# College Algebra

## Real Numbers, $\mathbb{R}$

Real numbers includes all the numbers in the number line. It is denoted by $\mathbb{R}$.

• Integers, $\mathbb{Z}$
Includes all positive and negative whole numbers. $\mathbb{Z} = \left\{ \ldots, -3, -2, -1, 0, 1, 2, 3, \ldots \right\}$

• Natural numbers, $\mathbb{N}$
All of positive integers including zero. Also called counting numbers. $\mathbb{N} = \left\{ 0, 1, 2, 3, \ldots \right\}$

• Rational numbers, $\mathbb{Q}$
If $a$ and $b$ are integers and $b \ne 0$, then $\dfrac{a}{b}$ is a rational number. As $b$ can take the value of $1$, all integers are rational numbers.

• Irrational numbers, $\mathbb{I}$
Real numbers that cannot be expressed as a fraction of integers are irrational numbers. Example, $\mathbb{I} = { \pi, e, \sqrt{7}, \dots }$

### Properties of Real Numbers

Let $a$, $b$, and $c$ be any real number.

$a + b$ is equal to another real number unique from $a$ and $b$.

• Closure Property of Multiplication
$a \times b$ is equal to another real number unique from $a$ and $b$.

$a + b = b + a$

• Commutative Property of Multiplication
$a \times b = b \times a$

$a + b + c = (a + b) + c = a + (b + c) = (a + c) + b$

• Associative Property of Multiplication
$a \times b \times c = (ab)c = a(bc) = c(ab)$

• Distributive Property
$a(b + c) = ab + ac$

### Absolute Value of Real Numbers

In the number line, the absolute value of any number is the distance between $0$ and the number.

The absolute value of $a$ is denoted by $\lvert a \rvert$.

#### Properties of Absolute Value

1. $\lvert \pm a \rvert = a$
2. $\lvert 0 \rvert = 0$
3. $\lvert a \rvert = \lvert -a \rvert$
4. $\lvert a - b \rvert = \lvert b - a \rvert$
5. If $\lvert x \rvert = a$ then $x = -a$ or $x = a$, where $a$ is a positive number.

### Operation of Real Numbers

If $a$ and $b$ are positive and $a \gt b$, then ...

1. $a + b = \lvert a + b \rvert$
2. $(-a) + (-b) = - \lvert a + b \rvert$

• Subtraction
1. $a - b = a + (-b)$
2. $a - b = \lvert a - b \rvert$
3. $b - a = - \lvert a - b \rvert$
4. $(-a) - (-b) = - \lvert a - b \rvert$
5. $(-b) - (-a) = \lvert a - b \rvert$

• Multiplication
1. $ab = (-a)(-b) = \lvert ab \rvert$
2. $a(-b) = (-a)b = - \lvert ab \rvert$
3. $ab = 0$ if $a = 0$ or $b = 0$ or both $a$ and $b$ are zero.

• Division
1. $\dfrac{a}{b} = \dfrac{-a}{-b} = \left| \dfrac{a}{b} \right|$

2. $\dfrac{-a}{b} = \dfrac{a}{-b} = - \left| \dfrac{a}{b} \right|$

3. $\dfrac{0}{\pm a} = 0$

4. $\dfrac{\pm a}{0} = \infty$

### Equality of Real Numbers

For all real numbers $a$, $b$, and $c$ ...

• Symmetric Property
$a = b$ then $b = a$

• Transitive Property
If $a = b$ and $b = c$ then $a = c$

$a = b$ iff $a + c = b + c$

• Multiplication Property of Equality
$a = b$ iff $ac = bc$, given that $c \ne 0$.

Note: iff means "if and only if".

### Inequality of Real Numbers

Symbol Statement
$a \gt b$ $a$ is greater than $b$
$a \lt b$ $a$ is less than $b$
$a \ge b$ • $a$ is greater than or equal to $b$
• $a$ is at least $b$
$a \le b$ • $a$ is less than or equal to $b$
• $a$ is at most $b$

#### Properties of Inequality

For all real numbers $a$, $b$, and $c$ ...

• If $a \gt b$ then $b \lt a$ and $-a \lt -b$

• If $a \gt 0$ then $-a \lt 0$ and if $a \lt 0$ then $-a \gt 0$

$a \gt b$ iff $a + c \gt b + c$

• Multiplication Property of Inequality
1. $a \gt b$ iff $ac \gt bc$ when $c \gt 0$
2. $a \gt b$ iff $ac \lt bc$ when $c \lt 0$
• If $\lvert x \rvert \lt a$ then $x \gt -a$ and $x \lt a$, where $a$ is a positive number.

• If $\lvert x \rvert \gt a$ then $x \lt -a$ or $x \gt a$, where $a$ is a positive number.

## Complex numbers, $\mathbb{C}$

Complex numbers are in the form $a + bi$, where $a$ is the real part and $bi$ is the imaginary part. Note that $a$ and $b$ are real numbers.

### Imaginary Numbers

The $\sqrt{-1}$ is called the imaginary unit and is denoted by $i$. Imaginary number is a product of a real number and $i$. Example: $5i$, $\sqrt{2} i$, $\pi i$.

$i = \sqrt{-1}$ or $i^2 = -1$

## Laws of Indices

1. $b^n = \underbrace{b \cdot b \cdot b \cdot \ldots}_{n \text{ factors of } b}$

2. $a^m \cdot a^n = a^{m + n}$

3. $(a^m)^n = a^{mn}$

4. $\dfrac{a^m}{a^n} = a^{m - n}$

5. $(abc)^n = a^n \cdot b^n \cdot c^n$

6. $\left( \dfrac{a}{b} \right)^n = \dfrac{a^n}{b^n}$

7. $a^{-n} = \dfrac{1}{a^n}$ and $\dfrac{1}{a^{-n}} = a^n$

8. $a^0 = 1$

9. If $a^m = a^n$ then $m = n$ provided $a \ne \left\{ 0, 1, -1 \right\}$

1. $a^{1/n} = \sqrt[n]{a}$

2. $a^{m/n} = \sqrt[n]{a^m} = \left( \sqrt[n]{a} \right)^m$

3. $\sqrt[m]{a} \cdot \sqrt[n]{a} = \sqrt[mn]{a^{m + n}}$

4. $\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{ab}$

5. $\dfrac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\dfrac{a}{b}}$ provided $b \ne 0$

6. $\left( \sqrt[n]{a} \right)^n = a$

## Properties of Logarithm

1. If $y = b^x$, then $x = \log_b y$

2. $\log_b (xyz) = \log_b x + \log_b y + \log_b z$

3. $\log_b \left( \dfrac{x}{y} \right) = \log_b x - \log_b y$

4. $\log_b x^n = n \log_b x$

5. $b^{\log_b x} = x$

6. $\log_b 1 = 0$

7. $\log_b b = 1$

8. Common Logarithm
$\log_{10} x = \log x$

9. Natural Logarithm
$\log_e x = \ln x$

The Euler's number $e$ is approximately 2.71828, an irrational number. The exact value of $e$ in infinite series is $\displaystyle e = \sum_{n = 0}^\infty \dfrac{1}{x!}$. In Calculus, the number $\displaystyle e = \lim_{n \to \infty} \left( 1 + \dfrac{1}{n} \right)^n$.

10. Change-base Rule
$\log_y x = \dfrac{\log x}{\log y} = \dfrac{\ln x}{\ln y}$

11. If $\log_b x = \log_b y$ then $x = y$

## Special Products and Factoring

### Special Products

1. $(x + y)(x - y) = x^2 - y^2$

2. $(x + y)^2 = x^2 + 2xy + y^2$

3. $(x - y)^2 = x^2 - 2xy + y^2$

4. $(x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3$

5. $(x - y)^3 = x^3 - 3x^2y + 3xy^2 - y^3$

6. $(x + a)(x + b) = x^2 + (a + b)x + ab$

7. $(ax + by)(cx + dy) = acx^2 + (ad + bc)xy + bdy^2$

### Factoring Polynomials

1. $ax + ay + az = a(x + y + z)$

2. $x^2 - y^2 = (x + y)(x - y)$

3. $x^3 + y^3 = (x + y)(x^2 - xy + y^2)$

4. $x^3 - y^3 = (x - y)(x^2 + xy + y^2)$

5. $x^2 + 2xy + y^2 = (x + y)^2$

6. $x^2 - 2xy + y^2 = (x - y)^2$

7. $x^2 + (a + b)x + ab = (x + a)(x + b)$

8. $acx^2 + (ad + bc)xy + bdy^2 = (ax + by)(cx + dy)$

### Expansion of $(a + b)^n$

This is called the Binomial Theorem. The rth term in the expansion of (a + b)n is given by the formula:

$r^{th} \, \text{term} = {_nC_m} \, a^{n - m} b^m$

where m = r - 1

The quadratic equation is inthe form $Ax^2 + Bx + C = 0$ and can be written in a factored form $(x - x_1)(x - x_2) = 0$, where $x_1$ and $x_2$ are the roots of the quadratic equation.

$x = \dfrac{-B \pm \sqrt{B^2 - 4AC}}{2A}$

#### Discriminant

The quantity $B^2 - 4AC$ inside the $\sqrt{~}$ of the quadratic formula is called the discriminant. The nature of roots of the quadratic equation according the value of discriminant are as follows:

• There is only 1 root if $B^2 - 4AC = 0$.
• The roots are two unequal numbers if $B^2 - 4AC \gt 0$.
• The roots are imaginary if $B^2 - 4AC \lt 0$.

#### Sum and Product of Roots

• Sum of roots
$x_1 + x_2 = -\dfrac{B}{A}$

• Product of roots
$x_1 \cdot x_2 = \dfrac{C}{A}$

## Sequences and Series

• Sequence is a succession of numbers formed according to some fixed rule.

$$1, 4, 9, 16, 25, \ldots$$

is a sequence whose $n^\text{th}$ term is equal to $n^2$

• Series is the indicated sum of a sequence of numbers.

$$1 + 4 + 9 + 16 + 25 + \ldots = \sum_{n = 1}^\infty n^2$$

is an example of a series.

### Arithmetic Progression, AP

A sequence of numbers is in AP if any number after the first is obtained by adding a fixed number to the one immediately preceding it. The fixed number that is added is called the common difference, $d$.

• Common difference
$d = a_2 - a_1 = a_3 - a_2 = a_4 - a_3 = \ldots$

• $n^\text{th}$ term
$a_n = a_1 + (n - 1)d$

• Sum of first $n$ terms
$S = \frac{1}{2}n(a_1 + a_n)$

$S = \frac{1}{2}n \left[ 2a_1 + (n - 1)d \right]$

### Geometric Progression, GP

A sequence of numbers is in GP if any number after the first is obtained by multiplying a fixed number to the one immediately preceding it. The fixed number that is multiplied is called the common ratio, $r$.

• Common ratio
$r = \dfrac{a_2}{a_1} = \dfrac{a_3}{a_2} = \dfrac{a_4}{a_3} = \ldots$

• $n^\text{th}$ term
$a_n = a_1 r^{n - 1}$

• Sum of first $n$ terms
$S = \dfrac{a_1(1 - r^n)}{1 - r}$   ←   for $r \lt 1$

$S = \dfrac{a_1(r^n - 1)}{r - 1}$   ←   for $r \gt 1$

#### Infinite Geometric Progression, IGP

Infinite Geometric Progression is a GP in which $-1 \lt r \lt 1$, $r \ne 0$ and $n \to \infty$. The sum of IGP is given by the formula

$$S = \dfrac{a_1}{1 - r}$$

### Harmonic Progression, HP

A sequence of numbers are in HP if their reciprocals form an AP.

• Common difference of reciprocals
$\dfrac{1}{a_2} - \dfrac{1}{a_1} = \dfrac{1}{a_3} - \dfrac{1}{a_2} = \dfrac{1}{a_4} - \dfrac{1}{a_3} = \ldots$

• $n^\text{th}$ term
$a_n = \dfrac{1}{a_1 + (n - 1)d}$

• Sum of first $n$ terms
$S = \dfrac{1}{d} ~ \ln \left[ \dfrac{2a_1 + (2n - 1)d}{2a_1 - d} \right]$