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

31

On the Importance of Mathematics

Filed Under Math Education | 3 Comments

Mathematics is the queen of science and the language of nature. Its importance should be clear to any reasonable person. It is easy however to diminish the value of certain areas of research because they’re currently thought as having little practical use. Evolutionary needs brought our mind to prefer knowledge that can be employed for the solution of specific problems in the real world, rather than deeply abstract ones. It is an understandable and even excusable fallacy that there are useful fields of math and useless ones, based on the perception of their applied or theoretical nature. But it’s still a misconception. Each theorem and discovery is a little piece of a larger puzzle that we conveniently categorize into aptly labeled macro-areas. Discoveries and mathematical ideas that are perceived as “useful” today because they’re applicable to engineering, for example, were at a certain point in time considered absolutely abstract and useless, or at least derived or intrinsically connected to some that were. Mathematics matters; all of it.

On the net I found an incredible lecture by the brilliant mathematician Timothy Gowers, entitled “The Importance of Mathematics”. In this keynote, Prof. Gowers makes a very strong case in favor of the value of math, of financing its relatively cheap research and its deep implications on human progress. You can watch the 8 parts that compose the whole video, in the following playlist:

For those who’d prefer it, a PDF transcript is also available.

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.

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