Numbers and Functions

1. What is a Number?

The study of calculus begins with understanding the types of numbers used in mathematics and their properties. This section introduces the different kinds of numbers and their significance in calculus.

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

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

• Negative Integers: ..., -3, -2, -1

• Rational Numbers: Numbers that can be written as the ratio of two integers, e.g., $\frac{1}{2}$, $-\frac{3}{4}$, $5$ (since $5 = \frac{5}{1}$).

• Irrational Numbers: Numbers that cannot be written as a ratio of two integers, such as $\sqrt{2}$ or $\pi$.

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

1.1 Decimal Expansions

• Every rational number has a decimal expansion that either terminates or repeats.

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

• Example: $\frac{1}{3} = 0.333...$ (repeating), $\sqrt{2} = 1.4142135...$ (non-repeating).

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 number line, such as $(a, b)$ (all numbers between $a$ and $b$, not including $a$ and $b$) or $[a, b]$ (including $a$ and $b$).

• Distance on the number line: The distance between two numbers $x$ and $y$ is $|x - y|$.

1.3 Set Notation

• Sets are collections of numbers or objects. Common notations include $\mathbb{R}$ (all real numbers), $\mathbb{Q}$ (all rational numbers), and $\mathbb{Z}$ (all integers).

• Example: $A = \{x \mid 1 < x < 6\}$ is the set of all real numbers $x$ such that $1 < x < 6$.

• Set operations include union ($A \cup B$), intersection ($A \cap B$), and set difference ($A - B$).

2. Exercises

Practice problems may include:

• Identifying the 200th digit after the period in the decimal expansion of $\frac{1}{7}$.

• Determining which fractions have finite decimal expansions.

• Describing sets using set notation.

• Comparing and combining sets using union and intersection.

3. Functions

Functions are fundamental objects in calculus, describing relationships between variables.

3.1 Definition of a Function

• A function $f$ assigns to each element $x$ in a set called the domain exactly one element $f(x)$ in a set called the range.

• Notation: $f: X \to Y$ means $f$ is a function from set $X$ (domain) to set $Y$ (range).

• Example: $f(x) = x^2$ is a function from $\mathbb{R}$ to $\mathbb{R}$.

3.2 Graphing a Function

• The graph of a function is the set of all points $(x, f(x))$ in the plane, where $x$ is in the domain of $f$.

• To graph a function, plot points for various $x$ values and connect them smoothly if possible.

• The domain is the set of all $x$ for which $f(x)$ is defined; the range is the set of all possible $f(x)$ values.

3.3 Linear Functions

• A linear function has the form $f(x) = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

• The graph of a linear function is a straight line.

• Example: $f(x) = 2x + 1$ has slope $2$ and y-intercept $1$.

3.4 Domain and "Biggest Possible Domain"

• When defining a function by a formula, the domain is all real numbers for which the formula makes sense (e.g., no division by zero, no square roots of negative numbers).

• Example: For $f(x) = \frac{1}{x}$, the domain is $x \neq 0$.

3.5 Example: Finding Domain and Range

• For $f(x) = \sqrt{x}$, the domain is $x \geq 0$ (since square roots of negative numbers are not real).

• The range is $f(x) \geq 0$.

3.6 Functions in Real Life

• Functions can model real-world relationships, such as distance as a function of time, or cost as a function of quantity.

• Example: The distance $d$ from a fixed point as a function of time $t$ if moving at constant speed $v$ is $d(t) = vt$.

3.7 The Vertical Line Property

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

• This ensures that for each $x$ in the domain, there is only one $f(x)$.

Summary Table: Types of Numbers

Additional info: These notes are based on the first chapter of a first-semester calculus course, focusing on foundational concepts of numbers, sets, and functions, which are essential for understanding calculus.