1.1: Basic Sets
It has been said 1 that “God invented the integers, all else is the work of Man.” This is a mistranslation. The term “integers” should actually be “whole numbers.” The concepts of zero and negative values seem (to many people) to be unnatural constructs. Indeed, otherwise intelligent people are still known to rail against the concept of a negative quantity – “How can you have negative three apples?” The concept of zero is also somewhat profound.
Probably most people will agree that the natural numbers are a natural construct – they are the numbers we use to count things. Traditionally, the natural numbers are denoted \(\mathbb{N}\).
At this point in time there seems to be no general agreement about the status of zero \(0\) as a natural number. Are there collections that we might possibly count that have no members? Well, yes – I’d invite you to consider the collection of gold bars that I keep in my basement…
The traditional view seems to be that
\[\mathbb{N} = \{1, 2, 3, 4, \ldots \}\]
i.e. that the naturals don’t include \(0\). My personal preference would be to make the other choice (i.e. to include \(0\) in the natural numbers), but for the moment, let’s be traditionalists.
Be advised that this is a choice. We are adopting a convention. If in some other course, or other mathematical setting you find that the other convention is preferred, well, it’s good to learn flexibility…
Perhaps the best way of saying what a set is, is to do as we have above. List all the elements. Of course, if a set has an infinite number of things in it, this is a difficult task – so we satisfy ourselves by listing enough of the elements that the pattern becomes clear.
Taking \(\mathbb{N}\) for granted, what is meant by the “all else” that humankind is responsible for? The basic sets of different types of “numbers” that every mathematics student should know are: \(\mathbb{N}\), \(\mathbb{Z}\), \(\mathbb{Q}\), \(\mathbb{R}\) and \(\mathbb{C}\). Respectively: the naturals, the integers, the rationals, the reals, and the complex numbers. The use of \(\mathbb{N}\), \(\mathbb{R}\) and \(\mathbb{C}\) is probably clear to an English speaker. The integers are denoted with a \(\mathbb{Z}\) because of the German word zählen which means “to count.” The rational numbers are probably denoted using \(\mathbb{Q}\), for “quotients.” Etymology aside, is it possible for us to provide precise descriptions of these remaining sets?
The integers (\(\mathbb{Z}\)) are just the set of natural numbers together with the negatives of naturals and zero. We can use a doubly infinite list to denote this set.
\[\mathbb{Z} = \{ ... -3, -2, -1, 0, 1, 2, 3, ....\}\]
To describe the rational numbers precisely we’ll have to wait until Section 1.6 . In the interim, we can use an intuitively appealing, but somewhat imprecise definition for the set of rationals. A rational number is a fraction built out of integers. This also provides us with a chance to give an example of using the main other way of describing the contents of a set – so-called set-builder notation.
\[\mathbb{Q} = \{ \dfrac{a}{b} | a \in \mathbb{Z} \text{ and } b \in \mathbb{Z} \text{ and } b \neq 0 \} \]
This is a good time to start building a “glossary” – a translation lexicon between the symbols of mathematics and plain language. In the line above we are defining the set \(\mathbb{Q}\) of rational numbers, so the first symbols that appear are “\(\mathbb{Q} =\).” It is interesting to note that the equals sign has two subtly different meanings: assignment and equality testing, in the mathematical sentence above we are making an assignment – that is, we are declaring that from now on the set \(\mathbb{Q}\) will be the set defined on the remainder of the line. 2 Let’s dissect the rest of that line now. There are only \(4\) characters whose meaning may be in doubt, \(\{\), \(\}\), \(∈\) and \(|\). The curly braces (a.k.a. french braces) are almost universally reserved to denote sets, anything appearing between curly braces is meant to define a set. In translating from “math” to English, replace the initial brace with the phrase “the set of all.” The next arcane symbol to appear is the vertical bar. As we will see in Section 1.4.3 this symbol has (at least) two meanings – it will always be clear from context which is meant. In the sentence we are analyzing, it stands for the words “such that.” The last bit of arcana to be deciphered is the symbol \(∈\), it stands for the English word “in” or, more formally, “is an element of.”
Let’s parse the entire mathematical sentence we’ve been discussing with an English translation in parallel.
| \(\mathbb{Q}\) | \(=\) | \(\{\) | \(\dfrac{a}{b}\) |
| The rational numbers | are defined to be | the set of all | fractions of the form \(a\) over \(b\) |
| \(|\) | \(a \in \mathbb{Z}\) | \(\text{and}\) | \(b \in \mathbb{Z}\) |
| such that | \(a\) is an element of the integers | and | \(b\) is an element of the integers |
| \(\text{and}\) | \(b \neq 0 \) | \(\}\) |
| and | \(b\) is nonzero | (the final curly brace is silent) |
It is quite apparent that the mathematical notation represents a huge improvement as regards brevity.
As mentioned previously, this definition is slightly flawed. We will have to wait ’til later to get a truly precise definition of the rationals, but we invite the reader to mull over what’s wrong with this one. Hint: think about the issue of whether a fraction is in lowest terms.
Let’s proceed with our menagerie of sets of numbers. The next set we’ll consider is \(\mathbb{R}\), the set of real numbers. To someone who has completed Calculus, the reals are perhaps the most obvious and natural notion of what is meant by “number.” It may be surprising to learn that the actual definition of what is meant by a real number is extremely difficult. In fact, the first reasonable formulation of a precise definition of the reals came around \(1858\), more than \(180\) years after the development of the Calculus 3 . A precise definition for the set R of real numbers is beyond the scope of this book, for the moment consider the following intuitive description. A real number is a number that measures some physical quantity. For example, if a circle has diameter \(1\) then its circumference is \(π\), thus \(π\) is a real number. The points \((0, 0)\) and \((1, 1)\) in the Cartesian plane have distance \(\sqrt{(0 − 1)^2 + (0 − 1)^2} = \sqrt{2}\), thus \(\sqrt{2}\) is a real number. Any rational number is clearly a real number – slope is a physical quantity, and the line from \((0, 0)\) to \((b, a)\) has slope \(\dfrac{a}{b}\). In ancient Greece, Pythagoras – who has sometimes been described as the first pure Mathematician, believed that every real quantity was in fact rational, a belief that we now know to be false. The numbers \(π\) and \(\sqrt{2}\) mentioned above are not rational numbers. For the moment it is useful to recall a practical method for distinguishing between rational numbers and real quantities that are not rational – consider their decimal expansions. If the reader is unfamiliar with the result to which we are alluding, we urge you to experiment. Use a calculator or (even better) a computer algebra package to find the decimal expansions of various quantities. Try \(π\), \(\sqrt{2}\), \(\dfrac{1}{7}\), \(\dfrac{2}{5}\), \(\dfrac{16}{17}\), \(\dfrac{1}{2}\) and a few other quantities of your own choice. Given that we have already said that the first two of these are not rational, try to determine the pattern. What is it about the decimal expansions that distinguish rational quantities from reals that aren’t rational?
Given that we can’t give a precise definition of a real number at this point it is perhaps surprising that we can define the set \(\mathbb{C}\) of complex numbers with precision (modulo the fact that we define them in terms of \(\mathbb{R}\)).
\[\mathbb{C} = \{a + bi | a ∈ R \text{ and } b ∈ R \text{ and } i^2 = −1\}\]
Translating this bit of mathematics into English we get:
| \(\mathbb{C}\) | \(=\) | \(\{\) | \(a+bi\) |
| The complex numbers | are defined to be | the set of all | expressions of the form \(a\) plus \(b\) times \(i\) |
| \(|\) | \(a \in \mathbb{R}\) | \(\text{and}\) | \(b \in \mathbb{R}\) |
| such that | \(a\) is an element of the reals | and | \(b\) is an element of the reals |
| \(\text{and}\) | \(i^2 = -1\) | \(\}\) |
| and | \(i\) has the property that its square is negative one. |
We sometimes denote a complex number using a single variable (by convention, either late alphabet Roman letters or Greek letters. Suppose that we’ve defined \(z = a + bi\). The single letter \(z\) denotes the entire complex number. We can extract the individual components of this complex number by talking about the real and imaginary parts of \(z\). Specifically, \(\text{Re}(z) = a\) is called the real part of \(z\), and \(\text{Im}(z) = b\) is called the imaginary part of \(z\).
Complex numbers are added and multiplied as if they were binomials (polynomials with just two terms) where \(i\) is treated as if it were the variable – except that we use the algebraic property that \(i\)’s square is \(-1\). For example, to add the complex numbers \(1 + 2i\) and \(3 − 6i\) we just think of the binomials \(1 + 2x\) and \(3 − 6x\). Of course, we normally write a binomial with the term involving the variable coming first, but this is just a convention. The sum of those binomials would be \(4−4x\) and so the sum of the given complex numbers is \(4 − 4i\). This sort of operation is fairly typical and is called component-wise addition. To multiply complex numbers we have to recall how it is that we multiply binomials. This is the well-known FOIL rule (first, outer, inner, last). For example the product of \(3 − 2x\) and \(4 + 3x\) is \((3 · 4) + (3 · 3x) + (−2x · 4)+ (−2x · 3x)\) this expression simplifies to \(12+x−6x^2\). The analogous calculation with complex numbers looks just the same, until we get to the very last stage where, in simplifying, we use the fact that \(i^2 = −1\).
\[\begin{equation} \begin{array} ((3 − 2i) · (4 + 3i) &= (3 · 4) + (3 · 3i) + (−2i · 4) + (−2i · 3i) \\ &=12 + 9i − 8i − 6i^2 \\ &= 12 + i + 6 \\ &= 18 + i \end{array} \end{equation} \]
The real numbers have a natural ordering, and hence, so do the other sets that are contained in \(\mathbb{R}\). The complex numbers can’t really be put into a well-defined order — which should be bigger, \(1\) or \(i\)? But we do have a way to, at least partially, accomplish this task. The modulus of a complex number is a real number that gives the distance from the origin \((0 + 0i)\) of the complex plane, to a given complex number. We indicate the modulus using absolute value bars, and you should note that if a complex number happens to be purely real, the modulus and the usual notion of absolute value coincide. If \(z = a+bi\) is a complex number, then its modulus, \(|| a+bi ||\), is given by the formula \(\sqrt{a^2 + b^2}\).
Several of the sets of numbers we’ve been discussing can be split up based on the so-called trichotomy property : every real number is either positive, negative or zero. In particular, \(\mathbb{Z}\), \(\mathbb{Q}\) and \(\mathbb{R}\) can have modifiers stuck on so that we can discuss (for example) the negative real numbers, or the positive rational numbers or the integers that aren’t negative. To do this, we put superscripts on the set symbols, either a \(+\) or a \(−\) or the word “noneg.”
So
\[\mathbb{Z}^+ = \{ x \in \mathbb{Z} | x > 0 \}\]
and
\[\mathbb{Z}^- = \{ x \in \mathbb{Z} | x < 0 \}\]
and
\[\mathbb{Z}^{\text{noneg}} = \{ x \in \mathbb{Z} | x \geq 0 \}\]
Presumably, we could also use “nonpos” as a superscript to indicate nonpositive integers, but this never seems to come up in practice. Also, you should note that \(\mathbb{Z}^+\) is really the same thing as \(\mathbb{N}\), but that \(\mathbb{Z}^{\text{noneg}}\) is different because it contains \(0\).
We would be remiss in closing this section without discussing the way the sets of numbers we’ve discussed fit together. Simply put, each is contained in the next. N is contained in \(\mathbb{Z}\), \(\mathbb{Z}\) is contained in \(\mathbb{Q}\), \(\mathbb{Q}\) is contained in \(\mathbb{R}\), and \(\mathbb{R}\) is contained in \(\mathbb{C}\). Geometrically the complex numbers are essentially a two-dimensional plane. The real numbers sit inside this plane just as the \(x\)-axis sits inside the usual Cartesian plane – in this context you may hear people talk about “the real line within the complex plane.” It is probably clear how \(\mathbb{N}\) lies within \(\mathbb{Z}\), and every integer is certainly a real number. The intermediate set \(\mathbb{Q}\) (which contains the integers, and is contained by the reals) has probably the most interesting relationship with the set that contains it. Think of the real line as being solid, like a dark pencil stroke. The rationals are like sand that has been sprinkled very evenly over that line. Every point on the line has bits of sand nearby, but not (necessarily) on top of it.
Exercises
Each of the quantities indexing the rows of the following table is in one or more of the sets which index the columns. Place a check mark in a table entry if the quantity is in the set.
| \(\mathbb{N}\) | \(\mathbb{Z}\) | \(\mathbb{Q}\) | \(\mathbb{R}\) | \(\mathbb{C}\) | ||
| \(17\) | ||||||
| \(\pi\) | ||||||
| \(\dfrac{22}{7}\) | ||||||
| \(-6\) | ||||||
| \(e^0\) | ||||||
| \(1+i\) | ||||||
| \(\sqrt{3}\) | ||||||
| \(i^2\) |
Write the set \(\mathbb{Z}\) of integers using a singly infinite listing.
Identify each as rational or irrational.
- \(5021.2121212121 . . .\)
- \(0.2340000000 . . .\)
- \(12.31331133311133331111 . . .\)
- \(π\)
- \(2.987654321987654321987654321 . . .\)
The “see and say” sequence is produced by first writing a \(1\), then iterating the following procedure: look at the previous entry and say how many entries there are of each integer and write down what you just said. The first several terms of the “see and say” sequence are \(1\), \(11\), \(21\), \(1112\), \(3112\), \(211213\), \(312213\), \(212223\), \(. . .\). Comment on the rationality (or irrationality) of the number whose decimal digits are obtained by concatenating the “see and say” sequence.
\(0.1112111123112211213...\)
Give a description of the set of rational numbers whose decimal expansions terminate. (Alternatively, you may think of their decimal expansions ending in an infinitely-long string of zeros.)
Find the first \(20\) decimal places of \(π\), \(\dfrac{3}{7}\), \(\sqrt{2}\), \(\dfrac{2}{5}\), \(\dfrac{16}{17}\), \(\sqrt{3}\), \(\dfrac{1}{2}\) and \(\dfrac{42}{100}\). Classify each of these quantity’s decimal expansion as: terminating, having a repeating pattern, or showing no discernible pattern.
Consider the process of long division. Does this algorithm give any insight as to why rational numbers have terminating or repeating decimal expansions? Explain.
Give an argument as to why the product of two rational numbers is again a rational.
Perform the following computations with complex numbers
- \((4 + 3i) − (3 + 2i)\)
- \((1 + i) + (1 − i)\)
- \((1 + i) · (1 − i)\)
- \((2 − 3i) · (3 − 2i)\)
The conjugate of a complex number is denoted with a superscript star, and is formed by negating the imaginary part. Thus if \(z = 3 + 4i\) then the conjugate of \(z\) is \(z^∗ = 3 − 4i\). Give an argument as to why the product of a complex number and its conjugate is a real quantity. (I.e. the imaginary part of \(z · z^∗\) is necessarily \(0\), no matter what complex number is used for \(z\).)