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.
- Closure Property of Addition
$a + b$ 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 $b$.
- Commutative Property of Addition
$a + b = b + a$
- Commutative Property of Multiplication
$a \times b = b \times a$
- Associative Property of Addition
$a + b + c = (a + b) + c = a + (b + c) = (a + c) + b$
- 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.
![000-the-number-line.png](/sites/default/files/reviewer-algebra/000-the-number-line.png)
The absolute value of $a$ is denoted by $\lvert a \rvert$.
Properties of Absolute Value
- $\lvert \pm a \rvert = a$
- $\lvert 0 \rvert = 0$
- $\lvert a \rvert = \lvert -a \rvert$
- $\lvert a - b \rvert = \lvert b - a \rvert$
- 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 ...
- Addition
- $a + b = \lvert a + b \rvert$
- $(-a) + (-b) = - \lvert a + b \rvert$
- Subtraction
- $a - b = a + (-b)$
- $a - b = \lvert a - b \rvert$
- $b - a = - \lvert a - b \rvert$
- $(-a) - (-b) = - \lvert a - b \rvert$
- $(-b) - (-a) = \lvert a - b \rvert$
- Multiplication
- $ab = (-a)(-b) = \lvert ab \rvert$
- $a(-b) = (-a)b = - \lvert ab \rvert$
- $ab = 0$ if $a = 0$ or $b = 0$ or both $a$ and $b$ are zero.
- Division
- $\dfrac{a}{b} = \dfrac{-a}{-b} = \left| \dfrac{a}{b} \right|$
- $\dfrac{-a}{b} = \dfrac{a}{-b} = - \left| \dfrac{a}{b} \right|$
- $\dfrac{0}{\pm a} = 0$
- $\dfrac{\pm a}{0} = \infty$
- $\dfrac{a}{b} = \dfrac{-a}{-b} = \left| \dfrac{a}{b} \right|$
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 $a = c$
- Addition Property of Equality
$a = b$ 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 $-a \gt 0$
- Addition Property of Inequality
$a \gt b$ iff $a + c \gt b + c$
- Multiplication Property of Inequality
- $a \gt b$ iff $ac \gt bc$ when $c \gt 0$
- $a \gt b$ 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.
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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: $5i$, $\sqrt{2} i$, $\pi i$.
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Laws of Indices
- $b^n = \underbrace{b \cdot b \cdot b \cdot \ldots}_{n \text{ factors of } b}$
- $a^m \cdot a^n = a^{m + n}$
- $(a^m)^n = a^{mn}$
- $\dfrac{a^m}{a^n} = a^{m - n}$
- $(abc)^n = a^n \cdot b^n \cdot c^n$
- $\left( \dfrac{a}{b} \right)^n = \dfrac{a^n}{b^n}$
- $a^{-n} = \dfrac{1}{a^n}$ and $\dfrac{1}{a^{-n}} = a^n$
- $a^0 = 1$
- If $a^m = a^n$ then $m = n$ provided $a \ne \left\{ 0, 1, -1 \right\}$
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Properties of Radicals
- $a^{1/n} = \sqrt[n]{a}$
- $a^{m/n} = \sqrt[n]{a^m} = \left( \sqrt[n]{a} \right)^m$
- $\sqrt[m]{a} \cdot \sqrt[n]{a} = \sqrt[mn]{a^{m + n}}$
- $\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{ab}$
- $\dfrac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\dfrac{a}{b}}$ provided $b \ne 0$
- $\left( \sqrt[n]{a} \right)^n = a$
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Properties of Logarithm
- If $y = b^x$, then $x = \log_b y$
- $\log_b (xyz) = \log_b x + \log_b y + \log_b z$
- $\log_b \left( \dfrac{x}{y} \right) = \log_b x - \log_b y$
- $\log_b x^n = n \log_b x$
- $b^{\log_b x} = x$
- $\log_b 1 = 0$
- $\log_b b = 1$
- Common Logarithm
$\log_{10} x = \log x$
- 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 $\displaystyle e = \lim_{n \to \infty} \left( 1 + \dfrac{1}{n} \right)^n$.
- Change-base Rule
$\log_y x = \dfrac{\log x}{\log y} = \dfrac{\ln x}{\ln y}$
- If $\log_b x = \log_b y$ then $x = y$
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Special Products and Factoring
Special Products
- $(x + y)(x - y) = x^2 - y^2$
- $(x + y)^2 = x^2 + 2xy + y^2$
- $(x - y)^2 = x^2 - 2xy + y^2$
- $(x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3$
- $(x - y)^3 = x^3 - 3x^2y + 3xy^2 - y^3$
- $(x + a)(x + b) = x^2 + (a + b)x + ab$
- $(ax + by)(cx + dy) = acx^2 + (ad + bc)xy + bdy^2$
Factoring Polynomials
- $ax + ay + az = a(x + y + z)$
- $x^2 - y^2 = (x + y)(x - y)$
- $x^3 + y^3 = (x + y)(x^2 - xy + y^2)$
- $x^3 - y^3 = (x - y)(x^2 + xy + y^2)$
- $x^2 + 2xy + y^2 = (x + y)^2$
- $x^2 - 2xy + y^2 = (x - y)^2$
- $x^2 + (a + b)x + ab = (x + a)(x + b)$
- $acx^2 + (ad + bc)xy + bdy^2 = (ax + by)(cx + dy)$
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:
where m = r - 1
The Quadratic Equation
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.
The Quadratic Formula
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
$x_1 \cdot x_2 = \dfrac{C}{A}$
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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
$S = \dfrac{1}{d} ~ \ln \left[ \dfrac{2a_1 + (2n - 1)d}{2a_1 - d} \right]$