A TREATISE ON ALGEBRA. SECTION I. DEFINITIONS AND NOTATION. 1. Quantity is anything that can be increased, diminished, or measured; as distance, space, weight, motion, time. A quantity is measured by finding how many times it contains a certain other quantity of the same kind, regarded as a standard. The conventional standard thus used is called the unit of measure. 2. Mathematics is the science which treats of the properties and relations of quantities. It employs a peculiar language, consisting of symbols, to express the values of quantities, and the operations to which these values are subjected. The symbols are of three kinds, as follows : 1st. Symbols of Quantity, consisting of figures or numerals used in arithmetical computations, letters and other characters used in general analysis, and graphic representations or drawings used in geometrical investigations. 2d. Symbols of Operation, consisting of the signs or characters employed to indicate those mathematical processes by which quantities are made to undergo changes of value, such as addition, subtraction, multiplication and division. 3d. Symbols of Relation, consisting of the signs used in comparing quantities with respect to their relative magnitudes, and certain abbreviations employed in the process of reasoning. 3. Algebra is that branch of mathematics in which quantities are represented by letters, and the operations and relations are indicated by signs. The object of algebraic notation is to abridge and generalize the analysis of mathematical problems. Algebra is therefore a species of universal arithmetic. SYMBOLS OF QUANTITY. 4. An Algebraic Quantity is a quantity expressed in algebraic language. There are two kinds of algebraic quantities—known and unknown. j. Known Quantities are those whose values are given; when these are not expressed by figures they are represented by the leading letters of the alphabet, as a, b, c, d. 6. Unknown Quantities are those whose values are to be determined ; they are represented by the final letters of the alphabet, as ", X, Y, 2. 7. The small italic letters just given are the more common symbols of quantity. In addition to these, capital letters are sometimes employed, as A, Ķ, C, D, X, Y, Z, etc. Quantities which have like relations to a series of quantities in any investigation, are sometimes represented by a single letter repeated with different accents, as a, a', a'', a'"' a''"', read, a, a prime, a second, a third, etc.; or by a letter repeated with different subscript figures, as a, 01, 02, 03, 4, etc., read, a, a sub one, a sub two, a sub three, etc. In certain investigations it is convenient to represent quantities by the initial letters of their names. Thus, S or s may represent sum; D or d, difference or diameter; R or r, ratio, remainder or rudius. In some cases the capital and the small letter may be used to. gether to distinguish between two quantities of the same kind. Thus, in a problem relating to two circles, r may represent the radius of the smaller, and R the radius of the larger circle. a SYMBOLS OF OPERATION. 8. The Sign of Addition is the perpendicular cross, +, called plus. It indicates that the quantity written after it is to be added to the other quantity or quantities in the expression. Thus, in a+b, the sign indicates that the quantity b is to be added to the quantity a ; and the expression is read, a plus b. 9. The Sign of Subtraction is a a short horizontal line, called minus. It indicates that the quantity written after it is to be subtracted from the other quantity or quantities in the expression. The sign Thus, in amb, or —6+a, the minus sign indicates that the quantity l is to be subtracted from the quantity a; and the expression is read, a minus b, or minus b plus a. ~ may be written between two quantities to indicate that their arithmetical difference is to be taken, when it is not known which is the greater. 10. The Double Sign, , is written before a quantity to indicate that it is to be both added and subtracted ; it serves to unite in a single expression two combinations of the same quantities. Thus, a £b is equivalent to a+b and a--b, and is read a plus or minus b. 11. The Sign of Multiplication is the oblique cross, X. It indicates that the quantity before it is to be multiplied by the quantity after it. Thus, in axb, the sign indicates that a is to be multiplied by b. Instead of the sign, X, a point is sometimes used to denote multiplication; as 3.xy, which signifies the same as 3 x x xy. The multiplication of quantities which are represented by letters, is generally indicated by writing the factors one after another without any intervening sign. Thus, 3abc signifies the same as 3 xa x bxc, or 3•a:bc. It is evident that this notation cannot be employed when the several factors are represented by figures. We cannot represent 3 times 4 by simply writing the factors together, thus, 3 4; for the product thus indicated could not be distinguished from the number 34. Notes. 1. The result of any multiplication is called a product, and the quantities multiplied are called factor's. 2. When the quantities to be multiplied are represented by letters, they are called literal factors ; when they are represented by figures, they are called numerical factors. 12. The Sign of Division is a short horizontal line with a point above and one below, ;. It indicates that the quantity before it is to be divided by the quantity after it. Thus, in a--b, the sign indicates that a is to be divided by b. Division is also expressed by writing the dividend above, and the divisor below, a short horizontal line; as b 13. The Sig. of Involution is a number written above and to the right of a quantity, to indicate how many times the quantity is a to be taken as a factor. Thus, in a', the number 5 indicates that a is to be taken 5 times as a factor; and the expression is equivalent to aaaaa. A factor repeated to form a product is called a root; the product itself is called a power ; and the figure which indicates how many times the root or factor is taken, is called the exponent of the power. Thus, in the indicated product a', a is the root, a' is the power, called the 5th power of a, and 5 is the exponent of this power. When no exponent is written over a quantity, the exponent 1 may always be understood. NOTE. -For the sake of brevity, the exponent of the power may be called the exponent of the letter or quantity over which it is placed. Thus in a', 5 may be called the exponent of a. 14. The Sign of Evolution, or Radical Sign, is the character V. It indicates that some root of the quantity after it is to be extracted. The name or index of the required root is the number written above the radical sign. Thus, Va denotes the cube root of a; Va denotes the 4th root of a; and so on. When no index is written over the sign, the index 2 is understood ; thus, va denotes the square root of a. Fractional Exponents are also used as the sign of evolution, the denominator being the index of the required root. Thus, in a3, the denominator, 3, indicates that the cube root of a is required, and the expression is equivalent to Va. Fractional exponents are used to denote both involution and evolution in the same expression, the numerator indicating the power to which the quantity is to be raised, and the denominator the required root of this power. Thus, the expression signifies the 4th root of the 3d power of a, and is equivalent to Va'. SYMBOLS OF RELATION. 15. The Sign of Equality is two short horizontal lines, =. » It indicates that the two quantities between which it is placed are equal. Thus, in a=b+c, the sign, =, indicates that a is equal to b plus c. An expression of equality between two quantities is called an equation. 16. The Sign of Inequality is the angle, >. It indicates that the quantities between which it is written are unequal, the opening being always turned toward the greater. When the opening is toward the left, it is read greater than ; wher the point or vertex is toward the left, it is read less than. Thus, a >b signifies that a is greater than b; x+y<z signifies that æ plus y is less than 2. 17. The Signs of Aggregation are the parenthesis, (), brackets, [ ], brace, { }, vinculum, and bar, |. They indicate that the quantities included within, or connected by them, are to be taken collectively and subjected to the same operation. Thus, (a+b—c)«, [a+bc] x, {a+b-c}x, a+b-cxx, ta and + b are expressions signifying that the whole quantity, a+b-c, is to be multiplied by x. Two or more of these signs may be used correlatively in the same expression, in which case the brackets should include the parenthesis or vinculum, and the brace should include the brackets; thus, {m—a[c-6m+d)]+z}. 18. The Sign of Continuation is a succession of points, indicating that a series of quantities may be continued indefinitely according to the same law. Thus, in the expression, ata? ta3 ta+ the points indicate that the series has an infinite number of terms, all formed according to the same law. 19. The Sign of Ratio is two points like the colon, : , placed between the quantities compared. Thus, the expression, a : 6, signifies the ratio of a to b. 20. The Sign of Proportion is a combination of the sign of ratio and the sign of equality, :=:; or a combination of points only, :::: Thus, a : b=c:d, signifies that the ratio of a to b is equal to the ratio of c to d; and the expression a : 6::C:d signifies the same, and may be read, a is be read, a is to b as c is to d. 21. The Sign of Variation is the character o. It signifies that the two quantities between which it is placed, whether equal or unequal, increase or diminish together, so as to preserve constantly the same ratio. NOTE.—The signs of ratio, proportion, and variation, will be more ful. ly explained hereafter. |