Axioms of Algebra
61. The object of mathematical inquiry is, generally, to investigate some unknown quantity, and discover how great it is. This is effected, by comparing it with some other quantity or quantities already known. The dimensions of a stick of timber, are found, by applying to it a measuring rule of known length. The weight of a body is ascertained, by placing it in one scale of a balance, and observing how many pounds in the opposite scale, will equal it. And any quantity is determined, when it is found to be equal to some known quantity or quantities.
Let a and b be known quantities, and y, one which is unknown. Then y will become known, if it be discovered to be equal to the sum of a and b: that is if
y = a + b.
An expression like this, representing the equality between one quantity or set of quantities, and another, is called an equation. It will be seen hereafter, that much of the business of algebra consists in finding equations, in which some unknown quantity is shown to be equal to others which are known. But it is not often the fact, that the first comparison of the quantities, furnishes the equation required. It will generally be necessary to make a number of additions, subtractions, multiplications, &c. before the unknown quantity is discovered. B ut in all these changes, a constant eq uality must be preserved, between the two sets of quantities compared. This will be done, if, in making the alterations, we are guided by the following axioms. These are not inserted here, for the purpose of being proved; for they are self-evident. (Art. 10.) But as they must be continually introduced or implied, in demonstrations and the solutions of problems, thev are placed together, for the convenience of reference.
62. 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 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. If to unequal quantities, equals be added, the greater will give the greater sum.
8. If from unequal quantities, equals be subtracted, the greater will give the greater remainder.
9. If unequal quantities be multiplied by equals, the greater will give the greater product.
10. If unequal quantities be divided by equals, the greater will give the greater quotient.
11. Quantities which are respectively equal to any other quantity are equal to each other.
12. The whole of a quantity is greater than a part.
This is, by no means, a complete list of the self-evident propositions, which are furnished by the It is not necessary to enumerate them all. Those have been selected, to which we shall have the most frequent occasion to refer.
63. The investigations in algebra are carried on, principally, by means of a series of equations and proportions. But instead of entering directly upon the3e, it will be necessary to attend in the first place, to a number of processes, on which the management of equations and proportions depends. These preparatory operations are similar to the calculations under the common rules of arithmetic. We have addition, multiplication, division, involution, &c. in algebra, as well as in arithmetic. But this application of a common name, to operations in these two branches of the mathemak. ics, is often the occasion of perplexity and mistake. The learner naturally expects to find addition in algebra the same as addition in arithmetic. They are in fact the same, in many respects: in all respects perhaps, in which the steps of \he one will admit of a direct comparison, with those of the other. But addition in algebra is more extensive, than in arithmetic. The same observation may be made concerning several other operations in They are, in many points of view, the same as those which bear the same names m arithmetic. But they are frequently extended farther, and comprehend processes which are unknown to arithmetic. This is commonly owing to the introduction of negative quantities. The management of these requires steps which are unnecessary, where quantities of one class only are concerned. It will be important, therefore, as we pass along, to mark the difference as well as the resemblance, between arithmetic and algebra; and, in some instances, to give a new definition, accommodated to the latter.