AN ELEMENTARY TREATISE ON ALGEBRA. INTRODUCTION, EXPLANATION OF THE ALGEBRAIC METHOD OF NOTATION DEFINITIONS AND AXIOMS. 1. Algebra is a general method of computation, in which abstract quantities and their several relations are made the subject of calculation, by means of alphabetical letters and other signs. 2. The letters of the alphabet may be employed at pleasure for denoting any quantities, as algebraical symbols or abbreviations; but, in general, quantities whose values are known or determined, are expressed by the first letters, a, b, c, &c.; and unknown or undetermined quantities are denoted by the last or final ones, u, v, w, x, &c. 3. Quantities are equal when they are of the same magnitude. The abbreviation ab implies that the quantity denoted by a is equal to the quantity denoted by b, and is read a equal to b; ab or a greater than b, that the quantity a is greater than the quantity b: and a<b_or a less than b, that the quantity a is less than the quantity b. The 4. Addition is the joining of magnitudes into one sum. sign of addition is an erect cross; thus, a+b implies the sum of a and b, and is called a plus b, if a represent 8 and b 4; then, a+b represents 12, or a+b=8+4=12. 5. Subtraction is the taking as much from one quantity as is equal to another. Subtraction is denoted by a single line; as a-b or a minus b, which is the part of a remaining, when a part equal to 6 has been taken from it; if a=9, and b=5; а- -b expresses 9 diminished by 5, which is equal to 4, or a-b-9-5=4. 6. Also, the difference of two quantities a and b; when it is not known which of them is the greater, is represented by the sign; thus, ab is a−b, or b-a; and ab signifies the sum or difference of a and b. 7. Multiplication is the adding together so many numbers or quantities equal to the multiplicand as there are units in the multiplier, into one sum called the product. Multiplication is expressed by an oblique cross, by a point, or by simple apposition; thus, a X b, a. b, or ab, signifies the quantity denoted by a, is to be multiplied by the quantity denoted by b; if a=5 and b=7; then aXb = 5X7 = 35, or a . b = 5.7=35, or ab=5X7=35. Scholium. The multiplication of numbers cannot be expressed by simple apposition. A unit is a magnitude considered as a whole complete within itself. And a whole number is composed of units by continued additions; thus, one plus one composes two, 2+1=3, 3+1=4, &c. 6. Division is the subtraction of one quantity from another as often as it is contained in it; or the finding of that quotient, which, when multiplied by a given divisor, produces a given dividend. Division is denoted by placing the dividend before the sign and the divisor after it; thus ab, implies that the quantity a is to be divided by the quantity b. Also, it is frequently denoted by placing one of the two quantities over the other, in the form of a fraction; thus, ab; if a = 12, b=4; α b 9. A simple fraction is a number which by continual addition composes a unit, and the number of such fractions contained in a unit, is denoted by the denominator, or the number below the line; thus, +1+3=1. A number composed of such simple fractions, by continual addition, may properly be termed a multiple fraction; the number of simple fractions composing it, is denoted by the upper figure or numerator. In this sense, ,,, are multiple fractions; and 3=1,‡=3+3=1+3=1. 10. When any quantities are enclosed in a parentheses, or have a line drawn over them, they are considered as one quantity with respect to other symbols; thus a- (b+c), or a-b+c; implies the excess of a above the sum of b and c; Let a 9, b=3, and c=2; then, a (b+c)=9-(3+2)=9 -5-4, or a-b+c=9-3+2=9-5-4. Also, (a+b) X (c+d), or a+bXc+d, denotes that the sum of a and b is to be multiplied by the sum of c and d; thus, let a=4, b=2,c=3, and d=5; then (a+b)×(c+d)=(4+2)X(3+5)=6 × 8=48, or a+bXc+d=4+2×3+5=6×8-48. And (a-b)÷(c a-b d) or implies the excess of a above b, is to be divic+d ; ded by the sum of c and d; if a=12, b=2, c=4, and d=1; then, = (ab)(cd)=(12—2)÷÷(4+1)=10÷÷5=2, or a-b c+d The line drawn over the quantities is sometimes called a vinculum. 11. Factors are the numbers or quantities, from the multiplication of which, the proposed numbers or quantities are produced; thus, the factors of 35 are 7 and 5, because 7X5=35; also, a and b are the factors of ab; 3, a3, b and c, are the fac tors of 3a2bc2; and a+b and a—b are the factors of the product (a+b)x(a−b). When a number or quantity is produced by the multiplication of two or more factors, it is called a composite number or quantity; thus, 35 is a composite number, being produced by the product of 7 and 5; also, 5acx is a composite quantity, the factors of which are 5, a, c, and x. 12. When the factors are all equal to each other, the product is called a power of one of the factors, and the factor is called the root of the product or the power. When there are two equal factors, the product is called the second power or square of either factor, and the factor is called the second root or square root of the power. When there are three equal factors, the product is called the third power or cube of either factor, and the factor is called the third root or cube root of the power. And so on for any number of equal factors. 13. Instead of setting down in the manner of other products, the equal factors which multiplied together constitute a power, it is evidently more convenient to set down only one of the equal factors, (or, in other words, the root of the power,) and to designate their number by small figures or letters placed near the root. These figures or letters are always placed at the upper and right side of the root, and are called the indices or exponents of the power. For example: axaxaxa or aaaa is denoted thus, a1; where a1 and y are the powers; a and y the roots, and 4 and 5 the indices or exponents of the powers. Again: 4ax2 X <4ax3×4ax3, is thus abridged, (4ax2)3; where (4ax2)3. is the power, 4ar the root, and 3 the index or exponent of the power. The same method is adopted, whatever be the form of the root: thus, (a2-x-y)X(a2 — x2—y3) × (a2 — x2 — y2) is written briefly thus, (a-x-y), where (a-x-y) is the power, a2—x2—y the root of the power, and 3 its index or exponent. N. B. Care must always be taken, to embrace the root in parentheses, except were it is expressed by a single charac ter. 14. The coefficient of a quantity is the number or letter prefixed to it; being that which shows how often the quantity is to be taken; thus, in the quantities 36 and 5x2, 3 and 5 are the coefficients of b and x2. Also, in the quantities 3ay and 5a2x, 3a and 5a2 are the coefficients of y and x. 15. When a quantity has no number prefixed to it, the quantity has unity for its coefficient, or it is supposed to be taken only once; thus, x is the same as 1; and when a quantity has no sign before it, the sign is always understood; thus, 3ab is the same as +3a2b, and 5a-36 is the same as +5a-3b. 16. Quantities which can be expressed in finite terms, or the roots of which can be accurately expressed, are rational quantities; thus, 3a, fa, and the square root of 4a2, are rational quantities; for if a=10; then, 3a=3x10=30; a= X10==4; and the square root of 4a2= the square root of 4 × 102= the square root of 4×10×10= the square root of 400-20. 17. An irrational quantity, or surd, is that of which the value cannotbee accurately expressed in numbers, as the square root of 3, 5, 7, &c.; the cube root of 7, 9, &c. 18. The roots of quantities are expressed by means of the radical sign, with the proper index annexed, or by fractional indices placed at the right-hand of the quantity; thus, 1 1 a, or a, expresses the square root of a; (a+x), or root of a; √(a+x), or (a+e)3, the cube root of (a+x); √(a+x), or (a), the fourth root of (a+x). When the roots of quantities are expressed by fractional indices; thus, a*, (a+x)*, (a+x); they are generally read a in the power (2), or a with (2) for an index; (ax) in the power (4), or (a+x) with (3) for an index; and (a) in the power (4), or (a+x) with (4) for an index. 19. Like quantities are such as consist of the same letters or the same combination of letters, or that differ only in their numeral coefficients; thus, 5a and 7a; 4ax and 9ax; +2ac and 9ac; -5ca; &c., are called like quantities; and unlike quantities are such as consist of different letters, or of different combination of letters; thus, 4a, 3b, 7ax, 5ay, &c. are unlike quantities. 20. Algebraic quantities have also different denominations, according to the sign+, or -. Positive, or affirmative quantities, are those that are additive, or such as have the sign + prefixed to them; as, a, +6ab, or 9ax. 21. Negative quantities are those that are subtractive, or such as have the sign - prefixed to them; as, −x, −3a2, -4ab, &c. A negative quantity is of an opposite nature to a positive one, with respect to addition and subtraction: the condition of its determination being such, that it must be subtracted when a positive quantity would be added, and the re verse. 22. Also quantities have different denominations, according to the number of terms (connected by the signsor) of which they consist; thus a, 3b, -4ad, &c., quantities consisting of one term, are called simple quantities, or monomials; az, a quantity consisting of two terms, a binomial; a-x is sometimes called a residual quantity. A trinomial is a quantity consisting of three terms; as, a+2x-3y; a quadrinomial of four; as, a-b+3x-4y; and a polynomial, or multinomial, consists of an indefinite number of terms. Quantities consisting of more than one term may be called compound quantities. 23. Quantities the signs of which are all positive or all negative, are said to have like signs; thus, +3a, +4x, +5ab, have like signs; also, -4a, -3b,-4ac; When some are positive, and others negative, they have unlike signs; thus, the quantities +3a and -5ab have unlike signs; also, the quantities 3ax, +3ax: and the quantities -b, b. 24. If the quotients of two pairs of numbers are equal, the numbers are proportional, and the first is to the second, as the third to the fourth; and any quantities, expressed by such numbers, are also proportional; thus, if; then a is to b as c to d. The abbreviation of the proportion; a: b::c: d; and it is sometimes written a : bc:d; if a=8, b=4, 8 12 c=12, and d=6; then, =2, and 8: 4:; 12: 6. 4 6 |