# 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
