number. 5a is the measure of 20a. 3x is the measure of 12x, or 12ax. A multiple of any quantity is that which is some exact number of times that quantity; thus 12 is a multiple of 3, or of 4, or of 6, and 30ab is a multiple of 3ab, of 5ab, &c. AXIOMS. Axioms are self-evident truths, and of course are above demonstration; no explanation can render them more clear. The following are those applicable to algebra, and are the principles on which the truth of all algebraical operations finally rests: Axiom 1. If the same quantity or equal quantities be added to equal quantities, their sums will be equal. 2. If the same quantity or equal quantities be subtracted from equal quantities, the remainders will be equal. 3. If equal quantities be multiplied into the same, or equal quantities, the products will be equal. 4. If equal quantities be divided by the same, or by equal quantities, the quotients will be equal. 5. If the same quantity be both added to and subtracted from another, the value of the latter will not be altered. 6. If a quantity be both multiplied and divided by another, the value of the former will not be altered. 7. Quantities which are respectively equal to any other quantity are equal to each other. 8. Like roots of equal quantities are equal. 9. Like powers of the same or equal quantities are equal. EXERCISES ON NOTATION. When definite values are given to the letters employed, we can at once determine the value of their combination in any algebraic expression. Then a+b-c=5+20-4 or a+b-c=21 (Art. 1.) Before we can make use of literal or algebraical quantities to aid us in any mathematical investigation, we must not only learn the nature of the quantities expressed, but how to add, subtract, multiply, and divide them, and subsequently learn how to raise them to powers, and extract roots. The pupil has undoubtedly learned in arithmetic, that quantities representing different things cannot be added together; for instance, dollars and yards of cloth cannot be put into one sum; but dollars can be added to dollars, and yards to yards; units can be added to units, tens to tens, &c. So in algebra, a can be added to a, making 2a; 3a can be added to 5a, making 8a. As a may represent a dollar, then 3a would be 3 dollars, and 5a would be 5 dollars, and the sum would be 8 dollars. Again, a may represent any number of dollars as well as one dollar; for example, suppose a to represent 6 dollars, then 3a would be 18 dollars, and 5a would be 30 dollars, and the whole sum would be 48 dollars. Also, 8a is 8 times 6 or 48 dollars; hence any num ber of a's may be added to any other number of a's by uniting their coefficients; but a cannot be added to b, or 4a to 3b, or to any other dissimilar quantity, because it would be adding unlike things, but we can write a+b and 3a+36, indicating the addi tion by the sign, making a compound quantity, Let the pupil observe that a broad generality, a wide latitude must be given to the term addition. In algebra, it rather means uniting, condensing, or reducing terms, and in some cases, the sum may appear like difference, owing to the difference of signs. Thus, 4a added to -a is 3a; that is, the quantities united can make only 3a, because the minus sign indicates that one a must be taken out. Again, 76+36-4b, when united, can give only 6b, which is in fact the sum of these quantities, as 4b has the minus sign, which demands that it should be taken out; hence to add similar quantities we have the following RULE. Add the affirmative coefficients into one sum and the negative ones into another, and take their difference with the sign of the greater, to which affix the common literal quantity. N. B. Like quantities, of whatever kind, whether of powers or roots, may be added together the same as more simple or rational quantities. Thus 3a2 and 8a2 are 11a2, and 763+363=1063. No matter what the terms may be, if they are only alike in kind. Let the reader observe that 2(a+b)+3(a+b) must be together 5(a+b), that is, 2 times any quantity whatever added to 3 times the same quantity must be 5 times that quantity. Therefore, 4√x+y+3↓x+y=7x+y, for √x+y, which represents the square root of x+y, may be considered a single quantity. (Art. 2.) To find the sum of various quantities we have the following B RULE. Collect together all those that are alike, by uniting their coefficients, and then write the different sums, one after another, with their proper signs. 7. Add 2xy-2a, 3a2+xy, a2+xy, 4a2-3xy, 2xy-2a3. Ans. 4a2+3xy. 8. Add 8a2x2-3xy, 5ax-5xy, 9xy-5ax, 2a2x2+xy, 5ax-3xy. Ans. 10a2x2+5ax—xy. 124 9. Add 2x2 2-10y, 3/xy+10x, 2x2y+25y, 12xy-√xy, -8y+17/xy. Ans. 2x2y+12xy+10x+2x2+19√xy+7y. 10. Add 2bx-12, 3x2-2bx, 5x2-3/x, 3/x+12, x2+3. Ans. 9x2+3. 11. Add 10b2--3bx2, 2b2x2-b2, 10-2bx2, b2x2-20, 3bx2+b2. Ans. 10b2+362x-2bx2-10. 12. Add 2a2-3ax2+x2, 2ax3-13xy+8, 10a2—xy—4. Ans. 12a-ax+x2-14xy+4. 13. Add 9bc3-18ac2, 15bc3+ac, 9ac2-24bc3, 9ac2-2. Ans. ac-2 15. Add 12a-13ab+16ax, 8-4m+2y, 6a+7ab2+ 12y-24, and 7ab-16ax+4m. Ans. 6a-6ab+14y+7ab2—16. 16. Add 72ax-8ay3, -38ax1-3ay'+7ay3, 8+12ay*, —6ay3+12—34ax1+5ay3—9ay1. Add a+b and 3a-5b together. Add 6x-5b+a+8 to -5a-4x+4b-3. Ans. -2ay3+20. Add a+2b-3c-10 to 3b-4a+5c+10 and 56-c. Add 3a2+262-c to 2ab-3a2bc-b. (Art. 3.) When similar quantities have literal coefficients, we may add them by putting their coefficients in a vinculum, and writing the term on the outside as a factor. Thus the sum of ax and bx is (a+b)x. Ans. 10ax+11x. 5. Add 8ax+2(x+a)+3b, 9ax+6(x+a)—9b, and 11x+ 6b-7ax-8(x+a). 6. Add (a+b)√x and (c+2a-b)x together. 7. Add 28a3(x+5y)+21, 18a-13a3(x+5y), —15a3(x+ Ans. 18a-13. 5y)-8. |