DEFINITIONS AND NOTATION. 1. QUANTITY is anything that can be measured; as distance, time, weight, and number. 2. The UNIT of quantity is one of the same kind as the quantity, assumed as a standard, or unit of measure. 3. The MEASUREMENT of quantity is accomplished by finding the number of times the quantity contains the assumed unit of quantity. 4. MATHEMATICS is the science of quantities and their relations. 5. ALGEBRA is that branch of mathematics in which the relations of quantities are investigated, and the reasoning is abridged and generalized by means of symbols. 6. The SYMBOLS employed in Algebra are of four kinds : symbols of quantity, symbols of operation, symbols of relation, and symbols of abbreviation. SYMBOLS OF QUANTITY. 7. The SYMBOLS OF QUANTITY generally used, are the figures of arithmetic and the letters of the alphabet. The figures are used to represent known quantities and determined values, and the letters, any quantity whatever, known or unknown. 8. KNOWN QUANTITIES, or those whose values are given, when not expressed by figures, are represented by the first letters of the alphabet, as a, b, c. 9. UNKNOWN QUANTITIES, or those whose values are not given, are represented by the last letters of the alphabet, as x, y, z. 10. ZERO, or the absence of quantity, or that which is less than any assignable quantity, is represented by the symbol 0. 11. INFINITY, or that which is greater than any assignable quantity, is represented by the symbol ∞. 12. Quantities occupying similar relations in different operations are often represented by the same letter, distinguished by different accents, as a', a", a"", read, a prime, a second, a third, etc.; or by different subscript figures, as α1, α, α, α, read, a sub one, a sub two, a sub three, etc. SYMBOLS OF OPERATION. 13. The SYMBOLS OF OPERATION are certain signs or characters used to indicate algebraic operations. Thus, 14. The SIGN OF ADDITION, +, is called plus. a+b, read a plus b, indicates that the quantity b is to be added to the quantity a. α 15. The SIGN OF SUBTRACTION, is called minus. Thus, b, read a minus b, indicates that the quantity b is to be subtracted from the quantity a. The sign ~ may be written between two quantities, when it is not known which of them is the greater. Thus, ab indicates the difference of the two quantities a and b. 16. The SIGN OF MULTIPLICATION, X, is read into, or multiplied by. Thus, ab indicates that the quantity a is multiplied by the quantity b. A simple point (.) is sometimes used in the place of the sign X. The sign of multiplication is, however, usually omitted, except between two arithmetical figures separated by no other sign, and multiplication is therefore indicated by the absence of any sign. Thus, 2 ab indicates the same as 2 X a × b, or 2 . a . b. 17. The quantities multiplied are called factors, and the result of the multiplication is called the product. 18. The SIGN OF DIVISION,, is read divided by. Thus, ab indicates that the quantity a is divided by the quantity b. α Ъ Division is otherwise often indicated by writing the dividend above and the divisor below a short horizontal line. Thus, indicates the same as a b. Also, the sign of division may be replaced in an operation by a straight or curved line. Thus, a b, or b) a, indicates the same as a ÷ b. 19. The SIGN OF INVOLUTION, or EXPONENTIAL SIGN, is a figure or letter written at the right and above a quantity, to indicate the number of times the quantity is taken as a factor. Thus, in x3, the indicates that x is to be taken three times as a factor; and 3 is equivalent to ххх. The product obtained by taking a factor one or more times is called a power. Thus, a2, read a square, indicates the second power of a; a1, read a fourth, indicates the fourth power of a; The figures or letters used to indicate powers are called exponents; and when no exponent is written, the first power is understood. Thus, a is equivalent to a1. If the minus sign is prefixed to an exponent, it indicates that the quantity is to be used as a divisor. Thus, a2 a2 b-1 c3 is the same as bc39 -2 and x2 is the same as * The root of a quantity is one of its equal factors. Thus the root of a2, a3, a1, is a. 20. The SIGN OF EVOLUTION, or RADICAL SIGN, ✔, when prefixed to a quantity, indicates that some root of the quantity is to be extracted. Thus, Va indicates the square or second root of a; Va indicates the fourth root of a; and so on. The index of the root is the figure or letter written over the radical. Thus, 2 is the index of the square root, 3 of the cube root, and so on. over When the radical sign has no index written it, the index 2 is understood. Thus, a is the same as Va. A fractional exponent is also used to indicate a root. Thus, a, a, a, indicate, severally, the square, cube, and fourth root of a. In fractional exponents the numerator denotes a power, and the denominator a root, thus, b indicates the fifth power of the fourth root of b, or the fourth root of the fifth power of b. SYMBOLS OF RELATION. 21. The SYMBOLS OF RELATION are signs used to indicate the relative magnitude of quantities. 22. The SIGN OF EQUALITY,, read equals, or equal to, indicates that the quantities between which it is written are equal. Thus, xy, indicates that the quantity x equals the quantity y. An expression in which quantities are connected by the sign is an equation, as x + 4 = 2 x − 1, read x plus 4 equals 2x minus 1. 23. The SIGN OF RATIO (:), read to, indicates that the two quantities between which it is placed are taken as the terms of a ratio. Thus, ab indicates the ratio of the quantity a to the quantity b; and is read the ratio of a to b. An equality of ratios, or a proportion, is expressed by writing the sign, or the sign (::), between equal ratios. Thus, 30: 6 = 25: 5 is read the ratio of 30 to 6 equals the ratio of 25 to 5, or 30 is to 6 as 25 to 5. 24. The SIGN OF INEQUALITY, > or <, read is greater than, or is less than, when placed between two quantities, indicates that the quantity toward which the opening of the sign turns is the greater. Thus, x > y is read x is greater than y, x 6y is read x minus 6 is less than y. 25. The SIGN OF VARIATION, OC, read varies as, indicates that the two quantities between which it is placed increase or diminish together. Thus, a oc is read a varies as с с d' SYMBOLS OF ABBREVIATION. 26. The SIGNS OF DEDUCTION (.. and .) stand the one for therefore or hence, and the other for since or be cause. 27. The SIGNS OF AGGREGATION, the vinculum the bar, the parenthesis (), the brackets [], and the braces indicate that the quantities connected, or inclosed, by them are to be taken as one quantity, and to be operated upon in the same way. Thus, +ax ‡“*, (a+b) x, [a+b] x, { a + b} a+b ×x, + b x, each indicate that the quantity a+b is to be multi plied by x. 28. The SIGN OF CONTINUATION (. .) stands for and so on, or continued by the same law. a, a + b, a + 2 b, a + 3 b, Thus, . . . . . is read a, a + b, a + 2b, a + 3b, and so on. |