Algebra Review

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Basic Concepts

We introduce variables to generalize the analysis of numbers and their relations. For example a variable a could represent the whole set of possible numbers (in our case we will only have real numbers) like a = 1 or a = 0.2332 or a = (2).

Such representations will allow us to study and use the properties of quantifiable variables for our analysis.

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Properties and Relations

Commutative properties: If a, b, and c are any numbers we have a + b = b + a. Distributive properties: c(a + b) = ca + cb. For example if a = 2.0, b = 4.0, and c = 2.5 we have 2.5(2 + 4) = 2.5 ∗ 6 = 15 = 5.0 + 10. The opposite is also true. Associative properties: If a, b, and c are any numbers we have c · a + c · b = c(a + b). Power relations: When a number is multiplied n times we have a · a · a · ... · a = an .

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This leads to the following relations and notations. a0 = 1.0; 1 a−1 = a 1 a−n = an √ 1 an = n a (a · b)n = an · bn (an ) m

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= am

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an am = an+m an bm = an+m when a = b a an ( )n = b bn m a = am−n . an The sum of fractions can be represented as: b+a 1 1 + = . a b ab Using the distributive and associative properties one gets: (a + b)2 = (a + b)(a + b) (a + b)2 = a2 + 2ab + b2 (a − b) · (a + b) = a2 − b2

Just like 1, there are other numbers that will be important to mention in our studies. Of particular importance is the number denoted by e, which has a value of e = 2.7182818 . . . .

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Functions

A function is a mapping of one set of numbers, x, into another set, y, by a specific relation. y = f (x). f (x) represents a function for which x is the dependant variable. One example will be: f (x) = a · x2 + ex or y = a · x2 + ex . With a = 10 the relation will give us the following relations for a number series of x: y 12.7183 47.3891 110.0855 214.5982 398.4132 x 1 2 3 4 5

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One function of particular importance will be the exponential. This is represented as: f (x) = ex . It simply denotes that e is raised to the...