We are in the middle of a series on conic sections. We started with Finding Refreshment in the Desert, in which I introduced us to what a conic section is and demonstrated that the general second degree equation in two variables represents a conic section. After that, in Circular Tangentiality we obtained the general equation of a circle and derived the equation of the tangent to a circle at a point. Last week, in Contacting Circular Polarity, I introduced us to the concepts of the chord of contact and polar of a point with respect to a circle. And toward the end of the post I teased the reader with a mention of a ‘family of circles’. Whatever could that be?

Recall that the equation of a circle with center at (p, q) and radius r is given by

When this is expanded and rearranged, we get

Recall also that the general second degree equation in two variables is

We can see that the equation of the circle is similar to the general second degree equation in two variables with

Hence, by convention we say that the general equation of a circle is

with center at (-g, –f) and radius given by

Now, suppose we are given the circles

The radical axis of these two circles is obtained by equating C₁ and C₂. Hence,

This is clearly the equation of a straight line and we can easily show that it is perpendicular to the line joining the centers of C₁ and C₂. Now consider the equation

When λ = -1, this represents the radical axis. However, when λ ≠ -1, we can write the equation as

Dividing the equation by 1 + λ we get

This clearly is the equation of a circle. Hence, by varying the value of λ we can obtain different circles. Suppose we attempted to find the radical axis of this circle and C₁. We would get

If we focused on the coefficient of x we would obtain

Hence, the equation of the radical axis would be

If we divide this equation by λ/(1 + λ) we will get

which is the radical axis of C₁ and C₂. In other words, any two members of this family of circles shares the same radical axis as the two circles used to generate the family.

While the study of family of circles began, as is the case for many concepts related to conic sections, simply as an exercise of playing with the mathematics, this obscure concept has found applications in the study of paths of light and designs of tires and gears. This has been the case with many mathematical concepts that were developed without any practical applications in mind. However, as the study proceeds, quite often unforeseen applications become possible.

There is much more that can be discussed related to the circle, even related to families of circles. However, that will take us really deep down this rabbit hole. Hence, we will leave the circle for now. In the next post, we will turn our focus to the parabola and see what properties this conic section has.

Arithmetic Blasphemy

In one of the first posts in this blog, My Unbounded Mathematical Trauma, I had shared about my trauma when students says something like, “One divided by zero equals infinity.” Unfortunately, this is something that I have heard repeated by mathematics teachers too. And I shudder when I hear such blasphemy! I say ‘blasphemy’ because division by zero is the ‘unforgiveable sin’ in mathematics and a statement like “One divided by zero equals infinity” gives the impression that it is possible to divide one by zero and obtain a number.

However, if it is possible to divide one by zero, then the result must be a number since operations performed on numbers must yield numbers. If the result is infinity, where do we locate it on the number line? After all, the number line is a mathematical artifact that is supposed to be a visual representation on which we can locate every real number. If we are able to locate infinity on the number line, then either that must be where the number line ends, meaning that the number line itself is not unbounded, or there are numbers to the right of infinity, meaning that infinity is not something that is greater than any number.

What we realize is that infinity is not a number, but an idea, a construct, if you will, to denote unboundedness rather than quantity, which is what every number must denote.

Strange Infinite Series

When we take this to the area of infinite series, we have seen that there are series that converge and those that diverge. While I haven’t discussed tests for convergence either formally or comprehensively, I think posts like Infinitely Expressed and Serially Expressed have given us a reasonable idea that infinite series need to have certain properties for them to be convergent.

With this in mind, let us consider the series

S = 1 – 1 + 1 – 1 + 1 – 1 + 1 – 1 +…

This is an infinite series. What is its sum? Does it converge? If so, to what value does it converge?

Now, we can add grouping symbols that should not alter the sum. Hence, we can have

S = (1 – 1) + (1 – 1) + (1 – 1) + (1 – 1) + … = 0 + 0 + 0 + 0 + … = 0

Alternately, we could have

S = 1 – (1 – 1) – (1 – 1) – (1 – 1) – (1 – 1) – … = 1 – 0 – 0 – 0 – … = 1

So grouping the terms changes the sum! Of course, we could use the formula for the sum of an infinite geometric series with the common ratio being -1. This would give us

S = 1 ÷ (1 – (-1)) = 1/2

We could have obtained the same result as follows

S = 1 – (1 – 1 + 1 – 1 + 1 -…) = 1 – S ⇒ 2S = 1 ⇒ S = 1/2

Clearly adding up the terms of this series is not straightforward. There is no clear indication that it diverges. But there is no clear value to which it converges. However, if we consider the partial sums of the series we get the following:

S₁ = 1, S₂ = 0, S₃ = 1, S₄ = 0, S₅ = 1, S₆ = 0

Since the partial sums alternate between 1 and 0, we say that the series is alternating.

However, we can get even curiouser results with such infinite series. Consider the following

S = 1 – 2 + 3 – 4 + 5 – 6 + …

Here, the partial sums are as follows

S₁ = 1, S₂ = -1, S₃ = 2, S₄ = -2, S₅ = 3, S₆ = -3

If we group the terms we can get

S = (1 – 2) + (3 – 4) + (5 – 6) + … = -1 – 1 – 1 -…

It is clear that this way the series diverges toward negative infinity. However, we can also group the terms as follows:

S = 1 – (2 – 3) – (4 – 5) – (6 – 7) – … = 1 + 1 + 1 + 1 + …

In this case, the series diverges to positive infinity! Of course, the partial sums did indicate this since they alternated in signs while increasing the magnitude.

Since all the methods give us different answers, what we realize is that, with divergent or alternating series, grouping of terms does not help a bit.

Of course, infinity leads us to some additional strange results. For example, we know that the set of natural numbers is

N = {1, 2, 3, 4, 5, 6, …}

We also know that the set of even numbers is

E = {2, 4, 6, 8, …}

However, for every natural number, it is possible to multiply it by 2 and obtain an even number. In this sense the set mapping elements in N to elements in E can be listed as

M = {(1,2), (2,4), (3,6), …}

It is clear that no element in N will be excluded and that every element in E will be included. In other words the number of elements in N is the same as the number of elements in E even though it is clear that E excludes the odd numbers!

The Infinity of Rational Numbers

In fact, even the rational numbers can be mapped to the natural numbers. Let us arrange the rational numbers in a grid as shown below.

Here, the numerator corresponds to the column while the denominator corresponds to the row. Hence, the element in row 5 and column 3 will be 3/5. Of course, we can have rational numbers that are equal but appear in different cells. This is because we have equivalent fractions. These are indicated above with the same color. Hence, all numbers in red are equivalent to , while all in light green are equivalent to 2, and so on. Now, since each row and each column has an infinite number of elements, we cannot simply go down a row or a column and hope to reach all the rational numbers. For example, if we simply choose to go down the first row, which simply includes all natural numbers, we will never get to row 2! However, as can use a process of diagonalization as shown below

Here we go down the diagonals in the order indicated by the numbers on the left. With this order, the rational numbers are reached in the following order

If we eliminate all the equivalent fractions, we will get

Since the numbers are being reached diagonally, every cell will be reached after a finite number of moves. It is crucial to recognize that the numbers are not arranged in any numerical order. If we include negative numbers, we could do something like

Hence, we can map every rational number, without repetition, to the natural numbers, meaning that the number of rational numbers and the number of natural numbers is the same!

This is obviously counter-intuitive. After all, suppose we consider two consecutive integers, say 1 and 2 for convenience. We can generate rational numbers between 1 and 2 as follows

This means that between any two consecutive natural numbers there are infinitely many rational numbers. Despite this, as we have shown, it is possible to map every rational number to a natural number. Let the weirdness of this sink in. Though there are infinitely many rational numbers between any two consecutive natural numbers, there is a way of mapping every rational number to a natural number.

Brief Introduction to Cardinality

The property that between any two rational numbers it is possible to generate at least one more rational number between them constitutes the set of rational numbers as a ‘dense’ set. However, it is clear that the natural numbers are not ‘dense’ since there is no natural number between consecutive natural numbers. Hence, it is strange that a one to one mapping from the rational numbers to the natural numbers exists. In set theory, all sets that share this one to one mapping with the natural numbers are said to have the same cardinality of ℵ₀, aleph-nought. Here, ‘cardinality’ gives us a measure of the ‘size’ of the set and ℵ₀ is the ‘smallest’ cardinality that a set with infinite members can have. Other sets that have cardinality of ℵ₀ are integers, square numbers, cube numbers, constructible numbers, algebraic numbers, etc. If we take the union of all these sets, even this union set would have cardinality of ℵ₀ since we can rotate between different sets, ignoring any duplications along the way.

However, cardinality of the real numbers is different because they include other numbers, like the transcendental numbers, making it impossible to map the real numbers to the natural numbers. The cardinality of the real numbers is hypothesized to be ℵ₁, the next larger element in the set of alephs. As of now, this is still a hypothesis.

The Perils of Infinity

What we can see is that ‘infinity’ is much weirder than we might ever have imagined. Not only have we seen that it is possible to map one ‘infinity’ (the natural numbers) to an ‘infinity of infinities’ (the rational numbers) but also that there are ‘infinities’ that are ‘infinitely larger’ than the intuitive ‘infinity’ obtained simply by counting upward without end. When students and (shudder) teachers treat ‘infinity’ in a trivial manner as though it were simply a very large number that can be located on the number line, for example by repeating or condoning statements like, “One divided by zero equals infinity,” they reveal that they have failed to understand the fact that the term ‘infinity’ is not used to denote a number but a conglomeration of idea that is itself ‘infinitely’ rich.

Starting the Critique

I don’t want you to get the wrong idea either while or after reading this post. At the risk of sounding like I’m blowing my own trumpet, let me just state that I have a really good test taking temperament. I have an excellent memory and am able to think on my feet. Moreover, I am rarely flustered by high pressure situations. However, I have reached a stage in life where I think that tests and exams are detrimental to student learning and health.

During my career spanning a few decades now, I have encountered students who were exceptionally brilliant. However, closed spaces and proximity to others gave them the heebie jeebies, which adversely affected how they fared during testing conditions. Someone may argue that most exam boards now have accommodations for students with special needs, be these learning needs or mental health needs. This is true. However, in many parts of the world, special needs are either not diagnosed effectively or are too expensive for the school to accommodate. If a school only has funds for one designated exam room, then expecting it to provide a separate room for each student with different needs is not just ridiculous, it is unjust.

Red Herrings

Some people may say that exams and tests help students develop time management, stress management, and critical thinking skills. I find these claims to be quite disingenuous. First, if the testing environment itself is generating the stress, then it’s not an environment that is developing stress management skills. That’s like saying that a bully helps you develop the skills to stand up for yourself! Indeed, this argument fails to address the issue of stress itself. Why are we okay with environments that create stress? After all, to say that a student must develop stress management skills in the context of exams is to say that the kind of stress developed in an exam is like what we would experience outside the exam. But this is evidently not true. I have experienced many stressful exams, but none of them were in any way helpful in managing the kind of stress that visits us uninvited in life. Indeed, in the real world, many stressful situations can be diffused precisely by walking away, something that is just not available in an exam.

Second, I do not know of any situation in the real world, apart from catching a flight or a train or making it for an interview or making an exquisite dessert, where management of time for a period of about 2 hours makes or breaks your future. Time management can be learned in many other ways. In fact, asking someone to make a batch of brownies would be a better way of teaching time management since you have to ensure that the butter is hot enough to stay melted but not so hot as to scramble the eggs when you break them into the batter! The idea of creating an artificial environment in which time management is critical is in my view only a way of exercising power over the student who has no say in the matter.

Third, exams may help people develop the ability to think on their toes. However, flying by the seat of one’s pants is not critical thinking! That is winging it. Critical thinking must include the ability to reflect on the solutions one has reached and to critique them dispassionately with the intention of discarding unrealistic solutions and improving inefficient ones, which is not possible in the context of an exam. Asking someone to think clearly while they have a gun to their head is not developing critical thinking skills, but allowing survival responses to rule the day.

Elevating Mediocrity

However, the problems with tests go much deeper. As mentioned, the environment created is artificial. Each candidate is expected to think on their own within a limited time with no opportunity of seeking the wisdom and guidance of those who have gone before them. This creates a mentality in which collaboration, which is critical in the real world, is thrown into the rubbish heap and declared to be off bounds. Further, an exam, in which no breaks are permitted, is not how anything in life works. In a day at work, most people will have time for a coffee break or lunch with colleagues and friends. These breaks rejuvenate the mind and can even stimulate it. They allow us to step away from the problem at hand, which is often essential to getting the creative juices flowing again. This is proscribed in the exam environment. However, expecting someone – even a strong introvert like me – to sit for 2+ hours on a series of disconnected tasks, is a recipe for mediocrity, not excellence, because the creative process is not allowed to develop and flourish. It conveys the idea that problems can be fixed by a single person in a matter of minutes. It is no wonder that patience and perseverance are in short supply these days. In my view, any real world problem that is worth solving requires investment of time and collaboration between humans that the exam system belies.

Choosing Conformity

Of course, we can understand that this would be the case if we just inquired about the origins of the exam system. The earliest known system of exams was the Imperial Examination, a civil examination administered in Imperial China for the purpose of selecting candidates who would serve in the state bureaucracy. A system of formal exams was introduced in England in the early nineteenth century, again with the intention of selecting civil servants. When the British took over in India, they introduced similar exams, all with the purpose of selecting civil servants.

Now, let us ask ourselves a very pertinent question: Do governments select civil servants for creativity or conformity? Quite obviously the latter. We do not want civil servants to be engaged in creative bookkeeping or flights of fancy while interpreting the law of the land! No! In such situations, we desire and demand strict conformity. And there is no denying that exams conducted by governments are good at selecting candidates who will conform.

However, human flourishing will not happen if conformity is given pride of place. Indeed, we almost certainly ensure that the creative impulses in most people are quenched when we tell them that the straitjacket of exams is the way to move ahead in life. It is no wonder that so many students emerge from high school and college completely jaded. After all, if we have killed in them what made them human, namely the impulse to create and speculate and imagine, then we should not be surprised if they emerge with a cynical view of the world.

Lipstick on a Pig

However, let us be brutally honest. The exam system was developed by governments to select people from the citizenry who would best execute and enforce what the government wanted in the nation. They were never designed to promote creative thinking or out of the box imagination. They were intended to be ways in which the nation controlled the masses, not empowered them. This is why memorization of arcane facts, accuracy with basic arithmetic, and repetition or identification of rules and regulations feature prominently in most exams.

Exams may have changed their face today. Many do not require mere memorization and regurgitation of facts. Many include attempts at promoting creativity. Many include attempts at introducing real world application. However, creativity cannot be dictated by the clock and application often requires a good night’s sleep before the lightning bolt of an idea strikes. Of course, exams still deny the fact that we are human and can develop as humans only in collaboration with others. Expecting someone to think of a bunch of new ideas thrown at them at random in a span of a couple of hours is inhumane. However, if control is what we desire, then exams are the best way to ensure it. And so, all the sweeping changes in exams over the past decades are in reality nothing more than putting lipstick on a pig. The changes are superficial precisely because what the exams are intended to do cannot be aligned with the needs of most humans.

Control Mechanism

You may ask me what the way ahead might be. If I am so strongly opposed to exams, how do I think we should assess students? The question presupposes that assessing students is necessary. But what are we assessing them for? And what is the purpose of assessment? Why are high school students, for example, expected to write a series of exams toward the end of their program? Why is the student who just wants to become a field anthropologist expected to display competence in business or mathematics? Why is the student who wants to become an artist expected to demonstrate acumen in biology or geography? Why is the student who wants to become a hairdresser expected to show knowledge of physics or history? Why is the student who wants to undertake biochemistry research expected to indicate skill with economics or a second language? We have straitjacketed programs that do not actually serve most students. And then we complain that they lack motivation and throw up our hands in wonder.

But someone may say that kids really don’t know what they want to do in life. Hence, we must give them exposure to a wide variety of options. This is simply a way of saying that we do not actually wish to invest the time and effort to mentor each child so that he/she can discover what most motivates him/her. We only want to give them a superficial overview of a whole gamut of human knowledge, but nothing to such a depth as would actually capture their imaginations and fire up their spirits. Expecting every student to learn the same thing is simply a way of throwing in the towel and reneging on our responsibility, as the adults in their lives, of providing meaningful guidance.

However, if we devote time with each student to help them discover what drives them, we will realize that exams are precisely not the way to do it. Indeed, nothing that provides an external reward, in this case grades or college admissions or a job, can ever fire up the spirits. Our spirits, you see, are not need driven. The basic animal needs – food, shelter, clothing – do not inspire the spirit. Hence, anything that promises to put food on the table or a roof above our heads or clothes on a backs can never become something that captures our imagination. This is why most people consider their jobs just a drudgery to be endured, something that provides options for the weekend. However, what the dichotomy between the week and the weekend develops is the sense that the latter exists just so that we can ‘get away’ from what the former involves. This causes confusion because, with this paradigm, we are always unsure which our real life is, the five or six days of toil or the day or two of getting away from it.

Straitjacket curriculums culminating in exams are designed to produce people who will conform to this dichotomy and confusion. They are designed precisely to ensure that most people will spend most of their time disliking what they are doing but doing it do that they have some respite from the drudgery for one or two days of each week. In other words, straitjacket curriculums and exams are precisely parts of a system of domination to ensure that very few of us will ever be able to truly flourish by enjoying what we do day in and day out.

Breaking Free

What I advocate is a system of mentorship between a student and a mentor in which the vast majority of adults play the part of mentor to some child or the other. The mentor is not a ‘know it all’ but someone who knows of people who have expertise in a wide variety of fields. If a student wants to learn about colonialism in India, the mentor can direct her to an expert. If the student wants to learn about the physiology of a horse, the mentor points her to an expert. In this way we get the modern equivalent of “it takes a village to raise a child.”

What this means is that industry takes the onus of training the next generation of young learners. They do not merely profit from the investment of parents and teachers and schools over the years, but must actually contribute to the learning of the students who choose to be pointed in their direction. Hence, if a student expresses the desire to learn about designing cars, the mentor points her to an automobile company. And the company is expected to take the student on and show her how cars are designed. If another students wants to learn to cook Korean food then the mentor directs him to a chef who then teaches him the art of making Korean food. I expect there will be push back from many people here. Why should companies play such a role. Well, in the past, a blacksmith who hoped to expand his trade would have taken on an apprentice and shown him the ropes of the blacksmithy. The blacksmith would have invested the time and effort to pass on his skills to the apprentice. Those who hope to benefit from the skills of someone should be expected to contribute to the development of those skills in that person.

Someone may say that this is impractical. Why? Are we not motivated enough to provide meaningful guidance to the next generation of humans, the ones who will carry the torch after our candles have been snuffed out? Some may say that it makes things very complicated. Of course it does! Each child is different! How can we say that a straitjacket curriculum will ever be beneficial to most of them? However, our responsibility to the next generation cannot be sacrificed on the altar of expediency. Expediency, however, is the way a system of dominance functions. Everyone has to be treated the same way. This ensures that control by the powers that be is possible. If everyone is made to pass through the same doors, then supervision of the majority by a minority is possible.

Of course, this is done, deceptively in my opinion, under the banner of equality. However, equality is not equity. Equality denies the idiosyncrasies and circumstances of each person while equity recognizes them. A unilateral decision by the powers that be that a child who wants to draw must learn his numbers or that another who wants to play with numbers must make her drawings is not equity. It is actually not even equality. It is oppression. And for too long we the masses have allowed governments, prestigious universities, large companies, and publishing houses to place us in situations of oppression whose quintessential element during our formative years is the examination.

For today’s post, I wish to briefly two books that delve into mathematical themes. So here goes.

Book Review: A Certain Ambiguity by Gaurav Suri and Hartosh Singh Bal

Gaurav Suri and Hartosh Singh Bal. A Certain Ambiguity: A Mathematical Novel. (New Delhi: Penguin, 2007). Available on Amazon.

Quite out of the blue, a colleague of mine gifted me a book toward the end of the 2018-2019 academic year. It was an unexpected gift, but quite timely since I did not have any holiday reading planned. I guess my colleague chose the book because I am a Mathematics teacher. I wonder, though, how many people would be attracted to a book that purports to be ‘a Mathematical novel’!

I was, however, quite eager to get started with the book. The protagonist of the novel is one Ravi Kapoor, who is a student at Stanford. His grandfather, Vijay Sahni, had been a Mathematician and had instilled in Ravi a love for Mathematics. When Ravi reaches Stanford, he enrolls in a Mathematics course called Thinking About Infinity taught by Professor Nico Aliprantis. During discussions related to the course, Ravi mentions his grandfather, whose name Nico recognizes. When Nico pulls out a paper written by Vijay, Ravi reads a note that indicates that Vijay had been in prison in New Jersey in the early twentieth century. That sets Ravi on a course to discovering why his grandfather had been imprisoned. His digging reveals that Vijay had been charged under an obscure blasphemy law in New Jersey.

The novel is like a braid, with three strands running through it. One strand, of course, is that related to the discussion in the course taught by Nico. The second consists of fictionalized memoirs of various Mathematicians ranging from Euclid to Gauss. The third consists of conversations between the grandfather, Vijay, and Judge John Taylor, appointed to decide whether Vijay should go on trial or be set free. Since Vijay was charged with blasphemy, these conversations also touch on religious themes.  The three strands are woven intricately and play off each other extremely well.

As the title suggests, the book narrates how Vijay and Ravi searched for certainty within Mathematics. I’ll just leave it at that without revealing the outcome of their search. That is for you to discover for I wholeheartedly recommend the book to anyone who either loves Mathematics or enjoys a well crafted tale.

Book Review: In Pursuit of the Unknown by Ian Stewart

Ian Stewart. In Pursuit of the Unknown: 17 Equations that Changed the World. (New York: Basic Books, 2012). Available on Amazon.

I stumbled across this book when a friend mentioned it to me, albeit in quite a vague manner. Once I read the subtitle, of course, I was hooked. As someone who has loved Mathematics for as long as he can remember, I could not resist the temptation to read a book about how Mathematical equations have played a key role in giving us the world we live in today.

As the subtitle suggests, Stewart describes the meaning and influence of seventeen Mathematical equations from all sorts of domains – from geometry to physics and from signal processing to economics. Some, such as the Pythagorean Theorem and E=mc², are what one may call ‘usual suspects’ given that they have entered common discourse even if most people have no clue about what the equations mean or of how to use them. Others are more elusive, such as the Navier-Stokes equation and the Black-Scholes equation, the knowledge of which is restricted only to a handful of people in the know. Still others, such as the second law of thermodynamics and the definitions of the derivative and the imaginary number i, might cause high school students around the world to shudder.

Stewart does a commendable job both of describing the meaning of each equation and of outlining its uses. While some knowledge of Mathematics is definitely helpful, Stewart has not targeted the book toward those with much knowledge of Mathematics. This makes the book far more accessible that it otherwise would have been.

Stewart also does not shy away from the downsides of using Mathematics. He shows that, when Mathematics is used as a tool within other disciplines, the ethical considerations must be dealt with within those disciplines for Mathematics itself is unconcerned about such issues. Thus the power of Mathematics is itself a major drawback for, in allowing itself to be used by other disciplines, it subjects itself to the whims of those disciplines.

The book, as a whole, is a remarkably good read and I could not put it down once I began. For those who benefit daily from the usefulness of Mathematics and who still wonder about why it is taught in schools, I would recommend this book as an eye-opener. Of course, I would recommend this book to anyone who has an inkling of curiosity coursing through their veins!