Numbers and Functions

1.1 Different Kinds of Numbers

Calculus relies on a solid understanding of different types of numbers. Here, we introduce the main categories:

• Positive integers:

• Zero:

• Negative integers:

• Rational numbers: Numbers that can be written as a fraction , where and are integers and .

• Irrational numbers: Numbers that cannot be written as a fraction of two integers (e.g., , ).

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

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

1.2 The Real Number Line and Intervals

The real number line is a geometric representation of all real numbers as points on a line. Intervals are subsets of the real line:

• Open interval : All such that .

• Closed interval : All such that .

• Half-open intervals: or .

Distance on the real line: The distance between two numbers and is .

1.3 Set Notation

Sets are collections of numbers. Common notations include:

• : The set of all real numbers.

• : The set of all rational numbers.

• : The set of all real numbers such that .

Other set operations:

• Intersection : Elements in both and .

• Union : Elements in or (or both).

1.4 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 input values for which the function is defined.

• Range: The set of all possible output values.

Example: has domain and range .

1.5 Graphing a Function

The graph of a function is the set of all points in the plane. The domain is the set of -values for which is defined, and the range is the set of -values that can take.

Vertical Line Test: A curve in the plane is the graph of a function if and only if no vertical line intersects the curve more than once.

1.6 Linear Functions

A linear function has the form:

• is the slope (rate of change).

• is the y-intercept (value when ).

Example: is a linear function with slope and y-intercept .

1.7 Domain and Range: Examples

• Example 1: has domain and range .

• Example 2: has domain and range .

1.8 Functions in Real Life

Functions are used to model relationships in science, engineering, and economics. For example, the distance traveled at constant speed over time is .

1.9 Piecewise Functions

Some functions are defined by different formulas on different intervals. These are called piecewise-defined functions.

Example:

1.10 Summary Table: Types of Numbers

1.11 Exercises (Selected)

• Which of the following fractions have finite decimal expansions?

• Draw the following sets of real numbers on the real line.

• Suppose and are intervals. Is it always true that is an interval? What about ?

Additional info: These notes are based on the first chapter of a standard college Calculus I course, covering foundational concepts necessary for further study in calculus, such as types of numbers, set notation, and the definition and properties of functions.