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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 Spring '12
 ellen
 Elementary algebra, Euclidean geometry, Mathematics in medieval Islam

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