At first glance we’re peering at a service that is able to provide objective information about a wide subset of human knowledge. But the idea behind Wolfram|Alpha is much more ambitious, as demonstrated by the first screencast to have been released (narrated by Stephen Wolfram).

Taking advantage of the popular and powerful mathematical software Wolfram Mathematica, this new engine is able to perform calculations on information requested by its users. Therefore there are three components at work: the ability to correctly interpret queries (in English) from the users, the ability to elaborate on the data source so as to reply with coherent results to somewhat complex questions, and finally the not-so-simple task of maintaining an up-to-date and accurate knowledge base for a very wide spectrum of human knowledge.

This is the theory behind - or at least the ultimate aim of - the service. And as such, unlike what has been reported elsewhere, Wolfram|Alpha should immediately be viewed as an addition to what Google already offers (not as a replacement for it).

As expected, the service is extremely good at mathematical calculations. In this case the only limit is the timeout imposed on each query to exclude those that require far too many resources to complete. An example of a calculation that is executed is “integrate e^-x^2”.

For the record, this service is currently under a fair bit of stress because of the initial curiosity of many worldwide, therefore the calculation of the integral above has shown an error message a couple of times (a tribute to HAL 9000):

All sorts of disciplines are represented by the examples on the site, with a particular focus on scientific and engineering ones. The results are elegantly presented and well organized, often illustrated and shown in table form, particularly when the user is asking for a comparison between different data sets. There are also fun tributes, including those to Monty Python and Douglas Adams. But how well does this system work when we step outside of the predefined examples provided by Stephen Wolfram’s team? The results range from exceptionally good to downright disappointing. Let’s see a few examples.

Let’s compare IBM, Google and Sun. As you can see, the results are definitely excellent. The comparison is almost exclusively numeric in nature, but the answer provided by WA jives with what I was hoping for. The same is true if you look for statistical information on a single large company like Telecom Italia.

Since this is a “computational engine” we can even try to perform a few calculations starting with the data that we found to be available in the comparison tables for the companies above. For example, “employees IBM/Google” will show us the ratio over the last few years between the number of IBM and Google employees (IBM currently hires almost 20 times as many people). We can calculate the revenue for each IBM employee by running “(market cap / employees IBM)”, but the engine fails to compare this parameter between companies: “(market cap / employees IBM) / (market cap / employees Google)”, despite the fact that the data for such a calculation is all there in the knowledge base (and as you can see there was already a revenue/employee row in the initial comparison table).

Moving on to something else, we can try to compare two cities like Toronto and Milan, and obtain very useful details. It’s also possible to calculate only certain attributes by running, for example, “population Toronto/Milan” or “distance Toronto Milan”. In the first case the data was updated to 2004, and is perhaps a little outdated now despite the fact that we are talking about demographic data. But the real problem arises when we try to compare the regions of Ontario and Lombardy. We’d expect to see nice geographical, demographical and economical comparisons between the two, yet instead we get nothing. Wolfram|Alpha will report information about little towns like Unionville (NC), but totally ignores a region like Lombardy or a Canadian province like Ontario (with a population of 13 million people). It would seem that this is a big hole in the knowledge of this service. Sure it’s fairly easy to fix, but it’s a symptomatic weakness nevertheless.

If we consider for a moment searches about famous people, we’ll find an excess of zealousness in trying to be concise and objective. In fact, when searching for Barack Obama, the results are limited to his place and date of birth, and stating that he is a head of state. We may be spoiled by Wikipedia, but a photo or a few more details at least were to be expected. For example, at a bare minimum, it could be indicated that we are reading the page of the 44th President of the United States of America. But Wolfram|Alpha reserves this type of treatment for all famous people, including Stephen Wolfram himself.

In the comparison between “Paul Erdos vs Euler” one would expect a nice parallel between these two great, prolific mathematicians (for example, you might expect comparisons on the number of publications, total number of pages, the most significant discoveries, and so on). Instead, the results are limited to a comparison of births and deaths. This is quite disappointing no matter how you look at it. On a side note however, it must be said that WA is quite good at interpreting misspelled names (e.g., Paul Erddsos).

The Natural Language Processing (NLP) capabilities of Wolfram|Alpha are good enough to use the service without encountering any major frustrations, but it doesn’t appear to be a particular revolution or advancement in the field of artificial intelligence either. It is also clear that we are not dealing with a Google-killer nor a Wikipedia-killer, but rather with an innovative new tool that can be used in addition to the existing ones. That said, elsewhere I mentioned that I personally think that this is a step forward for humanity. It may be a small step, but I stand behind that bold statement.

We are at the forefront of a service that will be useful for research and to anyone with a need for correct data as quickly as possible. The knowledge base will have to grow, some data will have to be updated, and the engine will need to permit more freedom in the kind of calculations that are allowed to be performed. There can be little doubt thought that we are witnesses to the birth of something ambitious that has the potential to accelerate the advancement of our civilization.

An Italian translation of this article is available on Stacktrace.it.

This tradition started in the late 80s and is now celebrated all over the world, particularly in North America, where parties and free pies are available on many campuses.

This year Pi Day is an ever bigger deal, because it’s no longer just a fun celebration of mathematics observed by a few incorrigible geeks. The US Congress approved the H.RES.224, sponsored by Rep. Bart Gordon and 15 cosponsors, titled “Supporting the designation of Pi Day, and for other purposes.”. Thanks to this, March 14, 2009 is now officially National Pi Day. More importantly the resolution includes the following statement:

Whereas Pi can be approximated as 3.14, and thus March 14, 2009, is an appropriate day for `National Pi Day’: Now, therefore, be it

Resolved, That the House of Representatives–
(1) supports the designation of a `Pi Day’ and its celebration around the world;
(2) recognizes the continuing importance of National Science Foundation’s math and science education programs; and
(3) encourages schools and educators to observe the day with appropriate activities that teach students about Pi and engage them about the study of mathematics.

Math-Blog applauds the sponsors of this resolution, which passed with 391 Yeas and 10 Nays. For once, both parties supported the initiative, and there is no doubt that the sponsors of this resolution will receive a great deal of thank you notes for acknowledging, albeit just symbolically, the importance of mathematics and science in our society. Sadly, 10 representatives felt the need to oppose this acknowledgment, and for those who are curious (without getting too political here) all 10 of them happen to be Republican.

There is now an official Pi Day website with cool merchandise, and a Facebook group you can join. To help you enjoy this day perhaps consider picking up a good book on the history of this fascinating transcendental number. While categorically rejecting any numerological implication regarding Pi Day, it’s a good occasion to celebrate mathematics and talk about it with those who otherwise usually wouldn’t be interested. And that could be the most important aspect to come out of the formal recognition of Pi Day.

It was “buzz worthy” for sure, and all the attention quickly attracted the interest of numerous scientists from many disciplines. As soon as the mathematicians, and particularly computer scientists, managed to get through its 1000+ pages, the first negative reviews began to pour in. Though, in all fairness, a few scientists had a little too much fun with this book and managed to showcase their comedic abilities by writing some of the most hilarious reviews known to humankind.

In this controversial best-seller, Stephen Wolfram primarily dissects the subject of cellular automata and its relevance to other scientific disciplines, in a systematic manner. It’s a book that covers a lot of ground and is arguably a remarkable piece of writing. Yet, the scientific community greeted the book with a fair dose of criticism.

So what went wrong? The main problem with A New Kind of Science is that it set very high expectations due to its author, title, and the numerous reminders of how important this material is, throughout the book.

The main accusations ranged from the book being called a display of Wolfram’s ego, to having very little “new” content, all the way to the more severe claims of not crediting other people’s work. For example, the idea of the universe as a cellular automaton was first presented by Konrad Zuse, so Wolfram’s “new” idea of a discrete, computable universe was anything but groundbreaking. On top of that, the most remarkable technical achievement revealed in this book was arguably the proof that the rule 110 cellular automaton is Turing complete. While this was conjectured by Wolfram, it was actually proven by his assistant Matthew Cook, who was refrained from publishing his results elsewhere by Wolfram’s lawyers.

It’s important to understand that, while perhaps not accepted as the breakthrough that Wolfram had hoped for, this book - and the methods for studying computational systems illustrated within it - is far from gibberish. Wolfram’s ambitious project failed in the eyes of the community due to the extremely high expectations that were set for this book. When you claim to have something radically new, you must be able to back that claim up in a convincing enough manner or else you’re bound to end up with egg on your face.

To be fair to Wolfram (for the few who are not familiar with his work) NKS is a controversial project, but he was already famous for having created the excellent program Mathematica (whose 7th version was recently released), one of the world’s most complete and advanced mathematical software.

Now Wolfram is at it again. According to his recent announcement, he is about to unleash something called WolframAlpha to the world, which combines both his work with Mathematica and NKS. In Wolfram’s own words:

I had two crucial ingredients: Mathematica and NKS. With Mathematica, I had a symbolic language to represent anything—as well as the algorithmic power to do any kind of computation. And with NKS, I had a paradigm for understanding how all sorts of complexity could arise from simple rules.

The project has been kept on the down-low for the past few years, while some of the brightest mathematicians and engineers employed by Wolfram Research, Inc. worked on it. It’s currently in private beta, but will go live in May of this year. From an initial glance, it would seem to be just another search engine a la Google.com. But is it? Not quite. It’s labeled as a “computational knowledge engine”, whose aim is to compute answers from the human knowledge available on the web. Whereas on Google you can search for strings and the results will be a series of relevant links, WolframAlpha will supposedly be able to parse and “understand” a query that’s inputted in English, and compute an answer based on the extensive knowledge stored in its system (assuming that a univocal answer exists). Conceptually speaking, it’s leaps and bounds more complex to get right than Google, which simply looks for matching strings and orders the results based on the popularity of the given keywords (For more information about the mathematics behind Google, read this book).

According to Nova Spivack, who had a chance to try out WolframAlpha, the service is able to compute factual answers to questions such as “What is the location of Timbuktu?”, “How many protons are in a hydrogen atom?,” “What was the average rainfall in Boston last year?,” “What is the 307th digit of Pi?,” “where is the ISS?” or “When was GOOG worth more than $300?”. This project has the potential to change the world as we know it, just like Google did. Several years ago Altavista was fine for most people’s search needs - or so we thought. It took Google to show us how much better off we could be search-wise, how much we needed Google, and ultimately how inadequate Altavista was. Unlike the case of Google and Altavista though, WolframAlpha would not replace Google, since the two services cover complimentary needs. Having access to a service that’s able to compute answers out of the chaos of the factual information that’s available to man would be a major breakthrough for humanity and computer science. And if an API (Application Programming Interface) were to become available, other developers would be able to tap into that with their applications.

Bold claims, high expectations. You understand why, two months away from experiencing something so potentially revolutionary, there is a lot of hype surrounding this project - but also major skepticism. For many this is A New Kind of Science all over again, especially since natural language processing and “computing knowledge” are extremely ambitious challenges in a realm where many have failed before. Pulling this one off would be a major accomplishment (that would dwarf Wolfram’s past achievements, including Mathematica), and, at long last, it would be the hard earned, practical validation of some of the methods and philosophies expressed in NKS by Wolfram.

I fully expect people to find bugs and have many simple questions, for which we will see bizarre answers. We’ll read blog posts about the whole thing and perhaps have a good laugh. But what interests me the most is whether, as Google did in the past, this new engine will be able to be practical and useful on an everyday level. Bugs are fair play and expected, but what we’re looking for here is a spark of true innovation thanks to the mathematical modelling of human knowledge.

I suspect that this engine will either have us in awe like Mathematica did, or leave us with mixed feelings - if not downright disappointment, like A New Kind of Science did for many. I can’t help but hope for the former, as I wait for my chance to try it out.

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.

And what was he?
Forsooth, a great arithmetician.
-Shakespeare Othello, I.i

Perhaps it should not be surprising, considering the vast libraries of published works, that mathematics should appear topically in works of fiction, but anyone who has sat through an introductory course in algebra can understand the difficulty that an author might have in captivating a reader if the subject is predominantly mathematical. This is not to say that casual readers should hold Jean-Pierre Serre up to the same literary standards of William Shakespeare, but that fiction can involve mathematics in complex and beautiful ways.

Mathematics is a rich ground for metaphor: the first scene of Tom Stoppard’s Arcadia is an excellent example of the intertwining of such unusual topics of conversation. In the very first scene, our young protagonist Thomasina is trying get her tutor, Septimus, to explain sexuality to her, whereas Septimus would rather encourage her studies:

Septimus: Carnal embrace is sexual congress, which is the insertion of the male genital organ into the female genital organ for purposes of procreation and pleasure. Fermat’s Last Theorem, by contrast, asserts that when x,y and z are whole numbers each raised to power of n, the sum of the first two can never equal the third when n is greater than 2.
(Pause)
Thomasina: Eurghhh!
Septimus: Nevertheless, that is the theorem.
Thomasina: It is disgusting and incomprehensible. Now when I am grown to practice it myself I shall never do so without thinking of you. (3)

Thomasina grunts out of mathematical frustration or coital pleasure, or perhaps both. And what precisely will she be doing when thinking of Septimus? This early comparison of algebraic theory with sexual self-discovery alters the tone of every mathematical reference uttered through the rest of the play, and double-entendres arise from otherwise innocent mathematical statements. Arcadia is laced with math, most prominently geometry and chaos theory. From carnal embrace to architecture, math is among the primary metaphors for examining the impermanence of forms and the struggle to find meaning in a world of background static.

While sex and death are compelling, authors also use math not as a metaphor for other things but as itself: a philosophical tool with which to pry off the lid of the universe. Neal Stephenson has written a number of historical fiction (or arguably science fiction) novels whose primary or secondary characters are mathematicians. The Baroque Cycle is a compendium of three novels centered around the dispute between Newton and Leibniz. In Quicksilver, the first book in the cycle, a character muses on the significance of conic forms (ie tracing the intersection of a cone and a plane) and gravitation:

Comets passed freely through space, their trajectories shaped only by (still mysterious) interactions with the Sun. If they moved on conic sections, it was no accident. A comet following a precise hyperbolic trajectory through the æther was a completely different thing from Daniel’s just happening to trace a roughly hyperbolic course through the English countryside. If comets and planets moved along conic sections, it had to be some kind of necessary truth, an intrinsic feature of the universe. It did mean something. What exactly? (676)

Daniel ponders an old question: is nature written according to the rules of Euclidean Geometry, or is geometry just the illusion of patterns in the fog? Newton’s “On the Motion of Bodies in Orbit” appears to answer the question, and in so doing open a window to the mind of God. The influence of mathematics on metaphysics has a tremendous effect on religious, and therefore political, thinking at the time. In this way mathematics drives a thousand plot lines spinning off from its philosophical implications.

But the subjects need not always be as weighty as the Universal Law of Gravitation. Math can appear within a sense of whimsy and joy in the beauty of solving puzzles and word games. Lewis Carroll’s Alice’s Adventures in Wonderland is full of subtle math jokes woven into the fabric of his fantastic tale. At the Mad Tea Party, when the Hatter wants another cup of tea, he has the whole party move around the table in a kind of infinite sequence. When Alice asks what happens when all the places are used up, the March Hare asks for a change of subject. In Through the Looking Glass, the chess queens grill Alice on her arithmetical skill:

“And you do Addition?” the White Queen asked. “What’s one and one and one and one and one and one and one and one and one and one?”
“I don’t know,” said Alice. “I lost count.”
“She can’t do Addition,” the Red Queen interrupted. “Can you do Subtraction? Take nine from eight.”
“Nine from eight I can’t, you know,” Alice replied very readily: “but – “
“She can’t do Subtraction,” said the White Queen. (222)

These are the joyous little jokes that make the Alice’s Adventures in Wonderland and Through the Looking Glass so endearing. In addition to writing children’s stories, Lewis Carroll was a mathematician; word games and clever exchanges such as these are the result of a mathematical attention to the precise meanings of words. Many excellent articles have been written on the logical puzzles hidden in his works.

Thus, contrary to the common perception that mathematics and literature occupy opposite ends on the spectrum of human thought, one can see how well the two disciplines may interweave. This is but a sampling, there is much more math to be found in fiction than just the few examples above. Whether heavy-handed or light-hearted, explicit or metaphorical, math appears in all sorts of ways throughout literary works. If you have but the patience to look into it and see, mathematics can be a light to brighten the many worlds you might visit.

References

1. Carroll, Lewis. Alice’s Adventures in Wonderland & Through the Looking Glass. New York: Penguin Putnam, 2000.
2. Stephenson, Neal. Quicksilver. New York: HarperCollins, 2003.
3. Stoppard, Tom. Arcadia. New York: Faber and Faber, 1993.

I’d like to claim that I’ve read it cover to cover. But I haven’t. At 1014 pages (and a hefty five and a half pounds), this recently published hardcover tome will probably take me a while to properly read through in its entirety. It can serve as a reference that can be randomly accessed if you so chose, but from what I’ve read so far, I’m extremely impressed and am eager to pursue my way through it systematically. It will be time really well spent, and dare I say, a life changing experience. This is the book to bring on a proverbial desert island, if you were allowed only one title.

The Princeton Companion to Mathematics is not only a beautiful book from an aesthetic standpoint, with its heavy, high quality pages and sturdy binding, but above all it’s a monumental piece of work. I have never seen a book like this before. It rigorously illustrates the (pure) mathematical field while remaining as accessible as possible to the general reader. There is no mathematician in the world who, upon reading this book cover to cover, would not have learned a great deal from it. And I’m sure this includes Timothy Gowers himself, who was the book’s Chief Editor and who brilliantly managed to coordinate a team of world class experts to cover (again in an accessible manner) their respective fields of expertise. Such experts not only are the best mathematicians alive today in their respective areas of expertise, but are also absolutely wonderful teachers who have the uncanny ability to divulge information in a understandable manner, under the editorial guidance of Professor Gowers. Timothy Gowers, by the way, also has a blog, which contains discussions about the book and a helpful errata.

This book is what I now consider to be the bible of mathematics, and unlike a typical reference or encyclopedia, The Princeton Companion to Mathematics never fails to provide a sense of unity and cohesion, both of which are essential if you want to truly provide an (nearly) complete panorama of a subject. While all the basics are well explained with the clarity and simplicity of really good popular science, this tome doesn’t skimp on details or theorems when it comes to highly advanced topics that few people are familiar with. The style remains geared towards providing a good introduction to each subject, as opposed to a PhD thesis, and as such it will prove useful to the ambitious high school student, as well as professional mathematicians or graduate students. And as if all this wasn’t enough, they managed to squeeze in a biographical overview of the most important historical mathematicians from Pythagoras to Bourbaki, as well as a respectable (yet not overly comprehensive section) about applied mathematics, and math’s influence on other disciplines.

I believe this is the kind of book that will still be is use a hundred years from now, even though by then it will be slightly outdated. This title is destined to be fully revered as a classic and monumental review of the subject of pure mathematics. I salivate like Pavlov’s dogs at the idea of the amount of fun I will have exploring this book, which will no doubt expose my ignorance about several key areas of math, and yet at the same time help me to remedy such things.

This Christmas, give yourself a great gift and get this book. If you are looking for the perfect gift for people who’re interested in mathematics, this is the right book. If you are a parent, I especially encourage you to pick it up for your son or daughter, it could change their lives. Ladies, your geeky boyfriend will likely propose to you if you put a copy under the tree. Jokes aside, The Princeton Companion to Mathematics makes for a great read to start 2009 off with.

I’m hip-deep in the teaching of Mathematics 152, a discrete mathematics course titled “The Mathematics of Symmetry” designed by Paul Bamberg and taught at Harvard University. The course is seminar-style: the students take turns presenting the material in 5-to-15-minute assigned topics during class. The design puts an emphasis on learning to communicate mathematics, and so as I took over the course this semester I considered what I might do to further this goal. I wanted to encourage class participation, discussion and a sense of community, as well as tie the mathematics of the course to the wider experience of the students. So I tried an experiment: an assigned community math blog. The blog is open to and in fact aimed at the layperson public, but also serves as a community discussion board for the students. The 23 students in the class have been assigned 4 posting dates each, spread throughout the semester, which means the blog is updated at least once and sometimes twice a day.

I was inspired by my mother and father, who both assign “journals” to their students in psychology, english, classics and philosophy. The journals, updated regularly by students, are a sort of private diary of reactions to the course. They serve to draw connections with the sudents’ world outside the classroom, and encourage reflection on the material. Blogging, it occurred to me, is a sort of public journaling, and provide some of what my parents sought from the course journals, but this time in the form of a community project.

We’re now approaching the halfway mark for the semester, and I’ve been incredibly impressed with the students’ posts. They range from amusing to historical to musical to magical—even social commentary. There’s been no shortage of topic ideas, although I had hoped there would be more discussion via comments. I hope you’ll take a look at the blog and post some responses, so the students see that they are really reaching an audience out there on the internet: reaching beyond the course itself.

Site: The Math 152 Weblog.

Hypatia of Alexandria (AD 350 to 370 – 415): Born nearly 17 centuries ago, Hypatia of Alexandria was a brazen, highly intelligent woman who excelled in the fields of science, math and philosophy, which at the time (and for hundreds upon hundreds of years further) were seen squarely as the domain of men. Hypatia’s foremost teacher was her father, Theon Alexandricus, a mathematician and philosopher, who she would later go on to contribute to several mathematical works with. Hypatia herself was a teacher, as well as being the inventor of the hydrometer. Though she forged ahead in a time when women were all but ignored in the realm of mathematics, this bright Greek woman eventually met with a tragic death when her chariot was attacked and she was brutally murdered by a gang of Christians. Though her life was cut short, while she was alive, through her accomplishments, Hypatia was able lay the groundwork for future female pioneers of mathematics.

Gabrielle Émilie Le Tonnelier de Breteuil, marquise du Châtelet (December 17, 1706 – September 10, 1749): A woman of many intellectual interests, Émilie was a mathematician, author, and physicist who hailed from France. Born into a well-to-do family, Châtelet was a gifted child with a natural penchant for linguistics. Given her family’s high social status, Émilie was able to receive a degree of education far above the vast majority of French women at the time. Her place in society also put her in a position wherein she was able to mingle with some of the leading minds of her time (such as Voltarie, who would go onto become one of her lovers). In 1740, Châtelet published a book entitled Institutions de Physique, which put forth some of her knowledge regarding both science and philosophy. In her last year of life, Émilie translated Newton’s well-known Principia Mathematica. In her early forties she became pregnant, and though she initially survived the pregnancy, a few days later both she and her newborn child passed away. Émilie was an independent, articulate and highly intelligent woman, who was somehow able to hold down both her role as a leading lady in French high society and as a mathematician, an equation which deserves respect in its own right.

Maria Gaetana Agnesi (May 16, 1718 – January 9, 1799): A woman of many skills, Agnesi was an Italian mathematician, linguist, and philosopher whose profound intelligence was evident from an early age. Born into a wealthy and large family (due in part to siblings which sprang from her father’s two subsequent marriages after Maria’s mother passed away), Agnesi was a devoted and studious woman who would go onto publish the first book that dealt with both integral and differential calculus. In 1750, Maria was appointed as chair of mathematics and natural philosophy at the Bologna Academy of Sciences, an incredible accomplishment for any woman in the mid eighteenth century, when exceptionally few universities in Europe allowed women to study, let alone hold teaching positions. Later in life, Agnesi, a deeply religious woman, joined a nunnery and ended her days tending to the less fortunate.

Marie-Sophie Germain (April 1, 1776 – June 27, 1831): Parisian born Germain was a passionate mathematician with a love of number theory and differential geometry. During her lifetime (which, in the context of both France and Europe in general, was a highly tumultuous era) Germain often corresponded under a pseudonym (Monsieur Le Blanc) as a means of hiding her gender when writing to leading male mathematicians of the time such as Lagrange and Gauss. In 1816 Sophie won a contest that was held by the French Academy of Science which dealt with the area of vibrations on elastic surfaces, that in turn lead her to become the first woman (short of some of the staffs’ wives) to attend classes at the Academy. In 1831, the University of Gottengen bestowed an honorary degree to Germain, however she died as a result of breast cancer before she was able to receive the degree. A self-taught mathematician who came of age during a truly unstable period in French history, Sophie will long be remembered for her mathematical contributions in the field of number theory.

Augusta Ada Byron King, Countess of Lovelace (December 10, 1815 – November 27, 1852): English born Ada was the daughter of famed poet Lord Byron, though he was not active in his daughter’s life. Aside from her famous father, Ada is primarily known for her programming work regarding Charles Babbage’s invention of the analytical engine, a very early mechanical general-purpose computer. Lovelace was ahead of her time in this field, as she believed that computers held the capacity to do more than just simply act as calculators. Like many of the women in this list, Ada met with an early death; she was only 36 when she died due to uterine cancer. Today Lovelace is remembered fondly as the first female computer programmer (in era before the modern computer came into existence), and the programming language Ada was named in her honor.

Sofia Vasilyevna Kovalevskaya (January 15, 1850 – February 10, 1891): Generally acknowledged as the first well-known Russian female mathematician, Kovalevskaya (portrayed above) began teaching herself advanced mathematics as a young teen, before going on to leave Russia so that she could attend university in mainland Europe (something that women were not allowed to do in Russia at the time). A very bright, quite and gentle person, Sofia loved to learn and was eager to share this passion with others by teaching math, though this proved to be very challenging for a woman in nineteenth century Russian and Kovalevskaya would again have to leave her homeland so as to take up a position lecturing at the University of Stockholm. Prior to her relatively young passing due to pneumonia, Kovalevskaya published numerous papers on topics pertaining to mathematics and mathematical physics, and won a prestigious award (the Prix Bordin) from the French Academy of Sciences. (Here you can find a mathematical book about her work.)

Amalie Emmy Noether (March 23, 1882 – April 14, 1935): Considered by Einstein to be most important woman in history of mathematics, Emmy (as she generally went by) was an early twentieth century German mathematician with a passion for such areas as theoretical physics and abstract algebra. Noether was both an accomplished university professor and a prolific writer of mathematical papers, as well as someone with a profound ability to grasp abstract thought. As the Nazi stronghold grew in Germany during the 1930s, Emmy found herself, like so many other Jewish professors, barred from teaching. Towards the end of 1933, Noether was able to escape Germany and take up a position at the American college of Bryn Mawr. However, sadly, two years later Emmy’s life was cut short when she died just days after undergoing surgery. To this day Noether’s many contributions towards mathematics and theoretical physics are highly revered, and many remain relevant to the math of the twenty-first century.

Dame Mary Lucy Cartwright (December 17, 1900 – April 3, 1998): An accomplished British mathematician, Cartwright led a long and distinguished career that focused on function theory. In her lifetime, Mary published in excess of 100 papers and was the first female mathematician to be elected as a Fellow of the Royal Society of England; a theorem regarding analytical function that she put forth, Cartwright’s theorem, shares her name. Cartwright received numerous awards and recognitions throughout her life including, the De Morgan Medal of the London Mathematical Society and the Sylvester Medal of the Royal Society.

Julia Hall Bowman Robinson (December 8, 1919 – July 30, 1985): An American mathematician who was born in St. Louis, Robinson is known for her work regarding Hilbert’s tenth problem and the field of decision problems. Though plagued by health problems for most of her life, Julia didn’t let this stand in the way of her love of math and the pursuit of knowledge. She taught as a professor at Berkley and was the first female mathematician to be elected to the National Academy of Sciences. An historical first in her career included becoming president of the American Mathematical Society. She would also go on to become elected to the American Academy of Arts and Sciences in the mid 1980s, just a few short years before she passed away from leukemia.

Shafi Goldwasser (b.1958 –): A native of New York (and the only living mathematician on our list), Goldwasser is both a professor of mathematics (at the Weizmann Institute of Science) and of computer science (at MIT, where she was the first person to hold an RSA Professorship). Shafi’s research focuses on areas such as cryptography, complexity theory and computation number theory, and she is well-known for her work with zero-knowledge proofs. For her work in the field of complexity theory, Goldwasser was awarded the Gödel Prize in theoretical computer science twice (1993 and 2001, respectively).

In this article we’ve taken a gander at ten well known and highly esteemed female mathematicians, but the list doesn’t stop here. Throughout history there have been numerous other women whose contributions to the field of mathematics have made significant impacts. In 1971 the Association for Women in Mathematics was formed with the intent of helping to establish and promote equal opportunities and treatment for girls and women in all areas of mathematics, while at the same time helping to encourage more to get involved with math.

1. General Math Cheat Sheet (iPaper and other formats)
2. Elementary Algebra Cheat Sheet (PDF)
3. Trigonometry Cheat Sheet (PDF)
4. Calculus Cheat Sheet (PDF)
5. Derivatives and Integrals Cheat Sheet (PDF)
6. Laplace Transforms Cheat Sheet (PDF)
7. Abstract Algebra Cheat Sheet (PDF)
8. Probability Theory Cheat Sheet (PDF)
9. Matlab Cheat Sheet (PDF)
10. Mathematica Cheat Sheet (PDF)
11. Maple Cheat Sheet (PDF)
12. Maxima Cheat Sheet (HTML web page)
13. LaTeX Cheat Sheet (several formats)

And since most of us like to show our math pride off when out and about as well, Amazon sells this awesome Math Cheat Sheet T-shirt with formulas on both sides (Also available for Science and Engineering). How awesome is this?

Flexible, powerful, and easy to use, gnuplot is not only handy as a stand-alone program but can be used successfully by a variety of programming languages, including but not limited to Ruby, Python and Perl. It has also been adopted as a plotting engine by Open Source programs like Maxima and GNU Octave. The minimal effort required to learn gnuplot is therefore a very worthwhile endeavor. Being able to visualize things as you analyze data and explore mathematics, is a very useful aid.

gnuplot is cross-platform, and you can download it from SourceForge.

Once you have installed gnuplot, you can start the tool by simply running gnuplot from the shell. Getting started with the program is straightforward, given that the most basic functionalities are fairly intuitive. For example, if you wanted to plot the sin(x) function, you could run:

gnuplot> plot sin(x)

gnuplot will automatically decide for you what portions of the function should be visualized (in our case between -10 and 10 on the x axis). If you want to overwrite this, you can specify otherwise, as shown below:

gnuplot> plot [-pi:pi] sin(x)

Similarly, you can customize the plotting, by specifying a range on the y axis:

gnuplot> plot [] [-0.5:0.5] sin(x)

It is also possible to plot several functions at once:

gnuplot> plot [-2:2] x, x**2, x**3

gnuplot can draw all sorts of functions and graphics. For example, the following plots two surfaces:

gnuplot> splot x**2+y**2, x**2-y**2

What you see plotted above, is based on a sensible set of defaults, but gnuplot offers countless options to customize the appearance of your graphics.

gnuplot isn’t just useful for plotting mathematical functions. I often use it for plotting data that is stored in tabular format in simple text formats. For example, assume that you have collected the following data on Deaths by Major Causes in the US, from a website:

stats.txt

That’s a big chunk of data, but it’s not very easy to analyze. It’d be nice to be able to visualize it. Assuming you saved it in a stats.txt file (you can include or remove the first few lines of comments) from gnuplot you can run the following:

gnuplot> plot "stats.txt"

That’s a meaningless mess. But gnuplot is very flexible, so we can easily do much better. Let’s use the first column for x values, and each of the remaining columns as a curve of its own by running:

gnuplot> plot "stats.txt" using 1:2 title "Heart disease" with lines, "stats.txt" using 1:3 title "Cancer" with lines, "stats.txt" using 1:4 title "Cerebro-vascular diseases" with lines, "stats.txt" using 1:5 title "Lower respiratory diseases" with lines, "stats.txt" using 1:6 title "Diabetes mellitus" with lines, "stats.txt" using 1:7 title "Influenza and pneumonia" with lines, "stats.txt" using 1:8 title "Chronic liver disease" with lines, "stats.txt" using 1:9 title "Accidents" with lines, "stats.txt" using 1:10 title "Suicide" with lines, "stats.txt" using 1:11 title "Homicide" with lines

Or it’s abbreviated version:

gnuplot> plot "stats.txt" u 1:2 title "Heart disease" w l, "" u 1:3 title "Cancer" w l, "" u 1:4 title "Cerebro-vascular diseases" w l, "" u 1:5 title "Lower respiratory diseases" w l, "" u 1:6 title "Diabetes mellitus" w l, "" u 1:7 title "Influenza and pneumonia" w l, "" u 1:8 title "Chronic liver disease" w l, "" u 1:9 title "Accidents" w l, "" u 1:10 title "Suicide" w l, "" u 1:11 title "Homicide" w l

Much better! Of course, the command above has a bit of repetition in it, but it’s justified by the flexibility of being able to pull data from different files if required, and plot some data with lines and others without (or in different styles) if we want to. To improve the appearance and legibility of the chart, let’s add a grid:

gnuplot> set grid
gnuplot> replot

Nice! And notice how the replot command runs the last plotting directive for us, saving us some typing (or scrolling through the previous commands). To improve this further, we can also set the labels for the axis and a title for the whole chart:

gnuplot> set xlabel "Years"
gnuplot> set ylabel "Death rates per 100,000 people"
gnuplot> set title "Deaths by Major Causes, 1960–2005"
gnuplot> replot

(Click here for a larger version)

Effortlessly, we went from having some data in a text file, to a meaningful, and professional looking chart that could be used in an article or a book. Not bad. You can save it in your favorite format, by setting the output file name and the terminal type. In the example below, I specify the font type, size and its location on my Mac OS X system, but it’s entirely optional. If missing, the default fonts apply.

gnuplot> set output "mychart.png"
gnuplot> set terminal png font "/Library/Fonts/Times New Roman.ttf, 11"
gnuplot> replot

This very short introduction should you give you a glimpse into what gnuplot can do for you, and even get you started with the tool. But gnuplot is much more than this and infinitely customizable, so you really may want to consider learning more about it. Unfortunately, despite being widely used for more than a decade, gnuplot has never had a book published about it.

Thankfully, Manning Publications will be putting out a book called Gnuplot in Action in October. The good news is that you don’t have to wait for the dead tree version, you can purchase it today through the Manning Early Access Program (MEAP). 12 out of the 15 chapters are already available, so the book is pretty much complete. If you’re interested in making the best out of gnuplot, this book is a must have. You don’t have to be a mathematician or a programmer to follow along, and it’s so much more than a bunch of options for customizing your graphics. The book guides you through the best techniques for taking advantage of the tool in several common scenarios. By reading, this easy to follow book, you’ll be able to generate colorful, nifty plots and have good mastery of the tool in a very short amount of time.