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 integers (e.g., , ).

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

Decimal Expansions: Rational numbers have either terminating or repeating decimal expansions. 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 defined by inequalities:

• Open interval:

• Closed interval:

• Half-open intervals: or

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

1.3 Set Notation

Sets are collections of numbers or objects. Common notations include:

• : The set of all real numbers

• : The set of all rational numbers

• : The set of all integers

• : The set of all natural numbers (positive integers)

Set-builder notation is used to describe sets, e.g., .

1.4 Functions

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

• Domain: The set of all input values for which the function is defined.

• Range: The set of all possible output values.

Example: The function 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 the function can take.

• Linear function: is a straight line with slope and -intercept .

• 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 Domain and Range from Formulas

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

Example: For , the domain is (all real numbers except ).

1.7 Functions in Real Life

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

1.8 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 ?

• Write the numbers , , as fractions.

1.9 Additional Info

• Irrational numbers such as can be constructed geometrically (e.g., as the diagonal of a unit square).

• Real numbers are called "real" to distinguish them from "imaginary" numbers, which involve .

Table: Types of Numbers

Additional info: This summary covers the main points from the first chapter of a first-semester calculus course, focusing on numbers, sets, and functions as foundational concepts for calculus.