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 a

  a + 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 that

  a + (-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 is

  a collection of constants and variables joined together by the algebraic operations of addition, substraction, multiplication, and division

• an equation is

  a mathematical statement indicating that two expressions are equal

• a formula is

  an 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