Showing posts with label Articles. Show all posts
Showing posts with label Articles. Show all posts

Sunday, March 24, 2019

Prime number conjectures from the Shapiro class structure

The first of hopefully many joint projects with my childhood friend Hartosh Singh Bal. For many reasons this has been a most exciting collaboration. For one thing, Hartosh and I have been discussing mathematical ideas since Class 11 in Modern School. So it was good to work on something which will lead to something new. For another, Shapiro was Hartosh's number theory Professor at NYU. And for three more reasons, you will have to look at the last section of this paper.

Here is the abstract:
The height $H(n)$ of  $n$, introduced by  Pillai in 1929,  is the  smallest positive integer $i$ such that the  $i$th iterate of Euler's totient function at $n$ is $1$.  H.  N. Shapiro (1943) studied the structure of the set of all numbers at a height. We provide a formula for the height function thereby extending a result of Shapiro. We list steps to generate numbers of any height which turns out to be a useful way to think of this construct. In particular, we extend some results of Shapiro regarding the largest odd numbers at a height. We present some theoretical and computational evidence to show that $H$ and its relatives are closely related to the important functions of number theory, namely $\pi(n)$ and the $n$th prime $p_n$. We conjecture formulas for $\pi(n)$ and $p_n$ in terms of the height function.  
Here is a link to a reprint of the paper.

Prime number conjectures from the Shapiro class structure (with Hartosh Singh Bal), 
UPDATE (Feb 14, 2020): The paper has appeared in INTEGERS: Electronic Journal of Combinatorial Number Theory (Volume 20), #A11, 23pp.


From left to right: Sonit, Hartosh, me, Punya in 1983 or so


Thursday, June 15, 2017

How to discover the exponential function

Another article on the "How to discover/guess/prove/..." series written for a high school audience. The basic idea is to find a function whose derivative is itself, and to find the power series which satisfies this. Then messing with it to guess it must be the exponential function. No proofs, in fact, it is outrageously un-rigourous.  I hope the editor allows it.

I try to include only the most beautiful items, and state facts which I feel every high school student should know, even if they doesn't appear formally in the syllabus. 

Update (Nov 2017). The article was published in the November issue of At Right Angles. A nice surprise was Shailesh Shirali's companion article which gives some graphical intuition to complement the algebraic computations in my article. Here is the link to a reprint

Abstract

If a function is such that its derivative is the function itself, then what would it be? Some interesting mathematical objects  appear while trying to answer this question, including a power series, the irrational number $e$ and the exponential function $e^x$. The article ends with a beautiful formula that  connects $e$, $\pi$, the complex number $i=\sqrt{-1}$, $1$ and $0$.

Update: 15/June/2017. I was wondering what happened to this article, and the editor said he had sent some comments from the referee which were yet to be incorporated. I resent the article after incorporating the referee's comments, and now this article is slated to appear in the November issue of At Right Angles. Time to think about the next article in the series.

Here is a link to the updated preprint. Please do give comments.

Sunday, February 14, 2016

Analogues of a Fibonacci-Lucas Identity

Recently, in 2014, Sury published a Fibonacci-Lucas  identity in the Monthly. It turned out that the identity had appeared earlier (as Identity 236 in Benjamin and Quinn's book: Proofs that count: The art of combinatorial proof). When I tried to prove it using my usual telescoping method, I found its connection with one of the oldest Fibonacci identities due to Lucas in 1876. I also found many generalizations and analogous identities for other Fibonacci type sequences and polynomials. This  small paper has been accepted in the Fibonacci Quarterly.

Here is a link to a preprint: Analogues of a Fibonacci-Lucas Identity

Update: Its has appeared. The ref is: Analogues of a Fibonacci-Lucas identity, Fibonacci Quart., 54 (no. 2) 166-171,  (2016)

I use the approach of my earlier paper on Telescoping: In praise of an elementary identity of Euler.

I am pleased, because I have thought of getting a paper in the Fibonacci Quarterly since I was in high school, and feel lucky I found something they found acceptable!

Tuesday, May 26, 2015

How to Discover the Rogers-Ramanujan Identities

Dec 22, 2012: It is Ramanujan's 125th birthday, but how many of his famous identities do you know? Here we examine a method to conjecture two very famous identities that were conjectured by Ramanujan, and later found to be known to Rogers.

Here is a link:  How to Discover the Rogers-Ramanujan Identities.

This was presented to a some high school math teachers in a conference. I tried to write it in a way that it could be understood by a motivated high school student.

Update (May 26, 2015): The article has been published. Here is a reference. Resonance, 20 (no. 5), 416-430, (May 2015).

Update (January 18, 2014): This article has been accepted for publication in Resonance, a popular science magazine aimed at the undergraduate level.


Tuesday, March 31, 2015

Of Art and Math: A series of articles with Punya Mishra


Right Angle: One of the many ambigrams made by Punya Mishra that appear in this series of articles appearing in "At Right Angles". All ambigrams are copyright Punya Mishra and cannot be used without permission. 

Punya and I are writing a series of articles on the subject of ambigrams. All the ambigrams are made by Punya. For this series, he has been making many new ambigrams, which communicate mathematical ideas. Already, in the space of working on a few articles, it looks like he has made the largest number of mathematical ambigrams.

Here is a longer blog entry from Punya's blog, about this series of articles. His blog has further links to his amazing ambigrams.

Updates

Dec, 2015. I presented Punya's and my work in TIME 2015, in Baramati, Maharashtra in my talk: On Punya Mishra's Mathematical Ambigrams. This was the seventh edition of TIME, which stands for 'Technology and innovation in Math education'.

July, 2015. The fifth article is Part 2 of 2 on the subject of paradoxes. It covers self-reference, Russell's Paradox and visual paradoxes. This article includes a 'new paradox', a version of Jourdain's card paradox by Punya. 

Mar, 2015. The fourth article is on Paradoxes. It is part 1 of 2 articles on this topic. Here we consider what TRUE and FALSE mean in the context of mathematics. Its an introduction to math philosophy. Again, it has many interesting ambigrams.

Feb 2015. The Michigan State Museum has launched an exhibit entitled "Deep Play: Creativity in Math and Art through Visual Wordplay." Check out: the exhibitions web-page

July 2014. The third article on Self-similarity. This one has some amazing ambigrams, and a graphic of the binary pascal's triangle I made many years ago.

Mar, 2014. The second article is on Introducing Symmetry. I think Punya outdid himself in some of the ambigrams here. The ambigram for sin (which is periodic, a sin wave, an odd function) and inverse (modeled on a hyperbola) and exp-log were my favorites. But this month's  puzzle ambigram is mind-blowing too.

Nov, 2013. The first article has come out. It is: Introducing Ambigrams. There is a hidden message in the article. See if you can find it.

Monday, March 31, 2014

How to discover 22/7 and other rational approximations to $\pi$


A short article written for a magazine that caters to high-school students. The basic idea is to use continued fractions found using Euclid's algorithm, and then to chop off the continued fraction to get rational approximations. Written at a high-school level. Some of the material was already present in my book Maths Concepts.

Update: March 2014. This article is published in "At Right Angles." Here is a link to the published article.

Friday, September 16, 2011

How to Guess the Binomial Theorem for any index


Download PDF



Newton extended the Binomial Theorem to the case where the index is no longer a non-negative integer. Newton did not provide a proof of the general case, where the index is a real number. We too will not provide a proof, but will motivate Newton's Binomial Theorem by showing some of the clues that lead to the statement of the general case.


We wish to generalize the identity
$$(1+x)^n=\sum_{k=0}^n {n\choose k} x^k$$
by replacing $n$ by a real number $a$. On the LHS, there is no problem, since the product $(1+x)^a$ makes sense for $a$ a real number. But on the RHS, there are two problems:


  1. The Binomial Coefficient ${n\choose k}$ is defined only when $n$ is a non-negative integer.
  2. The index of summation goes from $0$ to $n$, and thus $n$ has to be a non-negative integer.

The problems are easily solved. Note that ${n\choose k}$ may be written as
\begin{equation}\label{achoosek}
\frac{n(n-1)\cdots (n-k+1)}{k!},
\end{equation}
and \eqref{achoosek} makes sense if we replace $n$ by $a$.
Further, note that when $k>n$, then \eqref{achoosek} reduces to $0$. So we may as well write the Binomial Theorem as
$$(1+x)^n=\sum_{k=0}^{\infty} \frac{n(n-1)\cdots (n-k+1)}{k!} x^k.$$
Since all the terms of this series where $k$ is bigger than $n$ reduce to $0$, the series reduces to the finite sum of the familiar Binomial Theorem for non-negative integral index.

However, if we replace $n$ by a real number $a$, we may have to deal with an infinite series, and we need conditions for it to converge. It turns out the series converges whenever $|x|<1$. So finally, we are ready to state the Binomial Theorem for real index.
\begin{align}
(1+x)^a&=&\sum_{k=0}^{\infty} \frac{a(a-1)\cdots (a-k+1)}{k!} x^k, \text{ for $|x|<1$}\label{binseries} \\
&=& 1+ax+\frac{a(a-1)}{2!}x^2+\frac{a(a-1)(a-2)}{3!}x^3+\cdots\notag
\end{align}

The conditions we need on \( x\) are motivated by an example of the Binomial Theorem for real index that we have already seen. Recall the formula
$$\sum_{k=0}^\infty {x^k} = \frac{1}{1-x}, \text{ for $|x|<1$. }$$
for the sum of the geometric series with first term $1$ and common ratio $x$. This formula is a special case of \eqref{binseries}, where $a=-1$.

Friday, March 14, 2003

Experience Mathematics #29 -- Abstraction


The first time you encountered abstraction in mathematics was when you associated the number (say $5$) with five oranges and five apples. When we begin to learn mathematics, we associate numbers with specific objects. Soon we realize that we can think of the number $5$ as a concept removed from the apples and oranges. This is abstraction. Now we can apply the concept of $5$ (and also other numbers) in counting any set of objects.

Mathematics students become used to abstracting concepts into symbols that we can apply in many situations. The same skill in abstract thought helps in other domains also. For example, object oriented programming—the most useful of the programming paradigms is all about abstraction.

In Object-oriented programming, we think of everything as an object. For example, the button you press in most applications is an object. Now a button looks and behaves in much the same way in any application. So we would naturally wish to program it just once. So most language environments (like Java or Visual Basic) give us a “Class” that represents the object that is the button. (A Class is like the set we encounter in mathematics.)

However, when we use the button in a particular program, we may wish to add certain properties of our own. For example, we may like to put the word “OK” as a label on the button we wish to use. So we “instantiate” a button and set its properties that include a label “OK” that will appear on it. Further, when a user clicks on the button, the button performs an “Action”. You have to code this “Event” to tell the button what to do.

The Classes contain data (or “properties”) that are used to describe a particular member (or “instance”) of the class, functions (or “methods”) that do certain tasks, and have the ability to process messages (or “events”) that the rest of the application uses to tell the class to perform its tasks.

Friday, February 28, 2003

Experience Mathematics #28 -- How fast do functions grow?

In the last column, we talked about the explosive growth of the exponential function. The number of computers infected by the SQL Slammer worm increased dramatically, bringing the Internet crashing down in a couple of hours.

Computer scientists measure the speed of computer algorithms by comparing them to functions.
Some of the functions they use are logarithmic: $\log(n)$, linear: $n$; the power functions: $n^2, n^3, n^4,\dots $; and the exponential functions: $2^n, 10^n,$ etc.

Usually $n$ is the size of the input. Computer scientists make statements such as: The “order” of an algorithm is $n^2$. For example, if you have to sort $n$ numbers, the algorithm is of order $n^2$. This means that the computer has to make approximately $n^2$ calculations. To get an idea of which algorithms are faster, consider when $n=1000$. $\log(n)$ is just $3$. The linear function $n$ is also manageable, at $1000$. However, $n^2$ is $1,000,000$ (one million) and $n^3$, is one billion. And $10^n$ is a huge number, 1 followed by a thousand zeros. This number of calculations is more than what Deep Junior had to perform to defeat Kasparov in Chess.

All these functions go to infinity as $n$ goes to infinity. That is to say, they become bigger and bigger as $n$ becomes bigger. What matters (to computer scientists) is how fast or slow this increase is. The slower the increase, the faster the algorithm.

Logarithmic, linear and Polynomial time algorithms the only algorithms that are fast enough to work in practical situations.

Computer scientists are continuously finding faster and faster algorithms. Recently, Maninder Agarwal, Neeraj Kayal, and Nitin Saxena, of IIT, Kanpur, found a deterministic polynomial-time algorithm to determine whether a number is a prime number.

This solved a problem that mathematicians have been trying to solve for centuries.

Friday, February 07, 2003

Experience Mathematics # 27 -- The exponential function



“It only took 10 minutes for the SQL Slammer worm to race across the globe and wreak havoc on the Internet two weeks ago” is what newspapers reported on February 7, 2003. The number of computers doubled every 8.5 seconds in the first minute of the worm’s existence. So how many computers got infected in the first minute?

The number of computers infected in $t$ seconds can be modelled by the function $N(t)$, where
$$N(t)=2^{t/8.5}.$$

This is a reasonable model. In the beginning (when $t=0$) we assume that the worm has infected only $1$ computer, and indeed, $N(0)=1$. In 8.5 seconds, the number becomes $N(8.5)=2$, so it doubles. In another $8.5$ seconds, $N(17)=4$. This doesn’t sound like very fast growth, but at the end of one minute ($t=60$) the number becomes more than $133$. In two minutes, $28995$ computers are infected, and in $5$ minutes, the number of computers infected is in the billions. Which means that the rate of growth must have slowed down, because there aren’t so many computers in the world! I hope this helps you understand why it caused the slowdown of the Internet traffic in Korea.

$N(t)$ is an example of an exponential function, which increases very fast. The prototypical example is THE exponential function, $f(x)=e^x$. Here the number e is an irrational number (just like the famous $\pi$, whose value is approximately $2.7182818\dots$. The graph of the function (made using desmos.com) is as follows.


The exponential function is not symmetric across the $y$-axis, nor across the origin. That is to say, it is not an even function or an odd function. However, consider the functions:
$$E(x)=(e^x+e^{-x})/2,$$ and $$O(x)=(e^x-e^{-x})/2.$$ $E$ is an even function, and $O$ is an odd function, and the exponential function is the sum of these functions. Draw the function $E(x)$ from $x=-5$ to $x=5$ using MS-Excel (update: try www.desmos.com), and see what the graph looks like. Does it look like a clothesline secured at its two ends?

Friday, January 31, 2003

Experience Mathematics # 26 -- Symmetries


There are two kinds of symmetries in a function. A function may be symmetric across the $y$-axis, or symmetric across the origin. (If a curve is symmetric across the $x$-axis, it is not a function. Can you tell why?)

For example, the function $f(x)= x^2$ is an example of a function that is symmetric across the $y$-axis.


 This symmetry is obvious from the graph. An algebraic way to see that the function $f(x)= x^2$ is symmetric across the $y$-axis, is to replace $x$ by $–x$ in the formula, and note that:
$f(–x) =f(x)$ (since $(–x)^2=x^2$).
For example, the $y$-coordinate corresponding to the point $–2$ is the same as that corresponding to $2$.
The function $f(x)= x^3$ is an example of a function that is symmetric across the origin.


Each point (for example the point $(2, 8)$) maps to a symmetric point (the point $(-2, -8)$) in the graph. An algebraic way to notice that this function is symmetric across the origin is to note that
$f(–x) =–f(x)$ (because $(–x)^3= –x^3$).

Functions symmetric across the $y$-axis are called even functions, and functions symmetric across the origin are called odd functions.

What is remarkable is that any function defined on the set of real numbers can be written as a sum of an odd and an even function. Can you figure out a way to write the exponential function $f(x)=e^x$ as the sum of an even and an odd function? The curve formed by a hanging clothesline  appears in the answer to this question.

Thursday, January 23, 2003

Experience Mathematics # 25 - Functions


A ball thrown in the air follows the path of a parabola. Parabolas are modelled by a function of the form $p(x)=ax^2+bx+c$, where $a$, $b$ and $c$ are real numbers. This kind of function—a polynomial of degree 2—is called a Quadratic Function. While we will not formally define functions, it is helpful to get an intuitive idea of functions from several points of view.

One point of view is to think of functions as a rule. For example, consider the quadratic function:
$f(x)=1-x^2$. Every real number $a$ corresponds to a unique real number denoted by $f(a)$ obtained by replacing $x$ by $a$ in the above equation. For example,
$f(0)=1, f(1)=0, f(-2)=-3.$

This suggests that we can also think of a function as an input-output machine. For each input $a$ we have a unique output $f(a)$. The set of possible input values (in this case the set $R$ of real numbers) is called the domain of the function.

Imagine making a table of all the input-output values of the function. (There are an infinite number of elements in the domain, so you can only imagine making a table!) All these values can be plotted on the coordinate plane. The input values are the $x$-coordinates and the output values are the $y$-coordinates.

If we do this, we will get a graph of the function. We denote the graph by $y=f(x)$, (or $y= 1-x^2$).

This is the third way of thinking about a function: as a graph. The graph is shown below.


Note that this parabola is symmetric about the $y$-axis. It meets the $x$-axis when $x=1$ and when $x=-1.$ These are (graphically speaking) the solutions of the equation $1-x^2=0$. The function has a maximum when $x=0$, corresponding to the highest point a ball reaches, when it is thrown in the air.

Friday, January 10, 2003

Experience Mathematics # 24 -- The Calculus


Happy New Year. The earth has finished another revolution around the sun, taking a little more than 365 days to do so. Meanwhile, the moon continues to rotate around the earth, the planets around the sun, and the same forces that make these things move in an elliptical path ensure that a ball thrown up in the air always falls down, or that a ball thrown in the air (towards a friend) takes a parabolic path.

Over this and the next few columns, I will discuss these natural motivations that are behind the notions that you encounter as you study the Calculus.

The first concept is that of a function. Mathematicians were already familiar with curves from Euclidean and coordinate geometry by 1600 or so A.D. It was natural to begin modelling various physical phenomena with functions. For example, $y=1-x^2$ models the parabola. For each value of the input $x$, we get a unique output $y$. If you plot the curve in the coordinate plane, you obtain a parabola.

It was natural to do two things. To figure out laws that can explain why a ball thrown in the air follows a path traced by such a curve. This led to the laws of Gravitation. And the other thing is to use these laws to predict the answers to common questions that arise. For example: How high will the ball go? How far will the ball go? Given the curve, when does the curve go up (increase)? And when does it come down (decrease)? We will consider such questions and relate them to what you encounter in Calculus.

Curves such as the circle ($x^2+y^2=1$) are not functions since there is not one output $y$ for each input $x$. For example, for $x=0, y$ can be $1$ or $–1$. So, every curve does not give rise to a function.

Friday, December 20, 2002

Experience Mathematics #23 -- Ambigrams (by Punya Mishra)

Symmetry is important in mathematics and in art. Today we will look at a special kind of wordplay based on ideas of symmetry and figure and ground. Consider the word below: 



Can you read it? Now turn the page you are holding upside down and try reading it that way. The word stays the same. This image/word has rotational symmetry—essentially, it stays the same when rotated 1800.

Such visual wordplays are called ambigrams. The word “ambigram” was first coined by the cognitive scientist Douglas Hofstadter. Here is an ambigram of the word ambigram itself.


Ambigrams can be of many different kinds. For instance consider the word “logical” below.



This word has reflection symmetry i.e. it will read the same even when reflected in a mirror.

Some ambigrams are not about symmetry as much as they are about reading words in multiple ways. Here is one titled “good-evil” Can you see both words? Look carefully. This is similar to figure-ground paintings by M. C. Escher.



Creating ambigrams is great fun. Why don’t you try creating some yourself? If you want to see more examples of such wordplay you can search on Google or go to my wordplay gallery: http://punya.educ.msu.edu


This guest column has been written by Professor Punya Mishra, College of Education, Michigan State University, USA. You can email him at punya@msu.edu

Friday, December 13, 2002

Experience Mathematics # 22: The mobius strip

Take a long, thin strip of paper, give it a half twist, and paste the two ends together. What you get is a mobius strip (see picture).



Compare the mobius strip with the cylinder, which you get if you don’t give a half twist. A mobius strip has only one surface. Can you see why? Draw a line along the edge and keep going on. Eventually, you will arrive at the starting point. A cylinder, on the other hand, has an inside and an outside surface. The artist Escher portrayed this idea in Mobius Strip II (woodcut, 1963) (see picture).



If you cut the paper cylinder in half, you will get two cylinders. However, if you cut the mobius strip, you will get something very similar to a mobius strip. How many half-twists does the cut mobius strip have?

The mobius strip is one of the many surfaces that appear in Topology, a branch of mathematics. Topologists have been described as the mathematicians who cannot tell the difference between a coffee mug and a doughnut. This is because a mug can be “continuously deformed” to become a doughnut. A doughnut (which is a tyre tube, topologically speaking) cannot be transformed into a sphere. So, according to topologists, the tyre tube is not the same as a sphere, but a coffee mug is homotopic to a doughnut.

Pic credits: Both were stolen off the web, I don't know from where. 

Thursday, December 05, 2002

Experience Mathematics # 21 -Euclid's fifth axiom


Euclid’s fifth axiom says that given a line $l$ and a point $P$ not on the line, there is exactly one line parallel to $l$ passing through the point $P$. For centuries people thought that Euclid’s fifth axiom was “obvious”. But some mathematicians did not find it obvious. 

Finally, Reimann and Lobachevsky, both modified the axiom and tried to derive a new geometry. 

Reimann began with the axiom: Given a line $l$ and a point $P$ not on the line, there is no line parallel to $l$ passing through the point $P$. Reimann derived many geometrical theorems that are applicable on the surface of a sphere. For example, he showed that the sum of angles of a triangle is always greater than 180 degrees. Try drawing a triangle on a sphere and see why this has to be true.

Similarly, if we take a hyperbola ($y=1/x$) and rotate it around the y-axis, then we obtain a surface where Lobachevsky’s geometry holds. Lobachevsky’s geometry contains the axiom: Given a line $l$ and a point $P$ not on the line, there is more than one line parallel to $l$ passing through the point $P$. In this geometry, the sum of angles of a triangle is always less than $180$ degrees.

There is a property of the surface (known as curvature) that determines the geometry. Only surfaces with curvature zero follow the Euclidean geometry. Another example of a surface is that of a saddle (of a horse). Can you tell which geometry is applicable on this surface?

Friday, November 15, 2002

Experience Mathematics # 20 -- The sum of angles in a triangle

Euclidean or plane geometry begins with notions of points and lines, and the notion that a point lies on a line. Think of lines as sets, and a point as an element belonging to a set. Points and lines satisfy certain axioms. In Euclidean Geometry (or plane geometry), the axioms are based on Euclid’s original axioms. From these axioms, we can use the rules of logic to derive theorems (or propositions) that can be regarded as truthful statements that apply to the plane. Here a plane is a model, or a mini-universe where those axioms and theorems hold.

For example, consider the theorem: The sum of angles in a triangle is $180$ degrees. The various terms in this theorem (angle, triangle etc.) are constructs in the plane that we wish to study. The theorem itself is a property that will hold in our mini-universe. The proof should proceed from the axioms, use the definitions of the various constructs, and follow the rules of logic.

Even though the theorem is true, it does not imply that the sum all triangles is $180$ degrees. For example, consider the surface of the earth. Draw a triangle with a right angle at the North Pole. Suppose the two sides of this angle go down to the equator, and the third side of the triangle is the equator. The sum of angles of this triangle—made on the surface of the earth—is $270$ degrees!

In fact, in this non-euclidean geometry, the sum of angles in a triangle is always greater that $180$ degrees.

Can you find a surface where a sum of angles in a triangle is always less than $180$ degrees?

Friday, November 08, 2002

Experience Mathematics #19 -- Euclid's axioms

Just like elements and sets, Points and Lines are undefined notions.

We can think of a line as a set of points. These satisfy certain axioms, such as: Given a line $l$ and a point $P$ not on the line, there is only one line that is parallel to $l$ containing the point $P$. Axioms are considered to be self-evident truths.

However, several gaps were found in Euclid’s axioms. For example, consider Euclid’s proof that the base angles of an isosceles triangle are equal. Suppose we have an isosceles triangle $ABC$, where the side $AB$ is equal to the side $AC$. Drop a perpendicular $AD$ from a vertex to the side $BC$. There is nothing in Euclid’s axioms that says that the point $D$ is between the points $B$ and $C$. Nevertheless, Euclid proves that the triangles $ABD$ and $ACD$ are congruent. From this it is easy to see that the base angles of an isosceles triangle are equal.

The great mathematician Hilbert completed Euclid’s work by listing a few more axioms. These included the betweenness axioms. For example, given three points $A, B and C$, one of the axioms said either $B$ is between $A$ and $C$, or $C$ is between $A$ and $B$ or $A$ is between $C$ and $B$.

To return to Euclid’s proof, some steps need to be added to show that $D$ is between $B$ and $C$. 

But that is not all. We could consider a geometry where given a line $l$ and a point $P$ not on the line, there are no lines parallel to $l$ containing the point $P$. Such a non-euclidean geometry exists on the surface of the Earth. So one of Euclid’s axioms cannot be considered to be a self-evident truth after all.

Friday, October 25, 2002

Experience Mathematics #18 - All about itself


Russel’s Paradox shows that considering sets that contain themselves (or even asking whether they contain themselves or not) can lead to contradictory situations. But Real Life has many such self-referential situations. In this column, we will collect together many amusing (and not!) statements, such as this one.

“All Cretans are Liars”, said the Cretan Epimenides. Did Epimenides tell the truth? How can he, since he is a Cretan, and hence a liar? But if he lied, maybe he is telling the truth!

What about: This sentence is false. Is it true or false? Go through each sentence in this column and evaluate whether it is true or false.

This sentence has four words. This one, however, has six words. This one has one too too many words.

This sentence has no comma. This sentence does not describe itself.

This article is written by the author of this article. In other words, the author of Experience Mathematics writes Experience Mathematics. It is self-referential, since it refers to itself. In fact, the article refers to itself several times—but only once does the article refer to itself twice in one sentence. The author of this article is careful not to write self-referential statements.

Is this a question or not. How about this statement?

The above two statements beg the question. But what is the question? Was that the question? Does this answer the question?

The sentence below is false. The above sentence is true.

Lets not say any more, and end.

Friday, October 11, 2002

Experience Mathematics # 17 -- If it is, then it is not

A set can be thought of as a collection of objects. But what is it, really? The above sentence does not say: A set is a collection of objects. So is a set a collection of objects, or can it only be thought of as a collection of objects?

Sets can be of two types: those that contain themselves, and those that do not. For example, consider the set $F$ of fruits in your home. This set is not a fruit, so cannot contain itself. Now consider the set $A$. The set $A$ contains all sets that can be described in less than sixteen words. The above sentence has only $15$ words and describes $A$, so $A$ must be a member of itself.

Now consider the set $R$ of all sets that do not contain themselves as a member. In particular, $F$ is a member of $R$. The question is: Is $R$ a member of itself?

Well, if it is, then by definition $R$ consists of sets that do not contain themselves as a member. So $R$ is not a member of $R$. In short, if it is, then it is not.

Conversely, suppose $R$ is not a member of itself. Then since $R$ contains all sets that are not members of themselves, $R$ must be an element of $R$. Thus, if it is not, it is!

This paradox—pointed out the famous philosopher, Bertrand Russell—led to the formalization of set theory. Formally speaking, a ‘set’ and the relation ‘is an element of’ are undefined notions that satisfy certain axioms. However, we can continue to think of a set as a collection of objects. Just make sure that we consider only well defined sets—where we can decide whether any given object is an element of the set or not. That saves us from all Russellian disasters.