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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:
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:
The width of the gap between the longer rope and the object that’s taken into consideration is R-r.
Therefore:
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.
Quite 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.
Sep
9
Thought-provoking Mathematical Videos
Filed Under Essential Math, Math Education, Suggested reading | 14 Comments
1. The Tenth Dimension
2. Outside in (Turning a sphere inside out)
3. Flatland the film (Trailer)
The full movie is available on DVD
4. Math Education: An Inconvenient Truth
This short video shows what’s wrong with the current widely adopted methods of teaching mathematics (fortunately though, such practices have not caught on everywhere).
5. Math Education: A University View
You can consider this video a follow-up to the previous one. Clearly this education reform affects elementary school aged children, but the effects that it has on curricula at an early level also profoundly goes on to influences the education which is received by students at high school and even college levels.
Please note that we are now accepting authors and submissions for this website.
Jul
17
I love books with the ability to inspire readers. Many non-mathematicians consider mathematics as something abstruse and complicated, suitable only for ‘nerds’. Often I highlight the unfounded nature of this prejudice, but nothing is more effective at disproving this stigma than a good book. I was in fact able to quickly change many of my friends’ views on the topic, by just giving them a good book which shows the beauty and fascinating nature of mathematics and science in general. The following is a list of great titles, most of which are fairly cheap. Not all of them are suitable for the mathematically illiterate though, and thus cannot simply be considered popular science. In the description I’ll give you fair warning if a particular title is better suited to those who are more advanced when it comes to math.
- The Man Who Loved Only Numbers: an original biography of the genius Paul Erdős, who was arguably the most prolific mathematician of the last century , renowned for being just as much of an eccentric as a math whiz. This book won its author a 1999 Aventis Prize for Science Books and you can watch a lecture that the author, Paul Hoffman, gave about the subject. As you can expect for such a unique mathematician, this book is filled with anecdotes and fascinating facts. If you are like me, you’ll buy a copy for yourself and will end up buying copies for your friends as a means of providing them with a light and interesting reading.
- An Imaginary Tale: The Story of “i” [the square root of minus one]: complex numbers are what puzzle many non-mathematicians the most. It’s intuitively easy to explain Rational and Real numbers to the layman, but complex numbers are often seen as something mysterious. In this book, Nahin goes the extra mile in his attempt to provide historical details as well as insight into the motivation behind complex analysis, offering a serious introduction to the topic that will also serve many mathematically inclined high schoolers and freshmen well.
- Dr. Euler’s Fabulous Formula: Cures Many Mathematical Ills: the author, Nahin, follows up his first book above with this gem - with a somewhat ridiculous title - about the most beautiful equation in the history of mathematics:
. It’s a delightful read, but beware that the author cuts to the chase in this one, and expects from the reader a solid understanding of complex numbers, as he exposes the application in various fields and covers advanced topics such as Fourier Series and Integrals (dedicating a chapter to each of them). Therefore I would consider the book “An Imaginary Tale” above, a prerequisite before approaching this book. - Godel, Escher, Bach: An Eternal Golden Braid: this book is one of the most famous bestsellers in the world, and should have a spot in any technically minded person’s library. I would argue that it is a particularly good read for programmers. It’s really hard to give justice to this tome in a few lines, so if you want you can read more about it through the reviews on Amazon or through its wikipedia entry.
- Mathematics for the Nonmathematician: history and methodologies of mathematics are well covered in this very inexpensive title. It combines two aspects which are difficult to match: it’s a page-turner like many math popular titles, while being instructional as well as an effective introductory text to basic mathematics for students and amateur mathematicians alike .
- God Created the Integers: The Mathematical Breakthroughs That Changed History: the historic introduction to some of the greatest mathematicians who’ve ever walked the face of the earth is worth the cover price alone. But this book is so much more than that, covering a wide range of mathematical topics which have been developed throughout history, in an accessible but rigorous way. It is admittedly more challenging than your average popular math title, but if you already have some mathematical basics mastered and are willing to work through it, you’ll gain a lot of insight about the nature of mathematics and the discoveries made by the giants of math from this excellent book.
- Fermat’s Last Theorem: if you are interested in learning more about the history and fascinating tales which surround one of the most well known theorems, this book will provide you with a marvelous and entertaining way to spend a Sunday afternoon. My wife who is not a mathematician, simply loved this book for its rich story telling and coverage of a topic with substantial historical significance.
- The Code Book: The Science of Secrecy from Ancient Egypt to Quantum Cryptography: another intriguing title, by the same author of Fermat’s Last Theorem above. This book will get you excited about the topic of encryption and its history, from the time of Caesar right up to the future direction which encryption is taking in today’s technology based world. It currently has 5 stars on Amazon with 232 positive reviews out of 234.
- To Infinity and Beyond: What’s infinity? What it is its impact on mathematics and what are its cultural implications that it holds? These questions are clearly answered in this book which provides a beautiful exposition that is accessible to anyone. Read this book and chances are you’ll feel a sense of enlightenment as you soak up the words of this amazing writer. I would particularly recommend it as a gift for teenagers in high school, as a way of getting them interested in mathematics. This book will provide them with an essay on the reasons behind the study of Calculus and the practical implications within the areas of Art and Astronomy as well.
- Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics: unsolved number theory problems are a huge fascination for many mathematicians and hobbyists. It’s a fun field full of challenges and discoveries just waiting to spring forth. ‘Prime Obsession’ focuses on the Riemann’s Hypothesis, the most important unsolved problem in Mathematics. A reviewer on Amazon does an excellent job at describing the beauty of this book, quoting him: “Prime Obsession is a delight: a book about a hypothesis on the distribution of prime numbers that reads like a gripping mystery. Most fiction isn’t this vivid, moving, and well written, and this is no fiction. It is history, biography, philosophy, and, yes, mathematics brought to life with wit and wonder. You have to read this extraordinary book. This is the story of the Reimann Hypothesis, the greatest unsolved problem in mathematics today.“.
: an original biography of the genius 
: complex numbers are what puzzle many non-mathematicians the most. It’s intuitively easy to explain Rational and Real numbers to the layman, but complex numbers are often seen as something mysterious. In this book, Nahin goes the extra mile in his attempt to provide historical details as well as insight into the motivation behind complex analysis, offering a serious introduction to the topic that will also serve many mathematically inclined high schoolers and freshmen well.
: the author, Nahin, follows up his first book above with this gem - with a somewhat ridiculous title - about the most beautiful equation in the history of mathematics: 
: this book is one of the most famous bestsellers in the world, and should have a spot in any technically minded person’s library. I would argue that it is a particularly good read for programmers. It’s really hard to give justice to this tome in a few lines, so if you want you can read more about it through the reviews on Amazon or through its 
: the historic introduction to some of the greatest mathematicians who’ve ever walked the face of the earth is worth the cover price alone. But this book is so much more than that, covering a wide range of mathematical topics which have been developed throughout history, in an accessible but rigorous way. It is admittedly more challenging than your average popular math title, but if you already have some mathematical basics mastered and are willing to work through it, you’ll gain a lot of insight about the nature of mathematics and the discoveries made by the giants of math from this excellent book.
: if you are interested in learning more about the history and fascinating tales which surround one of the most well known theorems, this book will provide you with a marvelous and entertaining way to spend a Sunday afternoon. My wife who is not a mathematician, simply loved this book for its rich story telling and coverage of a topic with substantial historical significance.
: another intriguing title, by the same author of Fermat’s Last Theorem above. This book will get you excited about the topic of encryption and its history, from the time of Caesar right up to the future direction which encryption is taking in today’s technology based world. It currently has 5 stars on Amazon with 
: What’s infinity? What it is its impact on mathematics and what are its cultural implications that it holds? These questions are clearly answered in this book which provides a beautiful exposition that is accessible to anyone. Read this book and chances are you’ll feel a sense of enlightenment as you soak up the words of this amazing writer. I would particularly recommend it as a gift for teenagers in high school, as a way of getting them interested in mathematics. This book will provide them with an essay on the reasons behind the study of Calculus and the practical implications within the areas of Art and Astronomy as well.
: unsolved number theory problems are a huge fascination for many mathematicians and hobbyists. It’s a fun field full of challenges and discoveries just waiting to spring forth. ‘Prime Obsession’ focuses on the Riemann’s Hypothesis, the most important unsolved problem in Mathematics. A reviewer on Amazon does an excellent job at describing the beauty of this book, quoting him: “Prime Obsession is a delight: a book about a hypothesis on the distribution of prime numbers that reads like a gripping mystery. Most fiction isn’t this vivid, moving, and well written, and this is no fiction. It is history, biography, philosophy, and, yes, mathematics brought to life with wit and wonder. You have to read this extraordinary book. This is the story of the Reimann Hypothesis, the greatest unsolved problem in mathematics today.“.In the comments below feel free to share your thoughts on these books (if you have read any of them) and add other to the list which are near and dear to your own mathematical heart.
… No differential equations were harmed in the making of this post. 