01_Prerequisites
1.1 Real Numbers
1 Classifying a Real Number
natural numbers: {1, 2, 3, ...}
whole numbers: {0, 1, 2, 3, ...}
integers: {..., -3, -2, -1, 0, 1, 2, 3, ...}
**rational numbers: ** {$m\over n$ | m and n are integers and $n \neq$ 0}
irantional numbers: {h | h is not a rational number}
The sets of rational and irrational numbers together make up the set of real numbers.
2 Using Properties of Real Numbers
Commutative Properties
addition
a + b = b + a
multiplication
$a \cdot b = b \cdot a$
Associative Properties
addition: a + (b + c) = (a + b) + c
multiplication: a(bc) = (ab)c
Distributive Property
$a \cdot (b + c) = a \cdot b + a \cdot c$
Identity Properties
addition: There exists a unique real number called the
additive identity
, 0, such that, for any real number aa + 0 = a
multiplication: There exists a unique real number called the
multiplicative identity
, 1, such that, for any real number a$a \cdot 1 = a$
Inverse Property
addition: Every real number a has a
additive inverse, or opposite
, denoted -a, such thata + (-a) = 0
multiplication: Every nonzero real number a has a
multiplicative inverse, or reciprocal
, denoted $1 \over a$, such that$a \cdot (\frac{1}{a}) = 1$
3 Evaluating Algebraic Expressions
an
algebraic expression
isa collection of constants and variables
joined together by thealgebraic operations
of addition, substraction, multiplication, and divisionan
equation
isa mathematical statement indicating that two expressions are equal
a
formula
isan equation expressing a relationship between
constant and variable quanlities
e.g. $A=\pi r^2$ where A and r are variables and $\pi$ is a constant
1.2 Exponents and Scientific Notation
1.3 Radicals and Rational Exponents
the principal square root of a
is the nonnegative
number that, when multiplied by itself, equals a. It is written as a radical expression
, with a symbol called a radical
over the term called the radicand
: $\sqrt{a}$
$a^{\frac{m}{n}} = (\sqrt[n]{a})^m = \sqrt[n]{a^m}$
1.4 Polynomials
term of a polymomial
: $a_{i}x^{i}$monomial: a polynomial containing only one term
binomial: a polynomial containing two terms
trinomial: a polynomial containing three terms
The highest power of the variable that occurs in the polynomial is called the degree
of a polynomial. The leading term
is the term with the highest power, and its coefficient is called the leading coefficient
1.5 Factoring Polynomials
1.6 Rational Expressions
rational expression
the quotient of two polynomial expressions
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