Numbers and Functions

1. What is a Number?

Calculus is built on the concept of real numbers and functions of real variables. Understanding the types of numbers and their properties is foundational for further study.

• Positive Integers: The simplest numbers: 1, 2, 3, ...

• Zero: The number 0.

• Negative Integers: ..., -3, -2, -1

• Rational Numbers: Numbers that can be written as a ratio of two integers, e.g., , .

• Irrational Numbers: Numbers that cannot be written as a ratio of two integers, e.g., , .

• Real Numbers: All rational and irrational numbers together form the set of real numbers, denoted .

Decimal Expansions: Rational numbers have either terminating or repeating decimal expansions. Irrational numbers have non-terminating, non-repeating decimals.

Intervals: The set of real numbers between two values and is called an interval, denoted for open intervals or for closed intervals.

2. Set Notation

Sets are collections of numbers or objects. In calculus, we often use set notation to describe intervals and domains.

• Interval Notation: denotes all between and .

• Union: is the set of elements in or .

• Intersection: is the set of elements in both and .

3. Functions

A function is a rule that assigns to each element in a set (the domain) exactly one element (the range), often written as .

• Domain: The set of all for which is defined.

• Range: The set of all possible values can take.

• Graph of a Function: The set of points in the plane.

Example: The function has domain and range .

3.1 Linear Functions

Linear functions have the form , where is the slope and is the y-intercept.

• Slope:

• Graph: A straight line in the plane.

3.2 Domain and Range

To find the domain of a function, determine all for which the formula makes sense (e.g., avoid division by zero or taking square roots of negative numbers).

Example: For , the domain is .

3.3 The Vertical Line Test

A graph in the plane represents a function if and only if no vertical line intersects the graph at more than one point.

4. Exercises and Applications

• Identify whether a given decimal is rational or irrational.

• Find the domain and range of various functions.

• Use set notation to describe intervals and solution sets.

5. Summary Table: Types of Numbers

Additional info: This summary is based on the first chapter and table of contents, which indicate further topics such as limits, derivatives, integrals, and applications will be covered in subsequent chapters.