Tag Archives: integer

The most confusing notation in number theory

This is an observation in abstractmath that I think needs to be publicized more:

Two symbols used in the study of integers are notorious for their confusing similarity.

  • The expression “$m/n$” is a term denoting the number obtained by dividing $m$ by $n$. Thus “$12/3$” denotes $4$ and “$12/5$” denotes the number $2.4$.
  • The expression “$m|n$” is the assertion that “$m$ divides $n$ with no remainder”. So for example “$3|12$”, read “$3$ divides $12$” or “$12$ is a multiple of $3$”, is a true statement and “$5|12$” is a false statement.

Notice that $m/n$ is an integer if and only if $n|m$. Not only is $m/n$ a number and $n|m$ a statement, but the statement “the first one is an integer if and only if the second one is true” is correct only after the $m$ and $n$ are switched!

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The real numbers

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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Modules for mathematical objects

Notes on viewing.

A recent article in Scientific American mentions discusses the idea that concepts are represented in the brain by clumps of neurons.  Other neuroscientists have proposed that each concept is distributed among millions of neurons, or that each concept corresponds to one neuron.  

I have written many posts about the idea that:  

  • Each mathematical concept is embodied in some kind of module in the brain.
  • This idea is a useful metaphor for understanding how we think about mathematical objects.
  • You don't have to know the details of the method of storage for this metaphor to be useful.  
  • The metaphor clears up a number of paradoxes and conundrums that have agitated philosophers of math.

The SA article inspired me to write about just how such a module may work in some specific cases.  

Integers

Mathematicians normally thinks of a particular integer, say $42$, as some kind of abstract object, and the decimal representation "42" as a representation of the integer, along with XLII and 2A$_{16}$.  You can visualize the physical process like this: 

  • The mathematician has a module Int (clump of neurons or whatever) that represents integers, and a module FT that represents the particular integer $42$. 
  • There is some kind of asymmetric three-way connection from FT to Int and a module EO (for "element of" or "IS_A"). 
  • That the connection is "asymmetric" means that the three modules play different roles in the connection, meaning something like "$42$ IS_A Integer"
  • The connection is a physical connection, not a sentence, and when  FT is alerted ("fired"?), Int and EO are both alerted as well. 
  • That means that if someone asks the mathematician, "Is $42$ an integer?", they answer immediately without having to think about it — it is a random access concept like (for many people) knowing that September has 30 days.
  • The module for $42$ has many other connections to other modules in the brain, and these connections vary among mathematicians.

The preceding description gives no details about how the modules and interconnections are physically processed.  Neuroscientists probably would have lots of ideas about this (with no doubt considerable variation) and would criticize what I wrote as misrepresenting the physical details in some ways.  But the physical details are their job, not mine.  What I claim is that this way of thinking makes it plausible to view abstract objects and their properties and relationships as physical objects in the brain.  You don't have to know the details any more than you have to know the details of how a rainbow works to see it (but you know that a rainbow is a physical phenomenon).

This way of thinking provides a metaphor for thinking about math objects, a metaphor that is plausibly related to what happens in the real world.

Students

A student may have a rather different representation of $42$ in the brain.  For one thing, their module for $42$ may not distinguish the symbol "42" from the number $42$, which is an abstract object.   As a result they ask questions such as, "Is $42$ composite in hexadecimal?"  This phenomenon reveals a complicated situation. 

  • People think they are talking about the same thing when in fact their internal modules for that thing may be very differently connected to other concepts in their brain.
  • Mathematicians generally share many more similarities in their modules for $42$ than people in general do.  When they differ, the differences may be of the sort that one of them is a number theorist, so knows more about $42$ (for example, that it is a Catalan number) than another mathematician does.  Or has read The Hitchhiker's Guide to the Galaxy.
  • Mathematicians also share a stance that there are right and wrong beliefs about mathematical objects, and that there is a received method for distinguishing correct from erroneous statements about a particular kind of object. (I am not saying the method always gives an answer!).
  • Of course, this stance constitutes a module in the brain. 
  • Some philosophers of education believe that this stance is erroneous, that the truth or falsity of statements are merely a matter of social acceptance.
  • In fact, the statements in purple are true of nearly all mathematicians.  
  • The fact that the truth or falsity of statements is merely a matter of social acceptance is also true, but the word "merely" is misleading.
  • The fact is that overwhelming evidence provided by experience shows that the "received method" (proof) for determining the truth of math statements works well and can be depended on. Teachers need to convince their students of this by examples rather that imposing the received method as an authority figure.

Real numbers

A mathematician thinks of a real number as having a decimal representation.

  • The representation is an infinitely long list of decimal digits, together with a location for the decimal point. (Ignoring conventions about infinite strings of zeroes.)
  • There is a metaphor that you can go along the list from left to right and when you do you get a better approximation of the "value" of the real number. (The "value" is typically thought of in terms of the metaphor of a point on the real line.)
  • Mathematicians nevertheless think of the entries in the decimal expansion of a real number as already in existence, even though you may not be able to say what they all are.
  • There is no contradiction between the points of view expressed in the last two bullets.
  • Students frequently do not believe that the decimal entries are "already there".  As a result they may argue fiercely that $.999\ldots$ cannot possibly be the same number as $1$.  (The Wikipedia article on this topic has to be one of the most thoroughly reworked math articles in the encyclopedia.)

All these facts correspond to modules in mathematicians' and students' brains.  There are modules for real number, metaphor, infinite list, decimal digit, decimal expansion, and so on.  This does not mean that the module has a separate link to each one of the digits in the decimal expansion.  The idea that there is an entry at every one of the infinite number of locations is itself a module, and no one has ever discovered a contradiction resulting from holding that belief.

References

  • Brain cells for Grandmother, by Rodrigo Quian Quiroga, Itzhak Fried and Christof Koch.  Scientific American, February 2013, pages 31ff.

Gyre&Gimble posts on modules

Notes on Viewing  

This post uses MathJax. If you see mathematical expressions with dollar signs around them, or badly formatted formulas, try refreshing the screen. Sometimes you have to do it two or three times.

 

 

 
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Whole numbers

Sue Van Hattum wrote in response to a recent post:

I’d like to know what you think of my ‘abuse of terminology’. I teach at a community college, and I sometimes use incorrect terms (and tell the students I’m doing so), because they feel more aligned with common sense.

To me, and to most students, the phrase “whole numbers” sounds like it refers to anything that doesn’t need fractions to represent it, and should include negative numbers. (It then, of course, would mean the same thing that the word integers does.) So I try to avoid the phrase, mostly. But I sometimes say we’ll use it with the common sense meaning, not the official math meaning.

Her comments brought up a couple of things I want to blather about.

Official meaning

There is no such thing as an "official math meaning".  Mathematical notation has no governing authority and research mathematicians are too ornery to go along with one anyway.  There is a good reason for that attitude:  Mathematical research constantly causes us to rethink the relationship among different mathematical ideas, which can make us want to use names that show our new view of the ideas.  An excellent example of that is the evolution of the concept of "function" over the past 150 years, traced in the Wikipedia article.

What some "authorities" say about "whole number":

  • MathWorld  says that "whole number" is used to mean any of these:  Any positive integer, any nonnegative integer or any integer.
  • Wikipedia also allows all three meanings.
  • Webster's New World dictionary (of which I have been a consultant, but they didn't ask me about whole numbers!) gives "any integer" as a second meaning.
  • American Heritage Dictionary give "any integer" as the only meaning.
  • Someone stole my copy of Merriam Webster.

Common Sense Meaning

Mathematicians think about and talk any particular kind of math object using images and metaphors.  Sometimes (not very often) the name they give to a math object embodies a metaphor.  Examples:

  • A complex number is usually notated using two real parameters, so it looks more complicated than a real number.
  • "Rings" were originally called that because the first examples were integers (mod n) for some positive integer, and you can think of them as going around a clock showing n hours.

Unfortunately, much of the time the name of a kind of object contains a suggestive metaphor that is bad,  meaning that it suggests an erroneous picture or idea of what the object is like.

  • A "group" ought to be a bunch of things.  In other words, the word ought to mean "set".
  • The word "line" suggests that it ought to be a row of points.  That suggests that each point on a line ought to have one next to it.  But that's not true on the "real line"!

Sue's idea that the "common sense" meaning of "whole number" is "integer" refers, I think, to the built-in metaphor of the phrase "whole number" (unbroken number).

I urge math teachers to do these things:

  • Explain to your students that the same math word or phrase can mean different things in different books.
  • Convince your  students to avoid being fooled by the common-sense (metaphorical meaning) of a mathematical phrase.

 

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