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The reciprocal of any quantity, is that quantity inverted, or unity divided by it.
Thus, the reciprocal of a, or, is; and the reciprocal of ģisa
A function of one or more quantities, is an expression into which those quantities enter, in any manner whatever, either combined, or not, with known quantities.
Thus, d—2x, ax + 3x", 2x -a (a? _xo)*,axm, ax ,&c., are functions of x ; and axy + bx®, ay + x(ax2 - by:)), &c. are functions of x and y.
A vinculum, is a bar —, or parenthesis ( ), made use of to collect several quantities into one.
Thus, a + b Xc, or (a + b)c, denotes that the compound quantity a+b is to be multiplied by the simple quantity c; and vab+c?, or (ab+ca), is the square root of the compound quantity ab+c2.
Practical Examples for computing the numeral Values of various Algebraic Expressions, or Combinations
Supposing a=6, b=5, c=4, d=1, and eco,
Then 1. a3 + 2ab-c+d=36 +60—4+1=93. 2. 2a3_3a2b4c3=432-540+643-44. 3. a? Xatb-2abc=36X11-240=156. 4. 2a63 - act2ac+c=12X1+8=20. 5. 30./2actca, or 3a(2ac+ca)}=18/64=144.
6. v2a’ – V2ac+c=\.72–754=72-8=64 =8. 2a +3c
4bc 12+12 7. to 60+ 4e
Required the numeral values of the following quantities ; supposing a, b, c, d, e, to be 6, 5, 4, 1, and 0, respectively, as above.
1. 2a? +3bc-5d 2. 5a2 b -- 10ab2 +2e 3. 792 +b-cXdte 4. 5 vab+62 – 2ab-e?
Addition is the connecting of quantities together by means of their
proper signs, and incorporating such as are like, or that can be united, into one sum ; the rule for performing which is commonly divided into the three following cases (a):
When the Quantities are like, and have like Signs.
Add all the coefficients of the sereral quantities together, and to their sum annex the letter or letters belonging to each term, prefixing, when necessary, the common sign.
(a) The term Addition, which is generally used to denote this rule, is tvo scanty to express the nature of the operations that are to be performed in it; which are sometimes those of addition, and sometimes subtraction, according as the quantities are negative or positive. It should, therefore, be called by some name signifying incorporation, or striking a balance ; in which case, the inconigruity, here mentioned, would be removed,
When the Quantities are like, but have unlike signs.
Add all the affirmative coefficients into one sum, and those that are negative into another, when there are several of the same kind ; then subtract the least of these sums from the greatest, and to the difference prefix the sign of the greater, annexing the common letter or letters, as before.