## Category Archives: pure mathematics

### Architecture and Math: Fibonacci and Golden ratio

https://blogs.ams.org/blogonmathblogs/2018/01/22/on-seashells-spirals-and-solids/

On seashells, spirals and solids.

Just sharing for motivation towards math especially…

Nalin Pithwa.

### The infinite hotel paradox : due Jeff Dekofsky

This hardcore stuff about “infinity” , quite nicely, explained was pointed out to me by my ISC XII student, Mr. Utkarsh Malhotra! 🙂

### Genius of Srinivasa Ramanujan

1. In December 1914, Ramanujan was asked by his friend P.C. Mahalanobis to solve a puzzle that appeared in Strand magazine as “Puzzles at a Village Inn”. The puzzle stated that n houses on one side of the street are numbered sequentially starting from 1. The sum of the house numbers on the left of a particular house having the number m, equals that of the houses on the right of this particular house. It is given that n lies between 50 and 500 and one has to determine the values of m and n. Ramanujan immediately rattled out a continued fraction generating all possible values of m without having any restriction on the values of n. List the first five values of m and n.
2. Ramanujan had posed the following problem in a journal: $\sqrt{1+2\sqrt{1+3\sqrt{\ldots}}}=x$, find x. Without receiving an answer from the readers, after three months he gave answer as 3. This he could say because he had an earlier general result stating $1+x=\sqrt{1+x\sqrt{1+(x+1)\sqrt{1+(x+2)\sqrt{1+\ldots}}}}$ is true for all x. Prove this result, then $x=2$ will give the answer to Ramanujan’s problem.

Try try until you succeed!!

Nalin Pithwa.

### Limits that arise frequently

We continue our presentation of basic stuff from Calculus and Analytic Geometry, G B Thomas and Finney, Ninth Edition. My express purpose in presenting these few proofs is to emphasize that Calculus, is not just a recipe of calculation techniques. Or, even, a bit further, math is not just about calculation. I have a feeling that such thinking nurtured/developed at a young age, (while preparing for IITJEE Math, for example) makes one razor sharp.

We verify a few famous limits.

Formula 1:

If $|x|<1$, $\lim_{n \rightarrow \infty}x^{n}=0$

We need to show that to each $\in >0$ there corresponds an integer N so large that $|x^{n}|<\in$ for all n greater than N. Since $\in^{1/n}\rightarrow 1$, while $|x|<1$. there exists an integer N for which $\in^{1/n}>|x|$. In other words,

$|x^{N}|=|x|^{N}<\in$. Call this (I).

This is the integer we seek because, if $|x|<1$, then

$|x^{n}|<|x^{N}|$ for all $n>N$. Call this (II).

Combining I and II produces $|x^{n}|<\in$ for all $n>N$, concluding the proof.

Formula II:

For any number x, $\lim_{n \rightarrow \infty}(1+\frac{x}{n})^{n}=e^{x}$.

Let $a_{n}=(1+\frac{x}{n})^{n}$. Then, $\ln {a_{n}}=\ln{(1+\frac{x}{n})^{n}}=n\ln{(1+\frac{x}{n})}\rightarrow x$,

as we can see by the following application of l’Hopital’s rule, in which we differentiate with respect to n:

$\lim_{n \rightarrow \infty}n\ln{(1+\frac{x}{n})}=\lim_{n \rightarrow \infty}\frac{\ln{(1+x/n)}}{1/n}$, which in turn equals

$\lim_{n \rightarrow \infty}\frac{(\frac{1}{1+x/n}).(-\frac{x}{n^{2}})}{-1/n^{2}}=\lim_{n \rightarrow \infty}\frac{x}{1+x/n}=x$.

Now, let us apply the following theorem with $f(x)=e^{x}$ to the above:

(a theorem for calculating limits of sequences) the continuous function theorem for sequences:

Let $a_{n}$ be a sequence of real numbers. If $\{a_{n}\}$ be a sequence of real numbers. If $a_{n} \rightarrow L$ and if f is a function that is continu0us at L and defined at all $a_{n}$, then $f(a_{n}) \rightarrow f(L)$.

So, in this particular proof, we get the following:

$(1+\frac{x}{n})^{n}=a_{n}=e^{\ln{a_{n}}}\rightarrow e^{x}$.

Formula 3:

For any number x, $\lim_{n \rightarrow \infty}\frac{x^{n}}{n!}=0$

Since $-\frac{|x|^{n}}{n!} \leq \frac{x^{n}}{n!} \leq \frac{|x|^{n}}{n!}$,

all we need to show is that $\frac{|x|^{n}}{n!} \rightarrow 0$. We can then apply the Sandwich Theorem for Sequences (Let $\{a_{n}\}$, $\{b_{n}\}$ and $\{c_{n}\}$ be sequences of real numbers. if $a_{n}\leq b_{n}\leq c_{n}$ holds for all n beyond some index N, and if $\lim_{n\rightarrow \infty}a_{n}=\lim_{n\rightarrow \infty}c_{n}=L$,, then $\lim_{n\rightarrow \infty}b_{n}=L$ also) to  conclude that $\frac{x^{n}}{n!} \rightarrow 0$.

The first step in showing that $|x|^{n}/n! \rightarrow 0$ is to choose an integer $M>|x|$, so that $(|x|/M)<1$. Now, let us the rule (formula 1, mentioned above), so we conclude that:$(|x|/M)^{n}\rightarrow 0$. We then restrict our attention to values of $n>M$. For these values of n, we can write:

$\frac{|x|^{n}}{n!}=\frac{|x|^{n}}{1.2 \ldots M.(M+1)(M+2)\ldots n}$, where there are $(n-M)$ factors in the expression $(M+1)(M+2)\ldots n$, and

the RHS in the above expression is $\leq \frac{|x|^{n}}{M!M^{n-M}}=\frac{|x|^{n}M^{M}}{M!M^{n}}=\frac{M^{M}}{M!}(\frac{|x|}{M})^{n}$. Thus,

$0\leq \frac{|x|^{n}}{n!}\leq \frac{M^{M}}{M!}(\frac{|x|}{M})^{n}$. Now, the constant $\frac{M^{M}}{M!}$ does not change as n increases. Thus, the Sandwich theorem tells us that $\frac{|x|^{n}}{n!} \rightarrow 0$ because $(\frac{|x|}{M})^{n}\rightarrow 0$.

That’s all, folks !!

Aufwiedersehen,

Nalin Pithwa.

### Cauchy’s Mean Value Theorem and the Stronger Form of l’Hopital’s Rule

Reference: Thomas, Finney, 9th edition, Calculus and Analytic Geometry.

Continuing our previous discussion of “theoretical” calculus or “rigorous” calculus, I am reproducing below the proof of the finite limit case of the stronger form of l’Hopital’s Rule :

L’Hopital’s Rule (Stronger Form):

Suppose that

$f(x_{0})=g(x_{0})=0$

and that the functions f and g are both differentiable on an open interval $(a,b)$ that contains the point $x_{0}$. Suppose also that $g^{'} \neq 0$ at every point in $(a,b)$ except possibly at $x_{0}$. Then,

$\lim_{x \rightarrow x_{0}}\frac{f(x)}{g(x)}=\lim_{x \rightarrow x_{0}}\frac{f^{x}}{g^{x}}$ ….call this equation I,

provided the limit on the right exists.

The proof of the stronger form of l’Hopital’s Rule is based on Cauchy’s Mean Value Theorem, a mean value theorem that involves two functions instead of one. We prove Cauchy’s theorem first and then show how it leads to l’Hopital’s Rule.

Cauchy’s Mean Value Theorem:

Suppose that the functions f and g are continuous on $[a,b]$ and differentiable throughout $(a,b)$ and suppose also that $g^{'} \neq 0$ throughout $(a,b)$. Then there exists a number c in $(a,b)$ at which

$\frac{f^{'}(c)}{g^{'}(c)} = \frac{f(b)-f(a)}{g(b)-g(a)}$…call this II.

The ordinary Mean Value Theorem is the case where $g(x)=x$.

Proof of Cauchy’s Mean Value Theorem:

We apply the Mean Value Theorem twice. First we use it to show that $g(a) \neq g(b)$. For if $g(b)$ did equal to $g(a)$, then the Mean Value Theorem would give:

$g^{'}(c)=\frac{g(b)-g(a)}{b-a}=0$ for some c between a and b. This cannot happen because $g^{'}(x) \neq 0$ in $(a,b)$.

We next apply the Mean Value Theorem to the function:

$F(x) = f(x)-f(a)-\frac{f(b)-f(a)}{g(b)-g(a)}[g(x)-g(a)]$.

This function is continuous and differentiable where f and g are, and $F(b) = F(a)=0$. Therefore, there is a number c between a and b for which $F^{'}(c)=0$. In terms of f and g, this says:

$F^{'}(c) = f^{'}(c)-\frac{f(b)-f(a)}{g(b)-g(a)}[g^{'}(c)]=0$, or

$\frac{f^{'}(c)}{g^{'}(c)}=\frac{f(b)-f(a)}{g(b)-g(a)}$, which is II above. QED.

Proof of the Stronger Form of l’Hopital’s Rule:

We first prove I for the case $x \rightarrow x_{o}^{+}$. The method needs no  change to apply to $x \rightarrow x_{0}^{-}$, and the combination of those two cases establishes the result.

Suppose that x lies to the right of $x_{o}$. Then, $g^{'}(x) \neq 0$ and we can apply the Cauchy’s Mean Value Theorem to the closed interval from $x_{0}$ to x. This produces a number c between $x_{0}$ and x such that $\frac{f^{'}(c)}{g^{'}(c)}=\frac{f(x)-f(x_{0})}{g(x)-g(x_{0})}$.

But, $f(x_{0})=g(x_{0})=0$ so that $\frac{f^{'}(c)}{g^{'}(c)}=\frac{f(x)}{g(x)}$.

As x approaches $x_{0}$, c approaches $x_{0}$ because it lies between x and $x_{0}$. Therefore, $\lim_{x \rightarrow x_{0}^{+}}\frac{f(x)}{g(x)}=\lim_{x \rightarrow x_{0}^{+}}\frac{f^{'}(c)}{g^{'}(c)}=\lim_{x \rightarrow x_{0}^{+}}\frac{f^{'}(x)}{g^{'}(x)}$.

This establishes l’Hopital’s Rule for the case where x approaches $x_{0}$ from above. The case where x approaches $x_{0}$ from below is proved by applying Cauchy’s Mean Value Theorem to the closed interval $[x,x_{0}]$, where $x< x_{0}$QED.

### The Sandwich Theorem or Squeeze Play Theorem

It helps to think about the core concepts of Calculus from a young age, if you want to develop your expertise or talents further in math, pure or applied, engineering or mathematical sciences. At a tangible level, it helps you attack more or many questions of the IIT JEE Advanced Mathematics. Let us see if you like the following proof, or can absorb/digest it:

Reference: Calculus and Analytic Geometry by Thomas and Finney, 9th edition.

The Sandwich Theorem:

Suppose that $g(x) \leq f(x) \leq h(x)$ for all x in some open interval containing c, except possibly at $x=c$ itself. Suppose also that $\lim_{x \rightarrow c}g(x)= \lim_{x \rightarrow c}h(x)=L$. Then, $\lim_{x \rightarrow c}f(x)=c$.

Proof for Right Hand Limits:

Suppose $\lim_{x \rightarrow c^{+}}g(x)=\lim_{x \rightarrow c^{+}}h(x)=L$. Then, for any $\in >0$, there exists a $\delta >0$ such that for all x, the inequality $c implies $L-\in and $L-\in ….call this (I)

These inequalities combine with the inequality $g(x) \leq f(x) \leq h(x)$ to give

$L-\in

$L-\in

$-\in ….call this (II)

Therefore, for all x, the inequality $c implies $|f(x)-L|<\in$. …call this (III)

Proof for LeftHand Limits:

Suppose $\lim_{x \rightarrow c^{-}} g(x)=\lim_{x \rightarrow c^{-}}=L$. Then, for $\in >0$ there exists a $\delta >0$ such that for all x, the inequality $c-\delta implies $L-\in and $L-\in …call this (IV).

We conclude as before that for all x, $c-\delta implies $|f(x)-L|<\in$.

Proof for Two sided Limits:

If $\lim_{x \rightarrow c}g(x) = \lim_{x \rightarrow c}h(x)=L$, then $g(x)$ and $h(x)$ both approach L as $x \rightarrow c^{+}$ and as $x \rightarrow c^{-}$ so $\lim_{x \rightarrow c^{+}}f(x)=L$ and $\lim_{x \rightarrow c^{-}}f(x)=L$. Hence, $\lim_{x \rightarrow c}f(x)=L$. QED.

Let me know your feedback on such stuff,

Nalin Pithwa

### Mobius and his band

There are some pieces of mathematical folklore that you really should be reminded about, even though they are “well-known” — just in case. An excellent example is the Mobius band.

Augustus Mobius was a German mathematician, born 1790, died 1868. He worked in several areas of mathematics, including geometry, complex analysis and number theory. He is famous for his curious surface, the Mobius band. You can make a Mobius band by taking a strip of paper, say 2 cm wide and 20 cm long, bending it around until the ends meet, then twisting one end through 180 degrees, and finally, gluing the ends together. For comparison, make a cylinder in the same way, omitting the twist.

The Mobius band is famous for one surprising feature: it has only one side. If an ant crawls around a cylindrical band, it can cover only half the surface — one side of the band. But, if an ant crawls around on the Mobius band, it can cover the entire surface. The Mobius band has only one side.

You can check these statements by painting  the band. You can paint the cylinder so that one side is red and the other is blue, and the two sides are completely distinct, even though they are separated by only the thickness of the paper. But,, if you start to paint the Mobius band red, and keep going until you run out of band to paint, the whole thing ends up red.

In retrospect, this is not such a surprise, because the 180  degrees twist connects each side of the original paper strip to the other. If you  don’t twist before gluing, the two sides stay separate. But, until Mobius (and a few others) thought this one up, mathematicians didn’t appreciate that there are two distinct kinds of surface: those with two sides and those with one side only. This turned out to be important in topology. And, it showed how careful you have to be about making “obvious” assumptions.

There are lots of Mobius band recreations. Below are three of them:

• If you cut the cylindrical band along the middle with two scissors, it falls apart into two cylindrical bands. What happens if you try this with a Mobius band?
• Repeat, but this time, make the cut about one-thirds of the way across the width of the band. Now, what happens to the cylinder and to the band?
• Make a band like a Mobius band but with a 360 degrees twist. How many sides does it have? What happens if you cut it along the middle?

The Mobius band is also known as a Mobius strip, but this can lead to misunderstandings, as in aLimerick written by a science fiction author Cyril Kornbluth:

A burleycue dancer, a pip

Named Virginia, could peel in a zip,

and died of constriction

Attempting a Mobius strip.

A more politically correct Mobius limerick, which gives away one of the answers, is:

A mathematician confided,

That a Mobius strip  is one-sided,

You’ll get quite a laugh

if you cut it to half,

For it stays in one piece when divided.

Ref: Professor Stewart’s Cabinet of Mathematical Curiosities.

Note: There are lots of interesting properties of Mobius strip, which you can explore. There is a lot of recreational and pure mathematics literature on it. Kindly Google it. Perhaps, if explore well, you might discover your hidden talents for one of the richest areas of mathematics — topology. Topology is a foundation for Differential Geometry, which was used by Albert Einstein for his general theory of  relativity. Of course, there are other applications too…:-)

— Nalin Pithwa.

### Number theory has numerous uses

One of the fun ways to get started in mathematics at an early age s via number theory. It does not require deep, esoteric knowledge of concepts of mathematics to get started, but as you explore and experiment, you will learn a lot and also you will have a ball of time writing programs in basic number theory. One of the best references I have come across is “A Friendly Introduction to Number Theory” by Dr. Joseph Silverman. It is available on Amazon India.

Well, number theory is not just pure math; as we all know, it is the very core of cryptography and security in a world transforming itself to a totally digital commerce amongst other rapid changes. Witness, for example, the current intense debate about opening up an iPhone (Apple vs. FBI) and some time back, there was the problem with AES Encrypted Blackberry messaging services in India.

Number theory is also used in Digital Signal Processing, the way to filter out unwanted “noise” from an information signal or “communications signal.” Digital Signal Processing is at the heart of modem technology without which we would not be able to have any real computer networks.

There was a time when, as G H Hardy had claimed that number theory is the purest of all sciences as it is untouched by human desire. Not any more !!!

Can you imagine a world without numbers ?? That reminds me of a famous quote: “God created the natural numbers, all the rest is man-made.” (Kronecker).

More later,

Nalin Pithwa

### Careers in Mathematics

Most people already have a belief that the the only career possible with a degree in Mathematics is that of a teacher or a lecturer or a professor. Thanks to the co-founder(s) of Google, whose database search engine is based on the Perron-Frobenius Theorem, this notion is changing.

In particular, you might want to have a detailed look at the website of Australian mathematics/mathematicians —–

http://www.mathscareers.org.au/

I will cull more such stuff and post in this blog later…

Regards,

Nalin Pithwa