Real Numbers, Inequalities, and Integer Exponents: Precalculus Study Notes

Real Numbers and Their Classifications

Sets and Number Types

In mathematics, numbers are organized into sets based on their properties. Understanding these classifications is foundational for precalculus.

• Set: A collection of objects or numbers.

• Natural Numbers: Counting numbers: 1, 2, 3, ...

• Whole Numbers: Natural numbers plus zero: 0, 1, 2, 3, ...

• Integers: Whole numbers and their negatives: ..., -3, -2, -1, 0, 1, 2, 3, ...

• Rational Numbers: Any number that can be written as a fraction of two integers, $\frac{a}{b}$ where $b \neq 0$.

  • Includes terminating decimals (e.g., 0.6, 0.425978)

  • Includes repeating decimals (e.g., 0.555... = $\frac{5}{9}$)

• Irrational Numbers: Numbers that cannot be written as fractions. Their decimal expansions are non-terminating and non-repeating.

  • Examples: $\pi$, $e$, $\sqrt{2}$

  • Often called "messy" decimals

• Real Numbers: The set of all rational and irrational numbers.

Example: $\sqrt{2}$ is irrational, $\frac{3}{4}$ is rational, -5 is an integer.

Classification Diagram

The following table describes the relationships among the main number sets:

Set	Includes	Examples
Natural Numbers	Counting numbers	1, 2, 3, ...
Whole Numbers	Natural numbers and 0	0, 1, 2, ...
Integers	Whole numbers and negatives	-3, 0, 2
Rational Numbers	Fractions, terminating/repeating decimals	$\frac{2}{3}$, 0.5, -7
Irrational Numbers	Non-repeating, non-terminating decimals	$\pi$, $\sqrt{2}$
Real Numbers	All rational and irrational numbers	Any number on the number line

Inequalities and Interval Notation

Understanding Inequalities

Inequalities compare the relative size of two numbers. They are fundamental in expressing ranges and domains in precalculus.

• $a < b$: $a$ is less than $b$

• $a > b$: $a$ is greater than $b$

• $a \leq b$: $a$ is less than or equal to $b$

• $a \geq b$: $a$ is greater than or equal to $b$

• $a \geq 0$: $a$ is positive or zero

• $a < 0$: $a$ is negative

• $a \geq 0$: $a$ is non-negative

Example: If $x > 2$, $x$ is any number greater than 2.

Interval Notation and Number Lines

Intervals describe sets of numbers between endpoints. They are often represented on number lines and with interval notation.

• Open Interval: $(a, b)$ excludes endpoints $a$ and $b$.

• Closed Interval: $[a, b]$ includes endpoints $a$ and $b$.

• Half-Open Interval: $(a, b]$ or $[a, b)$ includes one endpoint.

• Infinity: $\infty$ is used to indicate unbounded intervals, always with an open bracket: $(a, \infty)$ or $(-\infty, b)$.

Example: $2.5 < x \leq 6$ is written as $(2.5, 6]$.

Set Operations: Union and Intersection

Sets can be combined or compared using union and intersection.

• Union ($\cup$): Combines all elements from both sets. Example: $(5,7) \cup (9, \infty)$

• Intersection ($\cap$): Includes only elements common to both sets. Example: $(5,7) \cap (6, \infty) = (6,7)$

Example: $x < 7$ and $x > 5$ is $x \in (5,7)$.

Algebraic Expressions and Exponents

Structure of Algebraic Expressions

Algebraic expressions consist of terms formed by variables and constants combined using operations.

• Terms: Parts of an expression separated by addition or subtraction.

• Factors: Quantities multiplied together within a term.

• Variables: Symbols (e.g., $x$, $y$) that stand in for numbers.

Example: $6x - 15xy + \frac{2}{3}y$ has three terms.

Properties of Operations

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

• Zero Product Property: If $ab = 0$, then $a = 0$ or $b = 0$

Exponent Rules

Exponents indicate repeated multiplication. Several rules govern their manipulation:

• Product of Powers: $a^m \cdot a^n = a^{m+n}$

• Power of a Power: $(a^m)^n = a^{mn}$

• Power of a Product: $(ab)^n = a^n b^n$

• Quotient of Powers: $\frac{a^m}{a^n} = a^{m-n}$

Example: $2^6 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2$

Simplifying Expressions with Exponents

To simplify expressions with exponents, apply the exponent rules and combine like terms.

• Example: Simplify $\frac{x^4(-2y)^5}{(x^3y^2)^2}$

Step-by-step solution:

• Expand numerator: $x^4 \cdot (-2)^5 \cdot y^5 = x^4 \cdot -32 \cdot y^5$

• Expand denominator: $(x^3y^2)^2 = x^{3 \cdot 2} y^{2 \cdot 2} = x^6 y^4$

• Combine: $\frac{-32x^4y^5}{x^6y^4} = -32 \cdot \frac{x^4}{x^6} \cdot \frac{y^5}{y^4} = -32 \cdot x^{-2} \cdot y^1 = \frac{-32y}{x^2}$

Interval Lengths

Finding the Length of an Interval

The length of an interval $[a, b]$ is the absolute value of the difference between its endpoints.

• Formula: $|b - a|$

• Example: The length of $[6, 12]$ is $|12 - 6| = 6$

Summary Table: Key Properties

Property	Description	Example
Distributive	$a(b + c) = ab + ac$	$2(x + 3) = 2x + 6$
Zero Product	If $ab = 0$, then $a = 0$ or $b = 0$	If $x(x-5) = 0$, $x = 0$ or $x = 5$
Exponent Product	$a^m \cdot a^n = a^{m+n}$	$x^2 \cdot x^3 = x^5$

Additional info: These notes provide foundational concepts for real numbers, inequalities, interval notation, set operations, algebraic expressions, and exponent rules, all of which are essential for success in precalculus.