1.3: Sets of Numbers
Sets of Numbers
-
The
Empty Set
: \( \emptyset=\{\}=\{x \mid x \neq x\}\).
This is the set with no elements, and it plays a vital role in mathematics. -
The
Natural Numbers
: \( \mathbb{N} = \{ 1, 2, 3, \ldots\}\).
The ellipsis (\(\ldots\)) indicate that the obvious pattern continues. That is, the natural numbers contain \(1\), \(2\), \(3\), and so forth. - The Whole Numbers : \( \mathbb{W} = \{ 0, 1, 2, \ldots \}\)
- The Integers : \( \mathbb{Z} =\{ \ldots, -3, -2, -1, 0, 1, 2, 3, \ldots \}\)
-
The
Rational Numbers
: \( \mathbb{Q} = \left\{\frac{a}{b} \mid a \in \mathbb{Z} \text{ and } b \in \mathbb{Z} \right\}\).
Rational numbers are the ratios of integers (provided the denominator is not zero). We can use set-builder notation as another way to describe the rational numbers:
\[\mathbb{Q} = \{ x \mid x \text{ possesses a repeating or terminating decimal representation.}\} \nonumber \] - The Irrational Numbers : \( \mathbb{P} = \{x \mid x \text { does not have a repeating or terminating decimal representation, and } x \text{ does not have an imaginary part}\}\).
-
The
Real Numbers
: \( \mathbb{R} = \mathbb{Q} \cup \mathbb{P} \).
The symbol \( \cup \) is the union of both sets. That is, the set of real numbers is the set comprised of joining the set of rational numbers with the set of irrational numbers. -
The
Complex Numbers
: \( \mathbb{C} = \{ a + b i \mid a, b \in \mathbb{R} \text { and } i = \sqrt{-1}\}\).
\( a \) is called the real part and \( b \) is called the imaginary part of the complex number. Despite their importance, the complex numbers play only a minor role in this text.
Get used to set notations like \( \mathbb{Z} \) and \( \mathbb{R} \) as we will use them extensively throughout this text. It is important to note that every natural number is a whole number, which, in turn, is an integer. Each integer is a rational number (take \(b =1\) in the above definition for \(\mathbb Q\)). The rational numbers are all real numbers since they possess decimal representations. If we take \(b=0\) in the above definition of \(\mathbb C\), we see that every real number is a complex number. In this sense, the sets \(\mathbb N\), \(\mathbb W\), \(\mathbb Z\), \(\mathbb Q\), \(\mathbb R\), and \(\mathbb C\) are nested like Matryoshka dolls (see Figure \( \PageIndex{1} \)).
Figure \( \PageIndex{1} \): A visualization of the number systems as a "container" model.
In Figure \( \PageIndex{1} \), the grey-shaded area of \( \mathbb{R} \) represents the irrational numbers \( \mathbb{P} \) and the orange-shaded area in \( \mathbb{C} \) represents complex numbers with nonzero imaginary parts.
To denote that all elements in set \(A\) also belong to set \(B\), we use either \(A \subseteq B\) or \(A \subset B\). These are read as, "\(A\) is a subset of \(B\)" and "\(A\) is a proper subset of \(B\)," respectively. We use the subset notation, \(\subseteq\), if each element in \(A\) is also in \(B\). If we know that \(A\) is a subset of \(B\), but \(B\) has some elements that are not in \(A\), we switch to the proper subset notation, \(\subset\), to let the reader know that the set \(B\) definitely has more elements than \(A\).
For example, as I am writing this section, I am hosting a final exam for my Linear Algebra students (a class I highly recommend taking one day). If \(S\) is the set of all shoes in the room and \(B\) is the set of all black shoes in the room, \(B \subset S\). It is a proper subset because Enoch (one of my students) is wearing brown shoes. Therefore, there are more elements in \(S\) than in \(B\).
As far as our number systems are concerned,\[ \mathbb{N} \subset \mathbb{W} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C}. \nonumber\]Notice how this matches beautifully with Figure \( \PageIndex{ 1 } \).
We use a line drawn over the repeating block of numbers instead of writing the group multiple times.
Write each of the following as a rational number.
- \(7\)
- \(0\)
- \(-8\)
- Solutions
-
- \(7= \frac{7}{1}\)
- \(0= \frac{0}{1}\)
- \(-8= \frac{-8}{1}\)
Write each of the following as a rational number.
- \(1\)
- \(3\)
- \(-4\)
- Answers
-
- \(\frac{11}{1}\)
- \(\frac{3}{1}\)
- \(-\frac{4}{1}\)
Write each of the following rational numbers as either a terminating or repeating decimal.
- \(-\frac{5}{7}\)
- \(\frac{15}{5}\)
- \(\frac{13}{25}\)
- Solutions
-
- a repeating decimal
- \(\frac{15}{5} = 3\)(or \(3.0\)), a terminating decimal
- \(\frac{13}{25} =0.52\), a terminating decimal
Write each of the following rational numbers as either a terminating or repeating decimal.
- \(\frac{68}{17}\)
- \(\frac{8}{13}\)
- \(-\frac{13}{25}\)
- Answers
-
- \(4\) (or \(4.0\)), terminating
- \(0.\overline{615384}\), repeating
- \(-0.85\), terminating
Determine whether each of the following numbers is rational or irrational. If it is rational, determine whether it is a terminating or repeating decimal.
- \(\sqrt{25}\)
- \(\frac{33}{9}\)
- \(\sqrt{11}\)
- \(\frac{17}{34}\)
- \(0.3033033303333 \ldots \)
- Solutions
-
- \(\sqrt{25}\): This can be simplified as \(\sqrt{25} = 5\). Therefore,\(\sqrt{25}\)is rational.
- \(\frac{33}{9}\): Because it is a fraction,\(\frac{33}{9}\) is a rational number. Next, simplify and divide. \[\frac{33}{9}=\cancel{\frac{33}{9}} \nonumber\] So, \(\frac{33}{9}\) is rational and a repeating decimal.
- \(\sqrt{11}\): This cannot be simplified any further. Therefore, \(\sqrt{11}\) is an irrational number.
- \(\frac{17}{34}\): Because it is a fraction, \(\frac{17}{34}\) is a rational number. Simplify and divide. \[\frac{17}{34} = 0.5 \nonumber\] So, \(\frac{17}{34}\) is rational and a terminating decimal.
- \(0.3033033303333 \ldots \) is not a terminating decimal. Also note that there is no repeating pattern because the group of \(3s\) increases each time. Therefore it is neither a terminating nor a repeating decimal and, hence, not a rational number. It is an irrational number.
Determine whether each of the following numbers is rational or irrational. If it is rational, determine whether it is a terminating or repeating decimal.
- \(\frac{7}{77}\)
- \(\sqrt{81}\)
- \(4.27027002700027 \ldots \)
- \(\frac{91}{13}\)
- \(\sqrt{39}\)
- Answers
-
- rational and repeating;
- rational and terminating;
- irrational;
- rational and terminating;
- irrational
Classify each number as either positive or negative and as either rational or irrational. Does the number lie to the left or the right of \(0\) on the number line?
- \(-\frac{10}{3}\)
- \(\sqrt{5}\)
- \(-\sqrt{289}\)
- \(-6 \pi \)
- \(0.615384615384 \ldots \)
- Solutions
-
- \(-\frac{10}{3}\) is negative and rational. It lies to the left of \(0\) on the number line.
- \(\sqrt{5}\) is positive and irrational. It lies to the right of \(0\).
- \(-\sqrt{289} = -\sqrt{17^2} = -17\) is negative and rational. It lies to the left of \(0\).
- \(-6 \pi \) is negative and irrational. It lies to the left of \(0\).
- \(0.615384615384 \ldots \) is a repeating decimal so it is rational and positive. It lies to the right of \(0\).
Classify each number as either positive or negative and as either rational or irrational. Does the number lie to the left or the right of \(0\) on the number line?
- \(\sqrt{73}\)
- \(-11.411411411 \ldots \)
- \(\frac{47}{19}\)
- \(-\frac{\sqrt{5}}{2}\)
- \(6.210735\)
- Answers
-
- positive, irrational; right
- negative, rational; left
- positive, rational; right
- negative, irrational; left
- positive, rational; right
Classify each number as being a natural number (N), whole number (W), integer (I), rational number (Q), and/or irrational number (Q′).
- \(\sqrt{36}\)
- \(\frac{8}{3}\)
- \(\sqrt{73}\)
- \(-6\)
- \(3.2121121112 \ldots \)
- Solutions
-
N W I Q Q' a. \(\sqrt{36} = 6\) X X X X b. \(\frac{8}{3} =2.\overline{6}\) X c. \(\sqrt{73}\) X d. \(-6\) X X e. \(3.2121121112...\) X
Classify each number as being a natural number (N), whole number (W), integer (I), rational number (Q), and/or irrational number (Q′).
- \(-\frac{35}{7}\)
- \(0\)
- \(\sqrt{169}\)
- \(\sqrt{24}\)
- \(4.763763763...\)
- Answers
-
N W I Q Q' a. \(-\frac{35}{7}\) X X b. \(0\) X X X c. \(\sqrt{169}\) X X X X d. \(\sqrt{24}\) X e. \(4.763763763...\) X