Numbers and Functions

1. What is a Number?

The study of calculus begins with a careful understanding of the types of numbers used in mathematics, especially the real numbers. This section introduces the different kinds of numbers and their properties.

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

• Zero: The number 0, which is neither positive nor negative.

• 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 of two integers, such as or .

• Real Numbers: The set of all rational and irrational numbers. Real numbers can be represented as points on the number line.

Decimal Expansions

• Rational numbers have decimal expansions that either terminate or repeat.

• Irrational numbers have non-terminating, non-repeating decimal expansions.

• Example: ,

Why are Real Numbers Called 'Real'?

• The term 'real' distinguishes these numbers from 'imaginary' numbers, which involve the square root of negative numbers.

• Real numbers are used to measure continuous quantities and can be visualized on the number line.

Intervals and the Number Line

• An interval is a set of real numbers between two endpoints. For example, the interval from to is written as (including endpoints) or (excluding endpoints).

• The distance between two numbers and is .

2. Set Notation

Sets are collections of objects, often numbers, that satisfy certain properties. Set notation is used to describe intervals and other collections of numbers.

• Interval Notation: denotes all between and .

• Union: is the set of elements in or .

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

• Example: is the set of all such that .

3. Functions

Functions are fundamental objects in calculus, describing how one quantity depends on another.

• Definition: A function assigns to each element in a set (the domain) exactly one element in another set (the range).

• Notation: means is a function from set to set .

• Example: assigns to each real number its square.

Graphing a Function

• The graph of a function is the set of all points in the plane.

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

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

• Example: The graph of is a parabola opening upwards.

Linear Functions

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

• The graph is a straight line.

• Example: is a line with slope 2 and y-intercept 1.

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).

• To find the range, determine all possible values can take as varies over the domain.

• Example: For , the domain is and the range is .

Functions in Real Life

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

• Example: The distance traveled at constant speed over time is .

The 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 passes the vertical line test, but the graph of a circle does not.

4. Additional Examples and Exercises

• Determine whether a given set is an interval.

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

• Classify numbers as rational or irrational based on their decimal expansions.

• Find the domain and range of various functions.

Additional info: These notes are based on the first chapter of a first-semester calculus course, focusing on foundational concepts such as types of numbers, set notation, and the definition and properties of functions. Later chapters (as seen in the table of contents) will cover limits, derivatives, integrals, and applications, which are standard topics in college calculus.