1. Some of our readers are likely familiar with your work, but could you tell us more about yourself and your mathematical background?

I majored in mathematics as an undergraduate at Yale, from which I graduated in 1976. I think I even won a couple of math prizes, but I have to confess that I wasn’t the top mathematician in my class. That distinction would surely have gone to Jonathan Rogawski, who last I knew was a professor of mathematics at UCLA. (Notice that I just created the impression that I was the second-best mathematician in my class. I don’t know whether that’s true, but I’ll take it.)

Anyway, I went on to get a Ph.D. in mathematics at M.I.T. and have remained in the Boston area ever since. I went into the investment business in the early 1980s, based on the assumption that quantitative expertise would be a good match. But the truth is that I got progressively more qualitative as time went by, going from securities analyst to investment writer. I don’t know whether that transition made complete sense, but it ultimately gave me the opportunity to write some books – first about investments and then about numbers, including several volumes of puzzle books.

2. What inspired you to write Number Freak?

I was asked by a publisher to come up with a concept that would do for mathematics what a slightly different concept did for the natural sciences. The idea I came up with was more of a coffee-table book than the sized-down version I now have in my hands, but that effort was considered too expensive. I subsequently cast a wider net for the project, and was fortunate enough to attract publishers in the U.S., the U.K., and Australia.

3. The book is chock-full of interesting facts about the first 200 natural numbers. What did you learn in the process of writing this book that you didn’t know before?

Well, I guess the pat answer is that I learned how little I actually knew. Some of the work on planar tilings was new to me, even though it probably shouldn’t have been – for example, the Archimedean and Laves tilings I discuss in #11 are quite beautiful but I hadn’t been aware of their categorization and duality. And I wasn’t familiar with the work of mathematicians such as Erich Friedman of Stetson University, somebody who surely could have pulled off a book like this: I was only too happy, for example, to include “Friedman numbers” such as 127.

In self-defense, I wasn’t a complete neophyte. One big advantage I had in writing the book – apart from doing it in the Internet age, which gave me an abundance of material – was that I have a good memory for mathematical and pop culture trivia. For example, I enjoyed reaching back and remembering that the ultra-high security “D” block at Alcatraz prison had precisely 42 individual cells, something that meshed quite nicely with the picture of the “magic cube” I displayed elsewhere in the discussion of #42.

4. Having read this book I feel that it’s accessible to virtually anyone. Who do you feel is the ideal target audience for the book?

Boy is that a good question. My answer is that it’s for absolutely anyone, but if that’s too mealy-mouthed a reply, I guess I would say that I’d be especially pleased if parents bought Number Freak to (successfully!) introduce their kids to the world of numbers in a way that maybe, just maybe, is friendlier than what those kids were getting elsewhere.

5. Was there anything that you wish you could have included in the book but didn’t?

Another good question, and I’m afraid a painful one. The book was originally slated to go from 1 to 300 — as in a perfect game in bowling, among other things — but the editorial powers-that-be eventually whittled that down to 200. Too bad, as my discussion of the infamous 256th level of Pac-Man was worth the price of admission. (Say, that’s a topic I didn’t know about when I started the book!) I also lost some precious photos, charts and diagrams along the way. And you can imagine how I felt when a friend berated me for not mentioning “77 Sunset Strip,” when of course my original manuscript mentioned the show – and I have a photograph of Efrem Zimbalist, Jr. to prove it! (Those of my vintage – I’m 54 – will remember the show’s catchy theme song, but not many are aware that 77 was a particular good choice for the street address because it is the smallest integer whose English pronunciation requires five syllables.)
Other than that, I deliberately went easy on the cult surrounding the number 23, for example, and left a bunch of numerology and religious interpretations for somebody else to ponder. That’s another book all by itself.

6. What’s the answer to life, the universe and everything?

Why it’s 42, of course. You know, I had already answered question #3 above before I saw this one!

7. What’s your favorite number and why?

When I started the book, 17 had the edge. First of all, “At 17” by Janis Ian is probably my favorite song of all time. It came out in 1975, which was my favorite music year of all time. (Perhaps I should have written it in 1975.) But 17 is famous in mathematics for Carl Friedrich Gauss’s famous straightedge-and-compass construction of a regular 17-gon, for the 17 “wallpaper” symmetries of the plane, and for the fact that if you connect 17 suitably spaced dots with a segment of red, blue, or green, you will automatically create a “monochromatic” triangle whose three vertices are among the original 17 dots. And nobody has yet created a solvable Sudoku puzzle with fewer than 17 original entries. How about that?

But by the time I finished Number Freak, my favorite number had become 36. What happened is that while doing research for the book I came across a conjecture from the 18th century called the 36 Officer Problem. I had never heard of it before (yet another example!), perhaps because the problem was resolved in the early 20th century and then ceased to be of interest. But there was a three-dimensional wrinkle to the problem that hadn’t been explored, and I used that wrinkle to design a puzzle with a gray base and 36 towers of various colors. I went to Toy Fair and showed the puzzle to ThinkFun, a great game and puzzle company out of Alexandria, Virginia. And guess what? They made me a deal for the puzzle and after a year tinkering with the basic model, they launched it as “36 Cube” in the fall of 2008—many months before Number Freak came out! I was thrilled that the lessons of the book came to life in such a tangible way, so I’d be lying if I didn’t admit that 36 holds a very special place in my heart.

Thank you very much, Derrick, for your insightful answers. And to our readers, if you haven’t already done so, check out his book.

The gambler’s fallacy, also known as the Monte Carlo fallacy or the fallacy of the maturity of chances, is the belief that if deviations from expected behavior are observed in repeated independent trials of some random process then these deviations are likely to be evened out by opposite deviations in the future.

This definition may seem a bit abstract, so let’s clarify it through a practical example. What’s the probability of flipping a fair coin 10 times in a row and obtaining heads consecutively each time? The answer is:

.

This would be very unlikely. How unlikely? One in 1,024 to be exact. So if we’ve just observed the coin appear as heads 9 times in a row, what are the odds that the same coin will land on heads on the 10th toss?

Many people would argue that the chance of this happening is less than one in a thousand, as we just calculated. However, that answer is blatantly wrong. The probability that the 10th fair coin toss is going to come up as heads is still 0.5, because each trial (toss) is statistically independent from those that preceded it. Tossing 9 heads in a row is very unlikely, however once it has happened, it doesn’t influence the outcome of the 10th toss in any way.

People who fall for this fallacy, do so because of a fundamental misunderstanding of how probability works. They combine the probability of past events (irrelevant for independent trials), with that of future events. With the example above, some people would also erroneously conclude that “tails is long due to come up” and as such would think that it’s more likely to occur.

This informal fallacy has contributed to the ruin of many gamblers over the years. A tragic example of what happens when you uphold this way of looking at odds occurs with many who play the game of “Lotto” in Italy, a very popular lottery game played amongst the general population.

The idea behind this game is very simple. Five distinct numbers between 1 and 90 are randomly selected in 10 different Italian cities, three times a week. Gamblers can place several types of bets, but the one we’re interested in, for the sake of this article, is called the “estratto semplice” (simple draw). This type of game requires gamblers to correctly predict that a specific number will be drawn in a particular city.

The probability of placing a winning bet is 1 in 18 (i.e., 5/90), while the payout is 11.232 times the amount that you put down (so if you bet 1 Euro and won, you’d walk away with 11.23 Euros before taxes). The odds are clearly stacked in favor of the house, of course. Incidentally, Lotto is run by the state and as is also known as “a tax on the stupid” for rather obvious reasons.

There are many “systems” and theories used by a large pool of gamblers who want to “beat the system”. More often then not such systems are based on some flawed understanding of how probability really works. A very popular theory is that of the “numeri ritardatari” (“late numbers”, as we will refer to them throughout this article). The basic principle behind late numbers is this: since it’s extremely unlikely that a given number will fail to appear at least once out of 150, 180 or 200 draws in a row, in a given city, you can identify what numbers are “due” to appear and thus bet on them. For example, if a number hasn’t been drawn in the past 140 trials, the number of bets on it will start to grow very quickly.

Of course, despite the fact that a number hasn’t come up in a given city 140 times in a row, its probability of occurring on the next draw is still just 1 in 18. So betting any of the other 89 numbers would yield the same probability of winning.

The application of this fallacy becomes extremely dangerous when coupled with Martingale betting systems, which are often adopted by “late number theorists”. The theory they use is very simple. Since they assume these late numbers are “due” very soon, they think they are going to be able to afford to put down double their previous wager on every bet until the number eventually appears. So when it does happen, the last sum they bet is multiplied 11 times (for the payout) and they will recoup all the money they’ve spent up until then, and end up netting a large additional payout, which is the (last wager x 9.232 + 1) Euros.

Martingale betting systems are guaranteed to work provided that the gambler has an infinite amount of capital and no limits are imposed on the maximum bet that’s allowed to be placed. In the real world, both of these requirements cannot be realistically met. The amount bet grows exponentially, so the Martingale system ends up being a surefire way to bankrupt those who employs it.

In the case of the Italian Lotto, both the fallacy that late numbers are “due” and the choice of betting systems (Martingale) are responsible for the ruin of many. The gambler’s fallacy plays an important role in this case because most people realize that they can’t sustain a Martingale type system for 200 consecutive draws. It’s their faith in the idea that late numbers are very likely to pop up soon, that tempts them into toying with this risky system.

If we assume these people are convinced that a very late number (say, one that hadn’t been drawn in the past 180 lottery draws) will be selected at some point during the next 5 weeks or so (15 trials), and that they’re starting with a bet of one Euro, we can see that the maximum amount they’d need to invest (according to their theory) would be 32,768 Euros, with a max bet of 16,384 Euros by the 15th draw. This is a sizable sum of money, but something that some people would still be able to put down, especially because they knew they payout would be 184,025.088 Euros (before taxes). A tempting prize indeed.

But what are the real odds that the number in question, the one that’s been eluding the gamblers, will not end up occurring at least once in the next 15 draws?

So there is a 42.43% risk that the punter will lose their 32,768 Euros, because they won’t have sufficient funds to double their wager at the next turn (assuming 32,768 Euros was the maximum amount they can afford to bet).

Bear in mind that with an exponential growth of the bet, a huge amount of capital will only afford our late number gamblers a few extra draws, thereby only slightly increasing their probability of making a profit. (With a payout of 11.232 times the wager, they could afford a smaller increase in the amount of money they put down draw by draw, but the overall principle remains the same.)

What has an adoption of this faulty theory led to in Italy? What kind of impact has it really had on those who adhere to it? The honest truth is that it’s gone so far as to contribute directly to things like suicides, people swindling their friends and employers, divorces, people betting their life savings and their homes, families being destroyed, and so on. Do such dire consequences occur to everyone who plays this game? No, of course not, but the fact that it’s happened to some people, and that these flawed theories are still employed today, is indicative of the misunderstanding about probability (and the risks of gambling) that occurs in the general population.

One could – and should – argue that such peoples’ demise is due to their gambling habits and to good old fashioned greed, yet I can’t help but feel that a solid understanding of probability theory would go a tremendous way in helping to cut down on the number of people who fall prey to these types of widespread theories.

An increased awareness of probability and statistics can only improve society and its ability to assess situations and make rational decisions. How do we begin to remedy this situation, not only in Italy, but around the world? We can start by devoting far more time in grade, middle and high school math classes, in order to teach students about this important subject and the implications that it can have on their everyday lives, understanding of society, and ability to make wise financial decisions.

Many parents face the challenge of helping their children with math homework, which for some stems in part to having developed a strong phobia or dislike of the subject themselves. Along with a psychological component, in many cases the challenge is augmented by a lack of basic skills (when it comes to knowing how to approach math problems and work their way through mathematical nomenclature). For some it’s like trying to help their child with French homework, when they don’t speak the language. Otherwise perfectly intelligent adults end up finding themselves worrying over problems that most math-savvy people would consider straightforward.

Math for Moms and Dads tries to solve this predicament by providing a vocabulary of essential terms, a very gentle introduction to problem solving and mathematical reasoning, fundamental concepts of elementary (primary) and middle school mathematics, and step-by-step solutions to basic exercises. It also stresses the importance of the parent-child and parent-teacher relationships when it comes to teaching and assisting with the learning of math. This book is very basic and relatively short, which means that it’s something most parents would be able to squeeze time into their schedule to read (which I feel is a positive element of this book). As someone with a passion for math, I’m biased and admit that I do not find this type of book terribly exciting myself, but I fully realize its usefulness for people who need a “less than scary” introduction (or refresher) to the subject.

The first chapter introduces the book and provides parents with a few pointers on how to use a calculator and when its usage is appropriate. The content on these pages will appear pretty obvious to a large number of readers, but this book tries not to make any assumptions, and as such it aims to cover concepts that many people might take for granted.

Chapter two details the mathematical vocabulary mentioned earlier in this article, and within this chapter parents will learn about fundamental math terminology, including terms such as absolute value, congruent, coordinate plane, diagonal, fraction, permutations and so on. The second part of the chapter provides the reader with more descriptive information about common, basic concepts like commutative and associative property, prime and composite numbers, rational and irrational numbers, union and intersection, linear and quadratic equations, etc.

Chapter three covers the basic rules necessary for resolving a variety of problems, including order of operations, exponents and their rules, properties of numbers, fraction and integer based arithmetic, expressions and equations, and so on.

Chapters four and five tackle the issue of solving homework exercises and preparing for math tests. Together these chapters help clarify how to approach mathematical problems, with examples that are solved in a step-by-step manner.

Chapter six is a pedagogical chapter about how to approach study, which covers topics such as how to create the right study conditions and find the ideal place in your house to turn into a homework area, as well as how to develop note taking and test preparation skills.

Chapter seven is entitled “When will I use this, anyway?”, and it attempts to convince both parents and their children that learning mathematics is an important and useful real world skill. I felt that this chapter (which is about a subject – the importance of math beyond the classroom – I believe strongly in) was on the weaker side, but it may still be useful to some.

Lastly chapter 8 deals with parent-teacher communication, a topic that I felt was important for this kind of book.

Should you feel that your own math skills are not your strongest suit or if you need a concise and easy to follow along with refresher course on numerous basic math topics, so that you can better assist your child with their studies, you will likely find this book right up your alley.

If you are a publisher and would like to have your books reviewed, please contact me at antonio@math-blog.com. As a policy, we will only publish reviews for book worth recommending, informing the publisher if a book doesn’t meet (in our opinion) the standard.

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.

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.

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: . 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.

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) 
(%i2) -17*19;
(%o2) 
(%i3) 10/2;
(%o3) 

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) 
(%i5) %o1 * 3;
(%o5) 

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;
=> 
8/2.0;
=> 
2/6;
=> 
float(1/3);
=> 
1/3.0;
=> 
26/4;
=> 
float(26/4);
=> 

As mentioned above, big numbers are not an issue:

13^26;
=> 
13.0^26
=> 
30!;
=> 
float((7/3)^35);
=> 

Constants and common functions

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

• %e – Euler’s Number
• %pi –
• %phi – the golden mean ()
• %i – the imaginary unit ()
• inf – real positive infinity ()
• minf – real minus infinity ()
• infinity – complex infinity

We can use some of these along with common functions:

sin(%pi/2) + cos(%pi/3);
=> 
tan(%pi/3) * cot(%pi/3);
=> 
float(sec(%pi/3) + csc(%pi/3));
=> 
sqrt(81);
=> 
log(%e);
=> 

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;
=> 
=> 
sqrt(a^2+b^2);
=> 
f(x):= x^2 -x + 1;
=> 
f(3);
=> 
f(a);
=> 
f(b);
=> 

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);
=> 
log10(10)
=> 

Symbolic Calculations

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

factor(30!);
=> 

We can also factor polynomials:

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

And expand them:

expand((x+3)^4);
=> 

Simplify rational expressions:

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

And simplify trigonometric expressions:

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

Similarly, we can expand trigonometric expressions:

trigexpand(sin(2*x)+cos(2*x));
=> 

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);
=> 
%[2]
=> 
solve(x^3=1,x);
=> 
trigsimp(solve([cos(x)^2-x=2-sin(x)^2], [x]));
=> 
solve([x - 2*y = 14,  x + 3*y = 9],[x,y]);
=> 

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]);

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

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

Limits

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

Differentiation

diff(sin(x), x);
=> 
diff(x^x, x);
=> 

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

diff(tan(x), x, 4);
=> 

Integration

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

integrate(1/x, x);
=> 

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

integrate(x+2/(x -3), x, 0,1);
=> 
integrate(%e^(-x^2),x,minf,inf);
=> 

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);
=> 
sum(k, k, 1, n), simpsum;
=> 
sum(1/k^4, k, 1, inf), simpsum;
=> 
fact(n):=product(k, k, 1, n);
=> 
fact(10);
=>  

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));
=> 
taylor(%e^x, x, 0, 5);
=> 

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

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

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.

It’s not all bad though; in fact I was already planning to switch to a more serious provider with VPS or dedicated server plans. After endless research I picked Liquid Web which is fully managed, provides root access and support 24/7 on the phone. So far so good, they worked quite a bit on tweaking my server and I’ve actually called them at 3 and 5 AM and got a person on the phone in less than 30 seconds. I don’t know how well they will be able to cope with the “Digg effect” though, only time will tell.

With my “Refresh your High School Math” article on the front page of several social websites, the amount of feedback received has been terrific. It also allowed me to confirm a theory that I’ve always thought about: there is little consistency and standardization in the teaching of mathematics. I say this because the reactions to my basic math test were very highly varied. Many people said they were not able to solve the problems. That’s sad given the admittedly basic nature of the questions, but it wasn’t a big surprise. You could consider it the FizzBuzz of mathematics. What strikes me the most though, is how many people considered the test to be rightfully “middle school” material and far too basic (except for some parts of it), while others argued that it was way too advanced and too difficult for high school mathematics. This variety of reactions shows that the topics and depth of coverage in math classes in high school are quite different throughout the world.

So I’d like to explicitly ask a question to my readers, what topics did you cover while studying mathematics in high school and in what state/country were you? Please use the comment section to answer, thank you.

One person commented on the blog, and another wrote me privately, to express their concern that “harder books are not necessarily better books” and that teaching which is geared towards only the smartest kids is a mistake. I want to point out that I’m in no way advocating teaching for the brightest minds only. Wide access to mathematics is something that should be encouraged all over the world, and I’m pretty sure it will help take society in a better direction. In fact, education in general – and mathematics, technical and scientific education in particular – are key for the development of every country and ultimately for good of humankind.

However my point was that with wider access to higher education mathematics, we should not reduce the expected and established testing standards. In other words, there is a fair level of understanding that we should expect from people who major in math or from students who strongly depend on mathematics for their future careers. Furthermore, the textbooks adopted should be mathematically sound and provide the right intellectual stimulation for those who could use it. That said, there is nothing wrong with teachers trying to use different styles of teaching to reach a wider audience, or for students who struggle with the level of math presented in the textbook, to supplement it with simpler books in order to get an easier start. Hence, it’s perfectly OK for a student (who for example is taking an undergraduate class in programming in C) to read C For Dummies if The C Programming Language by K&R is too hard for them off the bat. But that does not imply that the class should adopt “C for Dummies” as their textbook nor that the examination should be based on such a book. So to summarize this point, feel free to study any number of introductory books, as long as you know that if you plan to be serious about mathematics, you should be able to eventually read and understand standard books and be able to solve most of the exercises put forward in them.

Having clarified the first concern, I’d like to provide an answer for the second point, which actually interests me the most. A few readers wrote me emails about how they feel enthusiastic about the post and the opportunity to study mathematics again, but how those books are way too advanced for them, because they simply forgot all the mathematics taught at a high school level. So I’ve received a few “how can I get a refresher of high school math?” type of questions. The mathematics that you learned in high school is classified as pre-calculus, and as you can expect it is propaedeutic to learn math at an higher level. It is normal that you forgot quite a few formulas, but having a good grasp of the essentials of precalculus can make a big difference when trying to master calculus. You should have a decent knowledge of basic algebra, trigonometry, exponential, logarithmic, and analytic geometry. Calculus itself will provide you with a refresher of some of these topics and give you a deeper understanding not only of “how” but rather “why”. That said, Calculus without a decent precalculus base can be a big challenge for most people. Before proceeding to suggest a few resources, let’s try to establish if you actually need a refresher course or not. Here is a (simple and of course incomplete) list of some basic exercises. If you haven’t a clue or struggle to find a lot of the solutions for them, a refresher may be in order.

Simple Precalculus Questions:

1) Factor the following polynomials:

2) Solve for x:

3) Find the values of x for which:

4) Evaluate:

5) Solve for x:

6) Solve for x, where :

7) Write the equations of the following curves in the Cartesian plane:

1. Parabola
2. Hyperbola
3. Circle
4. Ellipse

8 ) Find the vertex, focus, and directrix of the parabolas given by the equations:

9) Find the center, vertices, foci, and eccentricity of the hyperbola given by the equation:

10) Find the equation of a circle whose center is at and radius .

11) Determine the center and radius of the circle with equation:

.

How did it go? Did you experience many struggles and the feeling that “I used to know this stuff”? If so, then it is a good idea to go for a refresher before attempting calculus right away. The following are two books that you may find useful to respectively learn and refresh basic math in a well organized manner:

• Precalculus by Michael Sullivan: a big book, which is quite extensive and thorough. If you want an all-in-one book that covers all you need to know about precalculus and more, in a clear but college oriented manner, than this is without doubt an excellent choice. It will likely make the step up to Calculus quite easy.
• Schaum’s Outline of Precalculus: it has a less prosaic approach but it’s still very clear and easy to read. If you were pretty good at math in high school and you just forgot a few things because you haven’t touched these topics in a while, then pick this book up. It is adequate for already mathematically inclined people who are in a rush to brush up the skills they once had.

If you feel entirely clueless and would like a “for dummies” type of book, the following two titles seem to have a good table of contents and excellent reviews:

• The Complete Idiot’s Guide to Precalculus
• Pre-Calculus Demystified

If you would like to use some free resources available online instead, here are some lessons:

• Precalculus Tutorial
• Topics in PRECALCULUS
• OJK’s Precalculus page
• Exploring Precalculus
• Precalculus problems
• Prof. Ward’s lecture notes (PDF, 23 pages)
• Dave’s Short Trig Course
• Precalculus Lessons
• Collection of links related to Precalculus
• Wikipedia entry on Precalculus

If you know of any other resources that are available for free, or if you successfully used other books for these purposes, please feel free to use the comment section to add to the discussion.