Precalculus: Lecture Notes

Section 1.1 The Set of Real Numbers

Calculus is the study of how things change. It provides a framework for modelling systems in which there is change, and a way to deduce the predictions of such models. — Daniel Kleitman, Emeritus Professor of Applied Mathematics, Combinatorics, Operations Research, MIT, 1934 –

The Real Numbers. It is likely that, up to this point in your studies of mathematics, you have used only real numbers, like \(0,1,-0.5, \pi, \frac{24}{7}, -\sqrt{11}\text{,}\) and so on.

There are two basic reasons why in mathematics we attach the adjective real to the noun numbers:

1. We use real numbers to deal with the real world, like for counting, measuring, and describing patterns and relationships among various quantities that we observe in the nature.

2. Later in your studies you will learn about the so-called imaginary numbers. As you may have already guessed, these numbers sometimes behave differently than the real numbers. Somewhat surprisingly, the imaginary numbers are sometimes a better mathematical tool to understand a real world phenomenon than the real numbers.

In this course we will deal only with the real numbers.

To be able to fully understand the real numbers, their structure and properties, it requires advanced mathematical knowledge. Our course is one of the initial steps in that direction.

Sets. Think about a set as a collection of objects. We think about each of those objects as an element of the set.

For example, I can think that all students enrolled in my current class form a set, call it the set \(J\text{.}\) A mathematical way to say the same thing is to write

We read: “\(J\) is the set of all \(x\) such that \(x\) is a student enrolled in Dr. J's class.”

This notation is something that we will use often in this course. To denote a set, we use the curly brackets “\(\{ \cdot \}\)” and the column “\(:\)” to give the property that we use to determine if something is an element of the given set or not:

and read it as: “the set of all \(x\) such that \(x\) satisfies the property.”

Another way to describe a set is to list all of its elements. For example, the set of all students who attended my office hour yesterday is given by

Again, notice the curly brackets “\(\{ \cdot \}\) .”

For any student at our university, based on the class list, I can determine if that student is enrolled in my class or not.

Say that I know that Xiao is enrolled in my class and that Serena is not. In other words, Xiao is an element of the set \(J\text{,}\) Serena is not. I use the mathematical symbols “\(\in\)” and “\(\not\in\)” to write those facts:

I read these statements as “Xiao belongs to the set \(J\)” and “Serena does not belong to the set \(J\text{.}\)”

The set with no elements is called the empty set and it is usually denoted by \(\emptyset\text{.}\)

Finally, recall the following three set operations:

1. Inclusion: The set \(A\) is a subset of the set \(B\) if any element of the set \(A\) is an element of the set \(B\) as well. In this case we write \(A\subseteq B\text{.}\)

2. Intersection: If \(A\) and \(B\) are two sets then the set \(A\cap B\) is called the intersection of the sets \(A\) and \(B\) and, by definition, contains all elements that belong to both \(A\) and \(B\text{.}\) In other words

   \begin{equation*} A\cap B=\{x:x\in A \mbox{ and } x\in B\}. \end{equation*}

3. Union: If \(A\) and \(B\) are two sets then the set \(A\cup B\) is called the union of the sets \(A\) and \(B\) and, by definition, contains all elements that belong to \(A\) or to \(B\text{.}\) In other words

   \begin{equation*} A\cup B=\{x:x\in A \mbox{ or } x\in B\}. \end{equation*}

For example, if \(A=\{1,2\}, B=\{1,2,3,4\}\) and \(C=\{3,4,5\}\) then

The Real Line. We will model the set of the real numbers by the set of points on the real line.

See Figure 1.1 and Figure 1.2.

More About the Set of Real Numbers. Throughout this course we will denote the set of real numbers by the symbol \(\mathbb{R}\text{.}\)

The following subsets of the set of real numbers will be of our interest during the course:

• The set of natural numbers: \(\mathbb{N}=\{ 1,2,3,\ldots\}\text{.}\)

• The set of whole numbers: \(\mathbb{N}_0=\{ 0, 1,2,3,\ldots\}\text{.}\)

• The set of integers: \(\mathbb{Z}=\{ \ldots, -3,-2,-1,0,1,2,3,\ldots\}\text{.}\)

• The set of rational numbers: \(\mathbb{Q}=\{ \frac{p}{q}: p\text{ and }q \text{ are integers, }q\not=0\}\text{.}\)

• The set of irrational numbers: \(\mathbb{I}=\{ x: x\text{ cannot be expressed as a ratio of two integers}\}\text{.}\)

• The set of real numbers: \(\mathbb{R}=\mathbb{Q}\cup \mathbb{I}\)

See Figure 1.3.

Example 1.1.1. Rational and irrational numbers.

Recall that every rational number can be represented as a ratio of two integers in many ways.

For example, we can write the number “one half” in the following ways:

\begin{equation*} \frac{1}{2}=\frac{2}{4}=\frac{3}{6}=\frac{5}{10}. \end{equation*}

Also, recall that it is common to write the number “one half” in the so–called decimal notation: \(0.5\text{.}\) Observe that \(0.5\) is just another way to represent the fraction \(\frac{5}{10}\text{.}\)

Below is a “visual” proof that \(\sqrt{2}\) is not rational. This way of proving the irrationality of \(\sqrt{2}\) is attributed to Theodorus of Cyrene.

Observe that \(\sqrt{2} \gt 1\text{.}\) Suppose that \(\sqrt{2}\) is a rational number. Then there are positive integer \(p\) and \(q\text{,}\) \(p\gt q\text{,}\) such that \(\sqrt{2}=\frac{p}{q}\text{.}\) This also means that we can construct an isosceles right triangle with its legs of the length \(q\) and its hypothenuse of the length \(p\text{.}\) See Figure 1.4. Thus, \(|\overline{AB}|=|\overline{AC}|=q\text{,}\) and \(|\overline{BC}|=p\text{.}\) Pick the point \(D\) on the line segment \(\overline{BC}\) so that \(|\overline{CD}|=q\) and then pick the point \(E\) on the line segment \(\overline{AM}\) so that \(\overline{ED}\perp \overline{BC}\text{.}\) Now observe that \(\triangle BDE\) is an isosceles right triangle such that \(|\overline{DE}|=|\overline{DB}|=p-q\) and \(|\overline{BE}|=\sqrt{2}(p-q)=\frac{p}{q}(p-q)=2q-p\text{.}\)

But this means that whenever we have an isosceles right triangle with all of its sides having integer lengths, then we can make a smaller isosceles right triangle with all of its sides having integer lengths. If we continue doing this, at one point the triangle will be so small that its legs will be of the length less than 1. This is a contradiction because 1 is the smallest positive integer.

Therefore, \(\sqrt{2}\) is an irrational number.

Probably the most celebrated among all numbers is the irrational number \(\pi\text{.}\)

Order of Operations. If necessary, review in which order a sequence of operations has to be performed to correctly evaluate the value of the given expression.

See Example 1.1.2.

Inequalities: If necessary, review the meaning of the symbols \(\lt, \leq, \gt, \geq\text{.}\)

See Example 1.1.3.

Example 1.1.3. Inequality symbols.

Which statement is NOT TRUE: \(2\lt 3\text{;}\) \(-1\gt-2\text{;}\) \(3\geq 2\text{;}\) \(-1\gt 0\lt 2\text{;}\) \(-1\leq 0\text{?}\)

See Figure 1.5.

Solution

\(-1\gt 0\lt 2\text{.}\) Why?

Variable: A variable in mathematics is a symbol or an object that represents the individual elements of a certain set.

See Figure 1.6.

Interval Notation: Throughout the course we will use the interval notation to denote a set of real numbers that contains all real numbers lying between two given numbers or between a given number and infinity, or the set of all real numbers. See Figure 1.6.