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, e.g., 1, 2, 3, ...

• Zero: The integer 0.

• Negative Integers: ..., -3, -2, -1

• Rational Numbers: Numbers that can be written as a fraction of two integers, , where .

• Irrational Numbers: Numbers that cannot be written as a fraction, e.g., , .

• Real Numbers: All rational and irrational numbers together.

Decimal Expansions: Every real number can be represented by a (possibly infinite) decimal expansion. Rational numbers have repeating or terminating decimals; irrational numbers have non-repeating, non-terminating decimals.

Intervals: Real numbers can be visualized on a number line. Intervals are subsets of real numbers between two endpoints, e.g., (closed interval), (open interval).

2. Set Notation

Sets are collections of numbers. Common notations include:

• : Set of all real numbers

• : Set of all rational numbers

• : Set of all integers

• : Set of all positive integers

• : Intersection of sets A and B (elements in both)

• : Union of sets A and B (elements in either)

Example: is the set of all real numbers between 1 and 6.

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 input values for which the function is defined.

• Range: The set of all possible output values .

Example: For , the domain is (since square roots of negative numbers are not real), and the range is .

3.1. 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.

• 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.

Example: The graph of is a parabola; the graph of is the right half of a parabola.

3.2. Linear Functions

A linear function has the form , where is the slope and is the -intercept.

• Slope: for two points and on the line.

• Graph: A straight line in the plane.

3.3. Piecewise Functions

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

3.4. Domain and Range from Formulas

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

Example: For , the domain is .

3.5. Functions in Real Life

Functions can model real-world relationships, such as distance over time, population growth, or temperature changes.

4. Intervals and the Real Number Line

Intervals are used to describe subsets of the real numbers:

• Open interval:

• Closed interval:

• Half-open intervals: or

The distance between two real numbers and is .

5. Decimal Expansions and Real Numbers

Every real number can be represented as a (possibly infinite) decimal expansion. Rational numbers have repeating or terminating decimals; irrational numbers do not.

Example: ,

6. Exercises and Applications

• Find the 200th digit after the period in the expansion of .

• Which of the following fractions have finite decimal expansions?

• Draw the following sets of real numbers and find their intersections and unions.

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

7. Table: Types of Numbers

Additional info: These notes cover the foundational concepts of numbers, sets, and functions, which are essential for understanding calculus. Later chapters (as seen in the table of contents) will cover limits, derivatives, integrals, and applications.