May

11

That’s impossible!

Filed Under Essential Math, Suggested reading | 3 Comments


Imagine a rope that was just long enough to wrap tightly around the equator of a perfectly spherical earth. Now imagine that the length of that rope is increased by one meter and again wrapped around the earth, supported in a regular way, so as to form an annulus. Doing things in this way will form a certain gap between the earth and the extended rope. Now imagine that you repeat the process with a golf ball. How do the two gaps compare?

Most people who understand the problem correctly, will immediately assume that adding a single meter to the rope surrounding the huge spherical earth won’t create much of a gap, while adding a single meter to the rope around the golf ball, will create a large gap. In reality, the two gaps are identical. It’s counterintuitive, but it can be proven easily with elementary mathematics.

We know that:

C = 2 \pi r
C + 1 = 2 \pi R

Where C is the length of the rope around a given spherical object, C+1 is obviously the length of the longer (by one meter) rope, r is the radius of the object (e.g. the golf ball) and R the radius of the annulus. We can express the two equations above as such:

\displaystyle r = \frac{C}{2\pi}
\displaystyle R = \frac{C+1}{2\pi}

The width of the gap between the longer rope and the object that’s taken into consideration is R-r.

Annulus’ gap

Therefore:

\displaystyle R - r = \frac{C+1}{2\pi} - \frac{C}{2\pi} = \frac{1}{2\pi} \approx 0.159

The gap, as you can see, is constant at about 16 cm, and it doesn’t depend in any way on the size of the radius (r) of the object at hand. That means that the gap between the extended rope and the golf ball is the same as in the case of the spherical earth.

impossible.jpgQuite a surprising result, isn’t it? This exact example opens up the book Impossible?: Surprising Solutions to Counterintuitive Conundrums which I received as a media copy in the mail. So far it’s been a very enjoyable and easy read, chocked full of surprising paradoxes and results which common sense would have you deem (practically) impossible or counterintuitive. To make things even nicer, the math involved is not overly advanced, and anyone who grasped high school level math, should have no problem following this engaging book.

Mar

21

An accessible Calculus book and Benjamin Franklin’s secret passion

Filed Under Suggested reading | 4 Comments

It has been a while since my last book recommendation. I’ll take the opportunity of the Easter longer weekend to write about two books that were recently released and I just finished reading. The first one is an atypical Calculus book, while the second one is a popular mathematics title which is historical and biographical in nature.

The first book is The Calculus Lifesaver: All the Tools You Need to Excel at Calculus.

Before I even start talking about the actual book, let me just tell you that this is a steal. I don’t know what the publisher was thinking, but a 750 page, recently published book on Calculus never sells for such a low price. On Amazon it sells for $16, which is a ridiculously low price for this 5 star tome. The average Calculus book is far from cheap, so this excellent guide is a pure bargain. Now, let’s talk about the content of the book.

I’m very exigent when it comes to Calculus books and usually like a very formal and rigorous style. Most people don’t. Many tend to like accessible books that speak to them in plain English. And this book is marketed as such. This is supposed to be an extra aid, on top of a regular textbook, to make Calculus more accessible. However, it stands on its own, thanks to its comprehensiveness and clarity. If commonly adopted Calculus books puzzle you, or if you are studying on your own, this is the book for you. Every step is clearly explained and it doesn’t fail when it comes to covering all the pre-requisites/fundamentals. Thanks to its style and approach, pretty much anyone who’s willing to learn, will. I’d even recommend it to high school students who wish to learn more about this subject, because I don’t think they would have any trouble following along. The tone is informal, friendly and often even funny, making it one of the least boring math books I’ve ever read. I highly recommend it to those who are struggling and would like to really understand the subject.

The second book is Benjamin Franklin’s Numbers: An Unsung Mathematical Odyssey.

Most geeks admire Ben Franklin, and not only for patriotic reasons. He was a brilliant, vibrant mind who made contributions to several fields. There isn’t a lack of biographies about the man. Or even good ones at that. What this short (and sweet) book does though, is to cover Franklin as a mathematician, a side of the genius that is often hidden or disputed. This hardcover focuses on Magic Squares and Franklin’s contribution to this field, even though he wrongly considered them as enjoyable, but useless in practice. Benjamin Franklin’s Numbers: An Unsung Mathematical Odyssey is filled with mathematical puzzles and will be a pleasure to read for those who can appreciate small challenges and the historical importance of Pasles’ research.

Oct

18

I’m reading a fantastic mathematical novel

Filed Under Suggested reading | 2 Comments


While I’m very busy at work and in my daily life, I’ve managed to start reading a new mathematical novel. It’s called A Certain Ambiguity: A Mathematical Novel and is published by the Princeton University Press. I’ve read only a few chapters but I must say that I’m really intrigued by this unusual and very interesting book. Besides being an easy read and an enjoyable page turner, this work of fiction manages to spark both interest in the illustration of thought-provoking mathematical concepts and at the same time, a genuine curiosity and interest towards the protagonist’s character.

It’s the kind of book that a mathematically inclined person would absolutely love, and in fact I’m having a hard time putting it down myself. On the other hand, anyone interested in an all around intelligent book, will be fascinated by its compelling narrative and the rather accessible mathematical insights, no matter what background they’re coming from.

A Certain Ambiguity is definitely a mathematical novel, but it’s not limited to that, because it’s a good philosophical novel to start with, so it can be appreciated when approached from different angles and mathematical skill levels. I plan to provide an in depth review as soon as I finish the book. Meanwhile it gets my positive recommendation; pick up a copy of this book to keep you company as “sweater weather” and a return to more time spent pursuing actives indoors arrives. The official page describes the book in the following terms:

“While taking a class on infinity at Stanford in the late 1980s, Ravi Kapoor discovers that he is confronting the same mathematical and philosophical dilemmas that his mathematician grandfather had faced many decades earlier–and that had landed him in jail. Charged under an obscure blasphemy law in a small New Jersey town in 1919, Vijay Sahni is challenged by a skeptical judge to defend his belief that the certainty of mathematics can be extended to all human knowledge–including religion. Together, the two men discover the power–and the fallibility–of what has long been considered the pinnacle of human certainty, Euclidean geometry.

As grandfather and grandson struggle with the question of whether there can ever be absolute certainty in mathematics or life, they are forced to reconsider their fundamental beliefs and choices. Their stories hinge on their explorations of parallel developments in the study of geometry and infinity–and the mathematics throughout is as rigorous and fascinating as the narrative and characters are compelling and complex. Moving and enlightening, A Certain Ambiguity is a story about what it means to face the extent–and the limits–of human knowledge.”

You can also download the first chapter here to whet your appetite.

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