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, and of course, you can also get “Flatland: A Romance of Many Dimensions”, dirt cheap (a classic geek novel for less than 4 bucks). If you prefer, you could pick up the annotated hardcover version: “The Annotated Flatland: A Romance of Many Dimensions”. Highly recommended.

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

Ten Must Read Books about Mathematics

Filed Under Essential Math, Suggested reading | 37 Comments

Euler’s FormulaI 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.

  1. 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.


  2. 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.


  3. 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: \displaystyle e^{\i\pi} + 1 = 0. 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.


  4. 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.


  5. 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 .


  6. 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.


  7. 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.


  8. 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.


  9. 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.


  10. 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.“.

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. :-)

Jun

4

A 10 minute tutorial for solving Math problems with Maxima

Filed Under Essential Math, Software | 81 Comments

About 50,000 people read my article 3 awesome free Math programs. Chances are that at least some of them downloaded and installed Maxima. If you are one of them but are not acquainted with CAS (Computer Algebra System) software, Maxima may appear very complicated and difficult to use, even for the resolution of simple high school or calculus problems. This doesn’t have to be the case though, whether you are looking for more math resources to use in your career or a student in an online bachelor’s degree in math looking for homework help, Maxima is very friendly and this 10 minute tutorial will get you started right away. Once you’ve got the first steps down, you can always look up the specific function that you need, or learn more from Maxima’s official manual. Alternatively, you can use the question mark followed by a string to obtain in-line documentation (e.g. ? integrate). This tutorial takes a practical approach, where simple examples are given to show you how to compute common tasks. Of course this is just the tip of the iceberg. Maxima is so much more than this, but scratching even just the surface should be enough to get you going. In the end you are only investing 10 minutes.

Maxima as a calculator

You can use Maxima as a fast and reliable calculator whose precision is arbitrary within the limits of your PC’s hardware. Maxima expects you to enter one or more commands and expressions separated by a semicolon character (;), just like you would do in many programming languages.

(%i1) 9+7;
(%o1) 16
(%i2) -17*19;
(%o2) -323
(%i3) 10/2;
(%o3) 5

Maxima allows you to refer to the latest result through the % character, and to any previous input or output by its respective prompted %i (input) or %o (output). For example:

(%i4) % - 10;
(%o4) -5
(%i5) %o1 * 3;
(%o5) 48

For the sake of simplicity, from now on we will omit the numbered input and output prompts produced by Maxima’s console, and indicate the output with a => sign. When the numerator and denominator are both integers, a reduced fraction or an integer value is returned. These can be evaluated in floating point by using the float function (or bfloat for big floating point numbers):

8/2;
=> 4
8/2.0;
=> 4.0
2/6;
=> \displaystyle \frac{1}{3}
float(1/3);
=> 0.33333333333333
1/3.0;
=> 0.33333333333333
26/4;
=> \displaystyle \frac{13}{2}
float(26/4);
=> 6.5

As mentioned above, big numbers are not an issue:

13^26;
=> 91733330193268616658399616009
13.0^26
=> \displaystyle 9.1733330193268623\text{ }10^_{+28}
30!;
=> 265252859812191058636308480000000
float((7/3)^35);
=> \displaystyle 7.5715969098311943\text{ }10^_{+12}

Constants and common functions

Here is a list of common constants in Maxima, which you should be aware of:

  • %e - Euler’s Number
  • %pi - \displaystyle \pi
  • %phi - the golden mean (\displaystyle \frac{1+\sqrt{5}}{2})
  • %i - the imaginary unit (\displaystyle \sqrt{-1})
  • inf - real positive infinity (\infty)
  • minf - real minus infinity (-\infty)
  • infinity - complex infinity

We can use some of these along with common functions:

sin(%pi/2) + cos(%pi/3);
=> \displaystyle \frac{3}{2}
tan(%pi/3) * cot(%pi/3);
=> 1
float(sec(%pi/3) + csc(%pi/3));
=> 3.154700538379252
sqrt(81);
=> 9
log(%e);
=> 1

Defining functions and variables

Variables can be assigned through a colon ‘:’ and functions through ‘:=’. The following code shows how to use them:

a:7; b:8;
=> 7
=> 8
sqrt(a^2+b^2);
=> \sqrt{113}
f(x):= x^2 -x + 1;
=> x^2 -x + 1
f(3);
=> 7
f(a);
=> 43
f(b);
=> 57

Please note that Maxima only offers the natural logarithm function log. log10 is not available by default but you can define it yourself as shown below:

log10(x):= log(x)/log(10);
=> \displaystyle log10(x):=\frac{log(x)}{log(10)};
log10(10)
=> 1

Symbolic Calculations

factor enables us to find the prime factorization of a number:

factor(30!);
=> \displaystyle 2^{26}\,3^{14}\,5^7\,7^4\,11^2\,13^2\,17\,19\,23\,29

We can also factor polynomials:

factor(x^2 + x -6);
=> (x-2)(x+3)

And expand them:

expand((x+3)^4);
=> \displaystyle x^4+12\,x^3+54\,x^2+108\,x+81

Simplify rational expressions:

ratsimp((x^2-1)/(x+1));
=> x-1

And simplify trigonometric expressions:

trigsimp(2*cos(x)^2 + sin(x)^2);
=> \displaystyle \cos ^2x+1

Similarly, we can expand trigonometric expressions:

trigexpand(sin(2*x)+cos(2*x));
=> \displaystyle -\sin ^2x+2\,\cos x\,\sin x+\cos ^2x

Please note that Maxima won’t accept 2x as a product, it requires you to explicitly specify 2*x. If you wish to obtain the TeX representation of a given expression, you can use the tex function:

tex(%);
=> $$-\sin ^2x+2\,\cos x\,\sin x+\cos ^2x$$

Solving Equations and Systems

We can easily solve equations and systems of equations through the function solve:

solve(x^2-4,x);
=> \displaystyle \left[ x=-2 , x=2 \right]
%[2]
=> x=2
solve(x^3=1,x);
=> \displaystyle \left[ x={{\sqrt{3}\,i-1}\over{2}} , x=-{{\sqrt{3}\,i+1}\over{2}}  , x=1 \right]
trigsimp(solve([cos(x)^2-x=2-sin(x)^2], [x]));
=> \displaystyle \left[ x=-1 \right]
solve([x - 2*y = 14,  x + 3*y = 9],[x,y]);
=> \left[ \left[ x=12 , y=-1 \right]  \right]

2D and 3D Plotting

Maxima enables us to plot 2D and 3D graphics, and even multiple functions in the same chart. The functions plot2d and plot3d are quite straightforward as you can see below. The second (and in the case of plot3d, the third) parameter, is just the range of values for x (and y) that define what portion of the chart gets plotted.

plot2d(x^2-x+3,[x,-10,10]);

2dplot.png

plot2d([x^2, x^3, x^4 -x +1] ,[x,-10,10]);

many_2dplot.png

f(x,y):= sin(x) + cos(y);
plot3d(f(x,y), [x,-5,5], [y,-5,5]);

3dplot.png

Limits

limit((1+1/x)^x,x,inf);
=> %e
limit(sin(x)/x,x,0);
=> 1
limit(2*(x^2-4)/(x-2),x,2);
=> 8
limit(log(x),x,0,plus);
=> -\infty
limit(sqrt(-x)/x,x,0,minus);
=> -\infty

Differentiation

diff(sin(x), x);
=> \displaystyle cos(x)
diff(x^x, x);
=> \displaystyle x^{x}\,\left(\log x+1\right)

We can calculate higher order derivatives by passing the order as an optional number to the diff function:

diff(tan(x), x, 4);
=> \displaystyle 8\,\sec ^2x\,\tan ^3x+16\,\sec ^4x\,\tan x

Integration

Maxima offers several types of integration. To symbolically solve indefinite integrals use integrate:

integrate(1/x, x);
=> \displaystyle log(x)

For definite integration, just specify the limits of integrations as the two last parameters:

integrate(x+2/(x -3), x, 0,1);
=> \displaystyle -2\,\log 3+2\,\log 2+{{1}\over{2}}
integrate(%e^(-x^2),x,minf,inf);
=> \sqrt{\% pi}

If the function integrate is unable to calculate an integral, you can do a numerical approximation through one of the methods available (e.g. romberg):

romberg(cos(sin(x+1)), x, 0, 1);
=> 0.57591750059682

Sums and Products

sum and product are two functions for summation and product calculation. The simpsum option simplifies the sum whenever possible. Notice how the product can be use to define your own version of the factorial function as well.

sum(k, k, 1, n);
=> \displaystyle \sum_{k=1}^{n}{k}
sum(k, k, 1, n), simpsum;
=> \displaystyle {{n^2+n}\over{2}}
sum(1/k^4, k, 1, inf), simpsum;
=> \displaystyle {{\%pi^{4}}\over{90}}
fact(n):=product(k, k, 1, n);
=> fact(n):=product(k,k,1,n)
fact(10);
=>  3628800

Series Expansions

Series expansions can be calculated through the taylor method (the last parameter specifies the depth), or through the method powerseries:

niceindices(powerseries(%e^x, x, 0));
=> \displaystyle \sum_{i=0}^{\infty }{{{x^{i}}\over{i!}}}
taylor(%e^x, x, 0, 5);
=> \displaystyle 1+x+{{x^2}\over{2}}+{{x^3}\over{6}}+{{x^4}\over{24}}+{{x^5}\over{120 }}+\cdots

The trunc method along with plot2d is used when taylor’s output needs to be plotted (to deal with the +\cdots in taylor’s output):

plot2d([trunc(%), %e^x], [x,-5,5]);

taylor.png

I hope you’ll find this useful and that it will help you get started with Maxima. CAS can be powerful tools and if you are willing to learn how to use them properly, you will soon discover that it was time well invested.

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