8. The difference of two quantities, when it is unknown which is the greater, is indicated by the sign, placed between them; thus, ab, or ba, denotes the difference of a and b. 9. indicate proportion; thus, ab:: cd, may be read, a has to b the same ratio that c has to d. = 10. equal to, signifies that the quantities between which it is placed, are equal to each other; thus, b+c =ad+e, is a combination, called an equation, which signifies, that when the operations indicated by the signs are performed on the quantities placed on each side of the sign of equality, the results are equal to each other. 11. A simple quantity is that which consists of one term only, viz. a quantity denoted by a single letter, or several letters and figures, connected by the sign of multiplication or division, expressed or understood, as a, abc, cb 5cd; d d' * 12. A compound quantity consists of two or more simple quantities connected by the signs of addition or subtraction, as a—b, ab-ac+bd. 13. A root is a number or quantity, from which a power is conceived to arise. 14. A power of a number or root, is the product of a unit, multiplied continually by the given root, any proposed number of times; and the figure or quantity which indicates the number of multiplications thus made, is called the index or exponent; thus, 1×5×5×5×5=625; and, 1×a×a×a×a×a, or aaaaa, are the 4th and 5th powers of 5 and a respectively, and are usually expressed by the root with the index of the power set over it; as 54, a5; hence, a°=1, whatever value may be assigned to a. 15. The second power is called the square; the third power, the cube; the fourth power, the biquadrate, &c. of their respective roots. * An absolute number, though containing numerous digits, is considered as a simple quantity. 16. The radical sign* ✔, prefixed to a quantity, indicates the square root;, the cube root; 4, the fourth root, &c. Roots are also expressed by fractional exponents; thus, a or a3, denotes the square root of a; ✅a, or a3, the cube root of a; and a2, or a3, the cube root of the square of a. 17. A root which cannot be accurately expressed in numbers, is called a surd, or irrational quantity, as ✓5, 35. 18. A quantity which has no radical sign, or which having a radical sign, admits of an accurate extraction of the root indicated by the sign, is called rational; thus a, ✓16, b3, are rational quantities. 19. When a compound quantity has a line drawn over it, or is enclosed in brackets, the operation indicated by a preceding or subsequent sign, is to be performed on the whole considered as a simple quantity; thus, a+b-cxd, and (b+c)×(d—e), signify that the compound quantities connected by the sign X, are multiplied together; and ab+dc, {ab-cd+ef,} signify the square root, and the cube root respectively of the compound quantities, which are preceded by the radical signs. 20. A number prefixed to a letter, or combination of letters, is called the co-efficient: thus, in the expression 3ab, 3 is the co-efficient. 21. A compound quantity, consisting of two terms, is called a binomial, as a+b; one of three terms, a trinomial, as ab+ac+de. 22.-Like quantities are those which consist of the same letters similarly involved; as ab, Sab, 5ab. 23. Unlike quantities consist of different letters, or different powers of the same letters, as a, ab, 3a2, 4ab3. In the solution of problems, it is usual to denote known or given quantities, by the initial letters, a, b, c, &c., and unknown ones, by the final letters, v, x, y, &c. * So called from radix, a root. The following examples are given for the purpose of exercising the student in the application of the algebraic signs. Required the numerical values of the following combinations, supposing a=7, b=6, c=5, d=3, e=2. a3+4bc-3de=343+120—18=445. (ab+cd)x(3bc+4ad)=(42+15)x(90+84)=57×174 =9918. 24. When the quantities are like, and have like signs. Add all the co-efficients together, and to their sum annex the given literal expression, prefixing the common sign.* The sign or, is frequently prefixed to quantities which are not preceded by others, to or from which, they are required to be added or subtracted: a preceding quantity, however, may always be supposed. Algebraic quantities are said to be positive or negative, according as they are preceded by the signor; the former being always understood, where none is expressed. |