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exposition, language of math, math, understanding math

The real numbers

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My website abstractmath.org contains separate short articles about certain number systems (natural numbers, integers, rationals, reals). The intent of each article is to discuss problems that students have when they begin studying abstract math. The articles do not give complete coverage of each system. They contain links when concepts are mentioned that the reader might not be familiar with.

This post is a revision of the abstractmath.org article on the real numbers. The other articles have also been recently revised.

Introduction

A real number is a number that can be represented as a (possibly infinite) decimal expansion, such as 2.56, -3 (which is -3.0), 1/3 (which has the infinite decimal expansion 0.333…), and $\pi$. Every integer and every rational number is a real number, but numbers such as $\sqrt{2}$ and $\pi$ are real numbers that are not rational.

  • I will not give a mathematical definition of “real number”.  There are several equivalent definitions of real number all of which are quite complicated.   Mathematicians rarely think about real numbers in terms of these definitions; what they have in mind when they work with them are their familiar algebraic and topological properties.
  • “Real number” is a technical term.  Real numbers are not any more “genuine” that any other numbers.
  • Integers and rational numbers are real numbers, but there are real numbers that are not integers or rationals. One such number is$\sqrt{2}$. Such numbers are called irrational numbers.

Properties of the real numbers

Closure

The real numbers are closed under addition, subtraction, and multiplication, as well as division by a nonzero number.

Notice that these are exactly the same arithmetic closure properties that rational numbers have. In the previous sections in this chapter on numbers, each new number system — natural numbers, integers and rational numbers — were closed under more arithmetic operations than the earlier ones. We don’t appear to have gained anything concerning arithmetic operations in going from the rationals to the reals.

The real numbers do allow you to find zeroes of some polynomials that don’t have rational zeroes. For example, the equation $x^2-2=0$ has the root $x=\sqrt{2}$, which is a real number but not a rational number. However, you get only some zeroes of polynomials by going to the reals — consider the equation $x^2+2=0$, which requires going to the complex numbers to get a root.

Closed under limits

The real numbers are closed under another operation (not an algebraic operation) that rational numbers are not closed under:

The real numbers are closed under taking limits.
That fact is the primary reason real numbers are so important
in math, science and engineering.

Consider: The concepts of continuous function, derivative and integral — the basic ideas in calculus and differential equations — are all defined in terms of limits. Those are the basic building blocks of mathematical analysis, which provides most of the mathematical tools used by scientists and engineers.

Some images and metaphors for real numbers

Line segments

The length of any line segment is given by a positive real number.

 

Example

The diagonal of the square above has length $2\sqrt{2}$.

Directed line segments

Measuring directed line segments requires the use of negative real numbers as well as positive ones. You can regard the diagonal above as a directed line segment. If you regard “left to right” as the positive direction (which is what we usually do), then if you measure it from right to left you get $-2\sqrt{2}$.

Real numbers are quantities

Real numbers are used to measure continuous variable quantities.

Examples
  • The temperature at a given place and a given time.
  • The speed of a moving car.
  • The amount of water in a particular jar.

Remarks

  • Temperature, speed, volume of water are thought of as quantities that can change, or be changed, which is why I called them “variable” quantities.
  • The name “continuous” for these quantities indicates that the quantity can change from one value to another without “jumping”. (This is a metaphor, not a mathematical definition!)
Example

If you have $1.334 \text{ cm}^3$ of water in a jar you can add any additional small amount into it or you can withdraw any small amount from it.  The volume does not suddenly jump from $1.334$ to $1.335$ – as you put in the water it goes up gradually from $1.334$ to $1.335$.

Caveat

This explanation of “continuous quantity” is done in terms of how we think about continuous quantities, not in terms of a mathematical definition.  In fact. since you can’t measure an amount smaller than one molecule of water, the volume does jump up in tiny discrete amounts.   Because of quantum phenomena, temperature and speed change in tiny jumps, too (much tinier than molecules). 

Quantum jumps and individual molecules are ignored in large-scale physical applications because the scale at which they occur is so tiny it doesn’t matter.  For such applications, physicists and chemists (and cooks and traffic policemen!) think of the quantities they are measuring as continuous, even though at tiny scales they are not.

The fact that scientists and engineers treat changes of physical quantities as continuous, ignoring the fact that they are not continuous at tiny scales, is sometimes called the “continuum hypothesis”. This is not what mathematicians mean by that phrase: see continuum hypothesis in Wikipedia.

The real line

It is useful to visualize the set of real numbers as the real line.

The real line goes off to infinity in both directions. Each real number represents a location on the real line. Some locations are shown here:

The locations are commonly called points on the real line.  This can lead to a seriously mistaken mental image of the reals as a row of points, like beads.  Just as in the case of the rationals, there is no real number “just to the right” of a given real number. 

Decimal representation of the real numbers

In this section, I will go into more detail about the decimal representation of the real numbers. There are two reasons for doing this.

  • People just beginning abstract math tend to think in terms of bad metaphors about the real numbers as decimals, and I want to introduce ways of thinking about them that are more helpful.
  • The real numbers can be defined in terms of the decimal representation. This is spelled out in a blog post by Tim Gowers. The definition requires some detail and in some ways is inelegant compared to the definitions usually used in analysis textbooks. But it means that the more you understand about the decimal representation, the better you understand real numbers, and in a pretty direct way.

The decimal representation of a real number is also called its decimal expansion.  A representation can be given to other bases besides $10$; more about that here.

Decimal representation as directed length.

The decimal representation of a real number gives the approximate location of the number on the real line as its directed distance from $0$.

Examples
  • The rational number $1/2$ is real and has the decimal representation $0.5$.
  • The rational number $-1/2$ has the representation $-0.5$.
  • The number $1/3$ is also real and has the infinite decimal representation $1.333\ldots$. Thereis an infinite number of $3$’s, or to put it another way, for every
    positive integer $n$, the $n$th decimal place of the decimal representation of $1/3$ is $3$.
  • The number $\pi $ has a decimal representation beginning $3.14159\ldots$. So you can locate $\pi$ approximately by going $3.14$ units to the right from $0$.  You can locate it more exactly by going $3.14159$ units to the right, if you can measure that accurately.  The decimal representation of $\pi$ is infinitely long so you can theoretically represent it with as much accuracy as you wish.  In practice, of course, it would take longer than the age of the universe to find the first ${{10}^{({{10}^{10}})}}$ digits.

Bar notation

It is customary to put a bar over a sequence of digits at the end of a decimal representation to indicate that the sequence is repeated forever. 

Examples
  • $42\frac{1}{3}=42.\overline{3}$
  • $52.71656565\ldots$ (the group $65$ repeating infinitely often) may be written $52.71\overline{65}$.
  • A decimal representation that is only finitely long, for example $5.477$, could also be written $5.477\overline{0}$.
  • In particular, $6=6.0=6.\overline{0}$, and that works for any integer.

Approximations

If you give the first few decimal places of a real number, you are giving an approximation to it.  Mathematicians on the one hand and scientists and engineers on the other tend to treat expressions such as $3.14159$ in two different ways:

  • The mathematician may think of it as a precisely given number, namely $\frac{314159}{100000}$, so in particular it represents a rational number. This number is not $\pi$, although it is close to it.
  • The scientistor engineer will probably treat it as the known part of the decimal representation of a real number. From their point of view, one knows $3.14159$ to six significant figures.
  • Abstractmath.org always takes the mathematician’s point of view.  If I refer to $3.14159$, I mean the rational number $\frac{314159}{100000}$.  I may also refer to $\pi$ as “approximately $3.15159$”.

Integers and reals in computer languages

Computer languages typically treat integers as if they were distinct from real numbers. In particular, many languages have the convention that the expression ‘$2$’ denotes the integer and the expression ‘$2.0$’ denotes the real number.   Mathematicians do not use this convention.  They usually regard the integer $2$ and the real number $2.0$ as the same mathematical object.

Decimal representation and infinite series

The decimal representation of a real number is shorthand for a particular infinite series.  Suppose the part before the decimal place is the integer $n$ and the part after the decimal place is\[{{d}_{1}}{{d}_{2}}{{d}_{3}}…\]where ${{d}_{i}}$ is the digit in the $i$th place.  (For example, for $\pi$, $n=3$, ${{d}_{1}}=1,\,\,\,{{d}_{2}}=4,\,\,\,{{d}_{3}}=1,$ and so forth.)  Then the decimal notation $n.{{d}_{1}}{{d}_{2}}{{d}_{3}}…$ represents the limit of the infinite series\[n+\sum\limits_{i=1}^{\infty }{\frac{{{d}_{i}}}{{{10}^{i}}}}\]

Example

             \[42\frac{1}{3}=42+\sum\limits_{i=1}^{\infty}{\frac{3}{{{10}^{i}}}}\]

The number $42\frac{1}{3}$ is exactly equal to the sum of the infinite series, which is represented by the expression $42.\overline{3}$.

If you stop the series after a finite number of terms, then the number is approximately equal to the resulting sum. For example, $42\frac{1}{3}$ is approximately equal to\[42+\frac{3}{10}+\frac{3}{100}+\frac{3}{1000}\]which is the same as $42.333$.

This inequality gives an estimate of the accuracy of this approximation:\[42.333\lt42\frac{1}{3}\lt42.334\]

How to think about infinite decimal representations

The expression $42.\overline{3}$ must be thought of as including all the $3$’s all at once rather than as gradually extending to the right over an infinite period of time.

In ordinary English, the “…” often indicates continuing through time, as in this example

“They climbed to the top of the ridge, and saw another, higher ridge in the distance, so they walked to that ridge and climbed it, only to see another one still further away…”

But the situation with decimal representations is different:

The decimal representation of $42\frac{1}{3}$ as $42.333\ldots$must be thought of as a complete, infinitely long sequence of decimal digits, every one of which (after the decimal point) is a “$3$” right now.

In the same way, you need to think of the decimal expansion of $\sqrt{2}$ as having all its decimal digits in place at once. Of course, in this case you have to calculate them in order. And note that calculating them is only finding out what they are. They are already there!

The preceding description is about how a mathematican thinks about infinite decimal expansions.  The thinking has some sort of physical representation in your head that allows you to think about to the hundred millionth decimal place of $\sqrt{2}$ or of $\pi$ even if you don’t know what it is. This does not mean that you have an infinite number of slots in your brain, one for each decimal place!  Nor does it mean that the infinite number of decimal places actually exist “somewhere”.  After all, you can think about unicorns and they don’t actually exist somewhere.

Exact definitions

Both the following statements are true:

  • The numbers $1/3$, $\sqrt{2}$and $\pi $ have infinitely long decimal representations, in contrast for example to $\frac{1}{2}$, whose decimal representation is exactly $0.5$.
  • The expressions “$1/3$”, “$\sqrt{2}$” and “$\pi $” exactly determine the numbers $1/3$, $\sqrt{2}$ and $\pi$:

These two statements don’t contradict each other. All three numbers have exact definitions.

  • $1/3$ is exactly the number that gives 1 when multiplied by $3$.
  • $\sqrt{2}$is exactly the unique positive real number whose square is 2.
  • $\pi $ is exactly the ratio of the circumference of a circle to its
    diameter.

The decimal representation of each one to a finite number of places provides an approximate location of that number on the real line On the other hand, the complete decimal representation of each one represents it exactly, although you can’t write it down.

Different decimal representations for the same number

The decimal representations of two different real numbers must be different. However, two different decimal representations can, in certain circumstances, represent the same real number. This happens when the decimal representation ends in an infinite sequence of $9$’s or an infinite sequence of $0$’s.

Examples

  • $0.\overline{9}=1.\overline{0}$. This means that $0.\overline{9}$ is exactly the same number as $1$. It is not just an approximation of $1$
  • $3.4\bar{9}=3.5\overline{0}$. Indeed, $3.4\overline{9}$, $3.5$, $35/10$, and $7/2$ are all different representations of the same number. 

The Wikipedia article “$0.\overline{9}$” is an elaborate discussion of the fact that $0.\overline{9}=1$, a fact that many students find hard to believe. 



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Notation for sets

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This is a revision of the section of abstractmath.org on notation for sets.

Sets of numbers

The following notation for sets of numbers is fairly standard.

Remarks

  • Some authors use $\mathbb{I}$ for $\mathbb{Z}$, but $\mathbb{I}$ is also used for the unit interval.
  • Many authors use $\mathbb{N}$ to denote the nonnegative integers instead
    of the positive ones.
  • To remember $\mathbb{Q}$, think “quotient”.
  • $\mathbb{Z}$ is used because the German word for “integer” is “Zahl”.

Until the 1930’s, Germany was the world center for scientific and mathematical study, and at least until the 1960’s, being able to read scientific German was was required of anyone who wanted a degree in science. A few years ago I was asked to transcribe some hymns from a German hymnbook — not into English, but merely from fraktur (the old German alphabet) into the Roman alphabet. I sometimes feel that I am the last living American to be able to read fraktur easily.

Element notation

The expression “$x\in A$” means that $x$ is an element of the set $A$. The expression “$x\notin A$” means that $x$ is not an element of $A$.

“$x\in A$” is pronounced in any of the following ways:

  • “$x$ is in $S$”.
  • “$x$ is an element of $S$”.
  • “$x$ is a member of $S$”.
  • “$S$ contains $x$”.
  • “$x$ is contained in $S$”.

Remarks

  • Warning: The math English phrase “$A$ contains $B$” can mean either “$B\in A$” or “$B\subseteq A$”.
  • The Greek letter epsilon occurs in two forms in math, namely $\epsilon$ and $\varepsilon$. Neither of them is the symbol for “element of”, which is “$\in$”. Nevertheless, it is not uncommon to see either “$\epsilon$” or “$\varepsilon$” being used to mean “element of”.
Examples
  • $4$ is an element of all the sets $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{R}$, $\mathbb{C}$.
  • $-5\notin \mathbb{N}$ but it is an element of all the others.

List notation

Definition: list notation

A set with a small number of elements may be denoted by listing the elements inside braces (curly brackets). The list must include exactly all of the elements of the set and nothing else.

Example

The set $\{1,\,3,\,\pi \}$ contains the numbers $1$, $3$ and $\pi $ as elements, and no others. So $3\in \{1,3,\pi \}$ but $-3\notin \{1,\,3,\,\pi \}$.

Properties of list notation

List notation shows every element and nothing else

If $a$ occurs in a list notation, then $a$ is in the set the notation defines.  If it does not occur, then it is not in the set.

Be careful

When I say “$a$ occurs” I don’t mean it necessarily occurs using that name. For example, $3\in\{3+5,2+3,1+2\}$.

The order in which the elements are listed is irrelevant

For example, $\{2,5,6\}$ and $\{5,2,6\}$ are the same set.

Repetitions don’t matter

$\{2,5,6\}$, $\{5,2,6\}$, $\{2,2,5,6 \}$ and $\{2,5,5,5,6,6\}$ are all different representations of the same set. That set has exactly three elements, no matter how many numbers you see in the list notation.

Multisets may be written with braces and repeated entries, but then the repetitions mean something.

When elements are sets

When (some of) the elements in list notation are themselves sets (more about that here), care is required.  For example, the numbers $1$ and $2$  are not elements of the set \[S:=\left\{ \left\{ 1,\,2,\,3 \right\},\,\,\left\{ 3,\,4 \right\},\,3,\,4 \right\}\]The elements listed include the set $\{1, 2, 3\}$ among others, but not the number $2$.  The set $S$ contains four elements, two sets and two numbers. 

Another way of saying this is that the element relation is not transitive: The facts that $A\in B$ and $B\in C$ do not imply that $A\in C$. 

Sets are arbitrary

  • Any mathematical object can be the element of a set.
  • The elements of a set do not have to have anything in common.
  • The elements of a set do not have to form a pattern.
Examples
  • $\{1,3,5,6,7,9,11,13,15,17,19\}$ is a set. There is no point in asking, “Why did you put that $6$ in there?” (Sets can be arbitrary.)
  • Let $f$ be the function on the reals for which $f(x)=x^3-2$. Then \[\left\{\pi^3,\mathbb{Q},f,42,\{1,2,7\}\right\}\] is a set. Sets do not have to be homogeneous in any sense.


Setbuilder notation

Definition:

Suppose $P$ is an assertion. Then the expression “$\left\{x|P(x) \right\}$” denotes the set of all objects $x$ for which $P(x)$ is true. It contains no other elements.

  • The notation “$\left\{ x|P(x) \right\}$” is called setbuilder notation.

  • The assertion $P$ is called the defining condition for the set.

  • The set $\left\{ x|P(x) \right\}$ is called the truth set of the assertion $P$.
Examples

In these examples, $n$ is an integer variable and $x$ is a real variable..

  • The expression “$\{n| 1\lt n\lt 6 \}$” denotes the set $\{2, 3, 4, 5\}$. The defining condition is “$1\lt n\lt 6$”.  The set $\{2, 3, 4, 5\}$ is the truth set of the assertion “n is an integer and $1\lt n\lt 6$”.
  • The notation $\left\{x|{{x}^{2}}-4=0 \right\}$ denotes the set $\{2,-2\}$.
  • $\left\{ x|x+1=x \right\}$ denotes the empty set.
  • $\left\{ x|x+0=x \right\}=\mathbb{R}$.
  • $\left\{ x|x\gt6 \right\}$ is the infinite set of all real numbers bigger than $6$.  For example, $6\notin \left\{ x|x\gt6 \right\}$ and $17\pi \in \left\{ x|x\gt6 \right\}$.
  • The set $\mathbb{I}$ defined by $\mathbb{I}=\left\{ x|0\le x\le 1 \right\}$ has among its elements $0$, $1/4$, $\pi /4$, $1$, and an infinite number of
    other numbers. $\mathbb{I}$ is fairly standard notation for this set – it is called the unit interval.

Usage and terminology

  • A colon may be used instead of “|”. So $\{x|x\gt6\}$ could be written $\{x:x\gt6\}$.
  • Logicians and some mathematicians called the truth set of $P$ the extension of $P$. This is not connected with the usual English meaning of “extension” as an add-on.
  • When the assertion $P$ is an equation, the truth set of $P$ is usually called the solution set of $P$. So $\{2,-2\}$ is the solution set of $x^2=4$.
  • The expression “$\{n|1\lt n\lt6\}$” is commonly pronounced as “The set of integers such that $1\lt n$ and $n\lt6$.” This means exactly the set $\{2,3,4,5\}$. Students whose native language is not English sometimes assume that a set such as $\{2,4,5\}$ fits the description.

Setbuilder notation is tricky

Looking different doesn’t mean they are different.

A set can be expressed in many different ways in setbuilder notation. For example, $\left\{ x|x\gt6 \right\}=\left\{ x|x\ge 6\text{ and }x\ne 6 \right\}$. Those two expressions denote exactly the same set. (But $\left\{x|x^2\gt36 \right\}$ is a different set.)

Russell’s Paradox

In certain areas of math research, setbuilder notation can go seriously wrong. See Russell’s Paradox if you are curious.

Variations on setbuilder notation

An expression may be used left of the vertical line in setbuilder notation, instead of a single variable.

Giving the type of the variable

You can use an expression on the left side of setbuilder notation to indicate the type of the variable.

Example

The unit interval $I$ could be defined as \[\mathbb{I}=\left\{x\in \mathrm{R}\,|\,0\le x\le 1 \right\}\]making it clear that it is a set of real numbers rather than, say rational numbers.  You can always get rid of the type expression to the left of the vertical line by complicating the defining condition, like this:\[\mathbb{I}=\left\{ x|x\in \mathrm{R}\text{ and }0\le x\le 1 \right\}\]

Other expressions on the left side

Other kinds of expressions occur before the vertical line in setbuilder notation as well.

Example

The set\[\left\{ {{n}^{2}}\,|\,n\in \mathbb{Z} \right\}\]consists of all the squares of integers; in other words its elements are 0,1,4,9,16,….  This definition could be rewritten as $\left\{m|\text{ there is an }n\in \mathrm{}\text{ such that }m={{n}^{2}} \right\}$.

Example

Let $A=\left\{1,3,6 \right\}$.  Then $\left\{ n-2\,|\,n\in A\right\}=\left\{ -1,1,4 \right\}$.

Warning

Be careful when you read such expressions.

Example

The integer $9$ is an element of the set \[\left\{{{n}^{2}}\,|\,n\in \text{ Z and }n\ne 3 \right\}\]It is true that $9={{3}^{2}}$ and that $3$ is excluded by the defining condition, but it is also true that $9={{(-3)}^{2}}$ and $-3$ is not an integer ruled out by the defining condition.

Reference

Sets. Previous post.

Acknowledgments

Toby Bartels for corrections.

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