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algebra review1 - Algebra Review In this appendix a review...

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Algebra Review In this appendix, a review of algebra skills will be provided. Students sometimes think that there are tricks needed to do algebra. Rather, algebra is a set of rules about what one may and may not do to an equation. The general idea is to move toward a solution to an equation, although that need not be the only goal. A primary factor in doing algebra is actually doing it. One cannot be successful in doing algebra if one does not grab one’s pencil and paper and use the pencil and paper to write out the steps. Only the very best students can do algebra in their head. A second fact is that student often do not want to write out each step; they want to combine steps and to two or more steps at the same time. This is bad practice. Do the algebra one step at a time. Otherwise, you may find yourself going back and doing the problem over and waste the time you thought you saved by doing more than one step at a time. The final piece of advice I would offer is that persistence pays off. If at first you don’t get the problem solved, come back later and try again. If you make a mistake, go back and try again. So how, then, do we begin? Start with the assumptions that we make for the real numbers. These are sometimes called field axioms. Algebra starts with a set of elements (real numbers in our case) and two operations on those numbers. The axioms tell us the allowed actions that we can take with these operations and the outcome of these actions. The two operations are addition and multiplication. AXIOMS Suppose we have the set of all real numbers, R, and two operations, +, and *. Then the following are assumed to be true. Assumptions for + (1) There exists an element 0, 0 ε R, so that for any a ε R, a + 0 = a (2) For each a ε R, there exists another element of R, -a, so that a + (-a) = 0 (3) For any two a, b ε R, a + b = b + a (4) For any then a, b, and c ε R, a + (b + c) = (a + b) + c Assumptions for * (5) There exists an element 1, 1 ε R, so that for any a ε R, a * 1 = a (6) For each a ε R (a 0) there exists a unique a -1 so that a * a -1 = 1 (7) For any two a, b ε R, a * b = b * a (8) For any three a, b, c ε R, a * (b * c) = (a * b) * c
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Distributive Principle (links +, *) (9) For any a, b, and c ε R, a * (b + c) = a * b + b * c Remark: First, note the symmetry. The axioms 1 – 4 are similar to axioms 5 – 8. Axiom 1 and 5 asserts the existence of an identity element. Zero is the identity for + and one is the identity for *. Axioms 2 and 6 assert the existence of an inverse element. For addition, the inverse element of a is minus a. This introduces the idea of subtraction which is addition with negative numbers. Similarly for multiplication, we have a multiplicative inverse which introduces division (except for the element zero). The next axioms (3 and 7) tell us that the order we add or multiply does not matter.
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