TAG: Mathematics
August 3, 2011– 2

In my last blog post on plotting functionality in Wolfram|Alpha, we looked at 2D and 3D Cartesian plotting. In this post, we will look at 2D polar and parametric plotting.

For those of you unfamiliar with polar plots, a point on a plane in polar coordinates is located by determining an angle θ and a radius r. For example, the Cartesian point (x, y) = (1, 1) has the polar coordinates (r, θ) = (√2,π/4). The following diagram illustrates the relationship between Cartesian and polar plots.

Relationship between Cartesian and polar plots

To generate a polar plot, we need to specify a function that, given an angle θ, returns a radius r that is a function r(θ). Making a polar plot in Wolfram|Alpha is very easy; for example, we can plot Archimedes’ spiral. More »

July 27, 2011– 7

In recent dinner conversation with my brother, I commented that I was “99.9 repeating” percent sure that my favorite author, Jorge Luis Borges, had lived into the 1980s (Wolfram|Alpha later showed me that he did, in fact, live through 1986!). I felt confident about my knowledge, but wanted to leave myself a little wiggle room, just in case. My brother grinned at me and said, “I know you know your geometric sequences. If you say you’re 99.9 repeating percent sure, then you’re 100 percent sure.”

I blushed, embarrassed. He was definitely right!

But just what did my brother mean? Well, he reminded me that:

99.999… = 99 + 9/10 + 9/100 + 9/1000 + 9/10000…

Glancing at the above, the series 9/10 + 9/100 + 9/1000 + 9/10000… stands out. Each term in this series is 1/10 times the term before it, making it a geometric series. A geometric series is a series wherein each term in the sequence is a constant number, r, multiplied by the term before it. Any geometric series whose r satisfies -1 < r < 1 is a convergent series, and we can say to what the series converges:

Sum[r^k, {k, 0, ∞}] = 1/(1-r)

More »

July 19, 2011– 3

Want to know if you and your 20 teammates can march in perfect triangular formation in a parade? Or maybe you want to know how many glasses you’ll need in order to break the world record for largest champagne pyramid (currently 60 stories high, with 37,820 glasses).

Whether you’re stacking watermelons or water-skiers or just playing with dots on paper, if you want to make a perfect geometrical formation, then you’re interested in “figurate” numbers. These are numbers for which that number of things can be arranged into a perfect geometrical shape, such as a triangle (in the case of the “triangular numbers”) or a triangular pyramid (AKA, a tetrahedron), which is the shape of champagne pyramids.

Wolfram|Alpha now has the ability to tell you if a given number is any figurate number. Entering our first question, “Can 21 people be put into a perfect triangle formation?”, into Wolfram|Alpha gives an affirmative, along with a diagram and other useful information:

Can 21 people be put into a perfect triangle formation? More »

July 15, 2011– 3

Periodic tilings (also known as tessellations) are often beautiful arrangements of one or more shapes, known as tiles, into regular patterns, which if extended infinitely are capable of covering the entire plane without gaps. Wolfram|Alpha has possessed detailed knowledge on more than 50 common (and uncommon) varieties of periodic tilings for some time, as illustrated, for example, in the case of the basketstitch tiling:

Periodic tilings possess an individual motif (more formally known as a primitive unit) that is repeated iteratively in a predictable (periodic) way. Such tilings are therefore intimately related to the set of symmetry groups of the plane, known as wallpaper groups. While the most general set of geometric similarity (i.e., shape-preserving) operations in the plane includes rotation (change in angle), dilation/expansion (change in size), reflection (flipping about an axis), and translation (change in position), only translation is needed to produce a periodic tiling from a correctly constructed primitive unit. More »

June 1, 2011– 4

What do you get when you cross a mountain climber with a mosquito? Nothing—you can’t cross a scalar with a vector!

But what do you get when you cross two vectors? Wolfram|Alpha can tell you. For example:

Cross product of (1, 2, 3) and (4, 5, 6)

And in fact, Wolfram|Alpha can give lots of information on vectors. A vector is commonly defined as a quantity with both magnitude and direction and is often represented as an arrow. The direction of the arrow matches the direction of the vector, while the length represents the magnitude of the vector. Wolfram|Alpha can now plot vectors with this arrow representation in 2D and 3D and return many other properties of the vector.
More »

April 25, 2011– 9

Wolfram|Alpha is written in Mathematica, which as its name suggests is a fantastic system for doing mathematics. Strong algorithms for algebraic simplification have always been a central feature of computer algebra systems, so it should come as no surprise to know that Mathematica excels at simplifying algebraic expressions. The main two commands for simplifying an expression in Mathematica are Simplify and FullSimplify. There are also many specific commands for expressing an algebraic expression in some form. For example, if you want to expand a product of linear polynomials, Expand is the appropriate function.

The good news is that everyone has access to the power of Mathematica‘s simplification and algebraic manipulation commands in Wolfram|Alpha. We will now outline some of these features in Wolfram|Alpha, starting with the expression:

Mathematical expression

and we will use Wolfram|Alpha to break it down to something significantly much simpler.
More »

March 23, 2011– 2

Do you need to work with numbers that are of the magnitude of thousands, millions, or even billions? How about the thousandths, millionths, or billionths? Scientists and engineers need to work with really large and really small numbers every day. Now Wolfram|Alpha can help put all of those large and small numbers into scientific notation. For example, the Earth’s mass is about 5973600000000000000000000 kg, but it is nicely represented in scientific notation as 5.9736×10^24 kg.

5973600000000000000000000 kg to scientific notation
More »

March 14, 2011– 7

Yes, it is once again the time of the year when the mathematically inclined gather together to celebrate Pi Day

Pi Day

…and, in the process, swap trivia of note on everyone’s (including Wolfram|Alpha’s) favorite number.

There have been no shortage of blog posts already written on the subject; see, for example, last year’s “Pi Day in Wolfram|Alpha” (or the Wolfram Blog Pi Day posts from 2008 or 2010). As already hinted at in last year’s blog, one would expect the pi to be ubiquitous in a computational knowledge engine—and so it is. Therefore, at the risk of beating a proven transcendental constant to death, this year we offer a few (well, OK: more than a few) additional pi-related esoterica courtesy of Wolfram|Alpha.
More »

March 9, 2011– 6

Remember those dynamic calculators we mentioned back in November?

Well, they’re back… with more features! We’ve released a whole new batch of calculators that cover some different areas of math.

For those who don’t remember, in November, we released calculators capable of computing integration, limits, date differences, and more. With this new slew of calculators, we are covering some other commonly asked math concepts, like vector and matrix manipulation, GCD, LCM, inverses of functions, linear approximation of functions, 3D plotting, base conversions, and currency conversion.
More »

March 3, 2011– 1

Students of the history of science will recall learning that Galileo observed and described the periodic motion of a simple pendulum around 1602. Until being supplanted by other technologies around the first third of the 20th century, this property of pendula has been indispensible in the creation of accurate timekeeping devices.

An idealized pendulum consisting of a weight (often called a “bob”) on the end of a massless cord and suspended from a frictionless pivot is called a simple pendulum or, more explicitly, a simple gravity pendulum. Wolfram|Alpha has known about simple pendula for some time, as you can verify by entering “pendulum”. In fact, doing so brings up not one but two pendulum results:

Pendulum More »

February 25, 2011– 3

Why did the mathematician name his dog Cauchy? Because he left a residue at every pole!

But could the mathematician find the poles and their residues for a given function? He certainly could, with the help of Wolfram|Alpha.

We are proud to announce that Wolfram|Alpha has added residues and poles to its ever-expanding library of mathematical data that it can compute! To showcase this behavior, let’s first recall just what a pole is.

In the study of complex analysis, a pole is a singularity of a function where the function behaves like 1/zn at z == 0 .

More technically, a point z_0 is a pole of a function if the Laurent series expansion of the function about the point z_0 has only finitely many terms with a negative degree of zz_0. As an example, let’s look at the Laurent expansion of 1/Sin[z] at z == 2 Pi:

Laurent expansion More »

January 28, 2011– 8

It is immediately clear to anyone who has used the site that Wolfram|Alpha knows a lot about mathematics. When computing integrals, sums, statistics, properties of mathematical objects, or a myriad of other mathematical and mathematics-related problems, it typically returns an extensive and exhaustively complete result. Which is of course not surprising, given that Wolfram|Alpha has the entire power and knowledge of Mathematica behind it, especially when combined with the fact that this native “smarts” is further augmented with large amounts of curated data and customized processing.

However, many visitors to the site have noted in the past that Wolfram|Alpha had relatively little computable knowledge about mathematical terms themselves, a state of affairs in contrast to its knowledge of words in the English language, and perhaps surprising in light of the existence of another Wolfram site devoted to the definition and description of terms in mathematics, namely MathWorld.

As readers of MathWorld likely already know, the entire MathWorld website is written and built using Mathematica. It has therefore been possible to programmatically process the entire 13,000+ entries comprising MathWorld into the native data format of Wolfram|Alpha, thus exposing its content in more computable form.

As an example of the sort of new knowledge this confluence brings to Wolfram|Alpha, consider the input “Lorenz attractor”. In the past, this would simply bring up a Wolfram|Alpha future topic page.

With the incorporation of MathWorld content, the default parse now goes to a description of the attractor, complete with an illustrative figure and some helpful typeset equations:

Lorenz attractor

More »

January 12, 2011– 44

Today we’re releasing the first three of a planned series of “course assistant” apps, built using Wolfram|Alpha technology.

Wolfram Algebra

The long-term goal is to have an assistant app for every major course, from elementary school to graduate school. And the good news is that Wolfram|Alpha has the breadth and depth of capabilities to make this possible—and not only in traditionally “computational” kinds of courses.

The concept of these apps is to make it as quick and easy as possible to access the particular capabilities of Wolfram|Alpha relevant for specific courses. Each app is organized according to the major curriculum units of a course. Then within each section of the app, there are parts that cover each of the particular types of problems relevant to that unit.
More »

January 7, 2011– 3

The real line runs from negative to positive infinity and consists of rational and irrational numbers. It generally appears horizontally, and every point corresponds to a real number. Also known as a number line in school, the real line is said to be one of the most useful ways to understand basic mathematics. Wolfram|Alpha can now aid you in learning the difference between x<-5 and x>5, or Abs[x]<2.

Wolfram|Alpha now graphs inequalities and points on the real line. This new feature in Wolfram|Alpha allows you to plot a single inequality or a list of multiple inequalities. Let’s start off simply and try “number line x<100”.

Number line x<100

You can easily see that this is the set of all real numbers from negative infinity to, but not including, 100.

What if you need to plot a more difficult inequality, like “number line 3x<7x^2+2”? This plot will show that the solutions to this inequality are all real numbers between negative and positive infinity.
More »

December 17, 2010– 2

Wolfram|Alpha already contains many extensive collections of mathematical data, including curves, surfaces, graphs, knots, and polyhedra. However, one type of object we had not systematically incorporated until recently was the class of plane geometric figures technically known as laminae:

Laminae

Most people (including the subset of small people who play with sorting toys such as the one illustrated below) are familiar with a number of laminae. A lamina is simply a bounded (and usually connected) region of the Euclidean plane. In the most general case, it has a surface density function ?(x, y) as a function of x- and y-coordinates, but with ?(x, y) = 1 in the simplest case.

Real world example of laminae

Examples of laminae, some of which are illustrated above, therefore include the disk (i.e., filled circle), equilateral triangle, square, trapezoid, and 5-point star. In the interest of completeness, it might be worth mentioning that laminae are always “filled” objects, so the ambiguity about whether the terms “polygon”, “square”, etc. refer to closed sets of line segments or those segments plus their interiors does not arise for laminae.
More »

December 6, 2010– 6

Wolfram|Alpha isn’t just the wolframalpha.com website; it’s a whole range of technologies. While the website may be the most familiar way to access these technologies, there are many potential uses and interfaces for the Wolfram|Alpha technology. We’ve already seen a few. Mobile apps for Google’s Android and Apple’s iOS make Wolfram|Alpha accessible anywhere. Widgets allow users to tap portions of Wolfram|Alpha and bring them into their own webpages. The Wolfram|Alpha API allows programmers to integrate Wolfram|Alpha’s data and computation abilities in their own programs. There are even private custom versions of Wolfram|Alpha used to analyze confidential corporate data.

But now there’s another interface to Wolfram|Alpha, one which brings with it a whole new set of capabilities: Mathematica. With the new Mathematica 8, you can access the Wolfram|Alpha engine directly from within Mathematica. Inside a Mathematica notebook document, just type == at the beginning of a line; you’ll get an orange Spikey icon indicating that Mathematica is ready to perform a Wolfram|Alpha query. Now simply type anything that you would type into the Wolfram|Alpha website. You’ll get back the same results as on the website—and more! Using the full power of the Mathematica software, this interface to Wolfram|Alpha allows new levels of interactivity and detail.

In Mathematica, all graphics can be resized, and three-dimensional graphics can be rotated. Moreover, since Mathematica receives the underlying vector graphic from Wolfram|Alpha and not simply a bit-mapped image, this means that enlarging a graphic provides greater detail instead of a boxy image. For example, let’s look at everyone’s favorite three-dimensional surface, the Mathematica Spikey.

Mathematica Spikey

By simply clicking and dragging, you can rotate the Spikey. To resize, click the resize points on the frame that appear after clicking on the graphic. More »

November 10, 2010– 10

As always, we are striving for better linguistic support of all things math, and over the past few months, we have made many improvements to that end.

We have significantly improved our support for queries involving polygons and circles being inscribed in and circumscribed about each other. Give these examples a spin:

Cube circumscribed about a sphere with volume 1

Another improvement of note is that we have gotten better accepting queries like “algebraically find the inverse function of f(x)=3-8e^x” and winnowing this down to the core question, that of  “inverse of f(x)=3-8e^x”. More »

November 1, 2010– 8

Wolfram|Alpha is still young and constantly improving. One of the biggest hurdles that our developers are constantly faced with is how to correctly interpret the meaning of general user inputs like “How do I factor an equation?”

Wolfram|Alpha is great at calculating answers that have specific inputs, but when general concepts are given (like “factoring”, for example), it becomes a bit tricky. How would one calculate a concept like that? Let us look at a specific example—Wolfram|Alpha can easily calculate integrals, as long as you ask it to integrate an explicit function. But what happens when you simply ask Wolfram|Alpha to “integrate”? Before, had you given this input, a number of examples using the word “integrate” would have been generated to show how to properly ask Wolfram|Alpha to calculate an integral. But now, when you enter a generic term or question related to a specific math function or formula, it provides a simple query-specific calculator.

For example, given the query “Show me how to integrate”, the following results appear:

Show me how to integrate

Notice that new input fields appear (as well as optional ones in case the integral is to be calculated within a range, and/or with multiple variables). Similarly, notice how the input fields differ depending on the query: More »

September 30, 2010– 26

In 1977, famed computer scientist Donald Knuth decided he didn’t like the typesetting of the second edition of The Art of Computer Programming. Rather than unhappily accept the results of photographic typesetting techniques, Knuth invented his own digital typesetting solution, TeX, which would eventually become the standard typesetting system for mathematical and academic content. Wikipedia displays math content using a variant of TeX, and research papers from a large range of fields are very commonly submitted in TeX format.

Our team recently added the ability to understand TeX notation and convert it to the Mathematica form used by the powerful Wolfram|Alpha engine. We’ve received many requests for this functionality from people who use Wolfram|Alpha for advanced math and physics. It’s often easy and natural to write mathematics using TeX, whereas it can otherwise be quite difficult to express clearly in plaintext notation.

int sin{x^2}+sqrt{x} dx

The beauty of this new capability is that one can now see, compute, and understand typeset mathematics all through the union of TeX notation and Wolfram|Alpha computation. Complicated expressions are now easily represented using the elegance of TeX: More »

September 10, 2010– 11

The concept of infinity has been fraught with paradox since antiquity. For this reason, Aristotle sought to banish it from his physics, claiming that there were no actual infinities in nature—only potential infinities. Over a millennium later, medieval scholars offered the following example when asked why infinity was forbidden.

Two concentric circles

Imagine two concentric circles. Each circle contains infinitely many points along its circumference, but since the outer circle has a greater circumference, it has more points than the inner circle. Now take any point A on the outer circle, and draw a line from A to the circle’s center. This line must intersect some point B on the circumference of the inner circle. Hence, for every point A on the outer circle, there is a corresponding point B on the inner circle, and vice versa. Therefore, both circles must have the same number of points, despite the fact that the outer circle appears to have more points than the inner circle. More »

September 2, 2010– 7

A new school year is here, and many students are diving into new levels of math. Fortunately, this year, you have Wolfram|Alpha to help you work through math problems and understand new concepts. Wolfram|Alpha contains information from the most basic math problems to advanced and even research-level mathematics. If you are not yet aware of Wolfram|Alpha’s math capabilities, you are about to have a “wow” moment. For the Wolfram|Alpha veterans, we have added many math features since the end of the last school year. In this post, we’re highlighting some existing Wolfram|Alpha math essentials, such as adding fractions, solving equations, statistics, and examples from new topics areas like cusps and corners, stationary points, asymptotes, and geometry.

You can access the computational power of Wolfram|Alpha through the free website, via Wolfram|Alpha Widgets, with the Wolfram|Alpha App for iPhone, iPod touch, and the iPad! Even better, the Wolfram|Alpha Apps for iPhone, and iPod touch, and the iPad are now on sale in the App Store for $0.99 though September 12.

If you need to brush up on adding fractions, solving equations, or finding a derivative, Wolfram|Alpha is the place to go. Wolfram|Alpha not only has the ability to find the solutions to these math problems, but also to show one way of reaching the solution with the “Show Steps” button. Check out the post “Step-by-Step Math” for more on this feature.

Using Wolfram|Alpha to solve for 3x2+7x-10

You can find this widget, and many others, in the Wolfram|Alpha Widget Gallery. Customize or build your own to help you work through common math problems. Then add these widgets to your website or blog, and share them with friends on Facebook and other social networks.

Of course, Wolfram|Alpha also covers statistics and probability. For example, Wolfram|Alpha can compute coin tossing probabilities such as “probability of 21 coin tosses“, and provides information on normal distribution: More »

August 19, 2010– 3

The Wolfram|Alpha Blog is not only your official news source for new data and features, but it’s also a great place to read how others are using Wolfram|Alpha in everyday life, for education and on the job. This week, a tweet linking to @drwetzel‘s latest blog post “How to Integrate Wolfram Alpha into Science and Math Classes” caught our attention. With a new school year upon us, we wanted to share his examples for using Wolfram|Alpha through the website, widgets, and mobile apps with educators who are looking for ways to incorporate Wolfram|Alpha into their math and science classes.

From the Teach Science and Math blog:

How to Integrate Wolfram Alpha into Science and Math Classes

“What is Wolfram Alpha? It is a supercomputing brain. It provides calculates [sic] and provides comprehensive answers to most any science or math question. Unlike other search sources, you and your students can ask questions in plain language or various forms of abbreviated notation.

Contrary to popular belief, Wolfram Alpha is not a search engine. Unlike popular search engines, which simply retrieve documents based on keyword searches, Wolfram computes answers based on known models of human knowledge. It provides answers which are complete with data and algorithms, representing real-world knowledge.

Teaching Strategies: Researching Facts and Information

Science and math teaching strategies with Wolfram begin with allowing students to search for information about specific facts and information. The following examples provide support for stimulating critical thinking using a digital lens.”

Click here to continue reading this post on the Teach Science and Math blog.

If you’re new to Wolfram|Alpha, we invite you to visit the Wolfram|Alpha for Educators site to browse our video gallery, download lesson plans, and more. Are you already using Wolfram|Alpha in your classroom? Share your story in the comment box below and you could be featured in an upcoming post on how educators are using Wolfram|Alpha as a learning tool in a variety of subjects.

June 16, 2010– 6

“It’s a quick and easy Saturday afternoon project!” We’ve all stood in the middle of our favorite home improvement store reciting that same line. Ty Pennington, Mike Holmes, Tim “The Tool Man” Taylor—they know how to make the most elaborate home improvement projects look as simple as tightening a bolt. Often the challenging part of the project is picking the perfect color of paint, or deciding between hardwood and tile flooring. But just as soon as those decisions are settled we’re faced with deciding how many feet of flooring to purchase for the kitchen or how many gallons of coral paint are needed for the north wall of the living room. But you don’t have to let a little bit of tricky math cut into your project time. Wolfram|Alpha has a number of math tools that come in handy for many common home improvement projects.

You can make quick computations and conversions from Wolfram|Alpha’s website or from the Wolfram|Alpha app for iPhone or iPad while standing in the flooring department. Wondering how many 8 x 8 square inch tiles you’ll need to cover a 12 x 14 square foot kitchen? Compute it with Wolfram|Alpha by entering “(12*14) square feet / (8*8) square inches”:

Computing the number of 8 x 8 square inch tiles you'll need to cover a 12 x 14 square foot kitchen

Need to know how many square feet you can cover with vinyl flooring that’s sold by the square yard? Tap into Wolfram|Alpha’s large collection of units to convert 60 square yards to square feet.

Converting 60 square yards to square feet

Thinking about livening up the living room with a splash of color? Query the name of your favorite hue and Wolfram|Alpha will give you a color swatch, properties, and a breakdown of related colors

Properties of the color coral in Wolfram|Alpha

Wondering how many gallons of paint you’d need per coat on a wall that’s 90 square feet? More »

May 27, 2010– 8

On May 22, 2010, Martin Gardner died, unexpectedly, at age 95. The previous sentence contains a paradox* explained within his book The Unexpected Hanging and Other Mathematical Diversions, one of 15 books known collectively as “the Canon,” comprising hundreds of the Mathematical Games columns Martin wrote for Scientific American between 1956 to 1981.

My fifth-grade science class had old copies of Scientific American available, and I read a few of those columns. From him I learned that math can be fascinating, perhaps one of the great lessons I’ve learned in life. I found out that the library had more issues, and whole books by Martin. I tracked down more of his columns on microfiche.

After reading all those columns, school-level math was easy. Years later, I tried to follow in Martin’s footsteps by putting recreational mathematics online. For example, I contributed a diagram of pentagon tiling to a very early version of MathWorld. “Tiling with Convex Polygons” was one of Martin’s columns, in his book Time Travel and Other Mathematical Bewilderments; today, you can explore these objects in Wolfram|Alpha.

Pentagon tiling in Wolfram|Alpha

Martin’s works influenced generations of mathematicians, and many of the topics he discussed can be found here at Wolfram|Alpha. For a Lewis Carroll expert like Martin, a snark was “something hard to find”, as in Carroll’s “The Hunting of the Snark” (for which Martin compiled a companion volume, The Annotated Snark). So he used the word “snark” to describe a graph with three edges attached to each node, but which could not be 3-colored without any clashes at a node. More »

May 7, 2010– 6

Hello, fellow readers of the Wolfram|Alpha Blog—my name’s Justin. In just a few short weeks, I’ll be graduating from the University of Illinois at Urbana-Champaign. Over the years I’ve found my own way of getting things done in regards to homework and studying routines. But this semester I realized there were tools available that would make studying and completing assignments easier and help me understand better. One tool that has become increasingly valuable in my routine and those of other students on my campus is Wolfram|Alpha. Recently, I was invited to share how Wolfram|Alpha is being used by students like myself.

Being a marketing major, I had to take some finance and accounting courses, but I was a bit rusty with the required formulas and the overall understanding of the cash flow concepts, such as future cash flows and the net present values of a future investment. A friend recommended I check out Wolfram|Alpha’s finance tools, and they’ve became indispensable in my group’s casework for the semester. Each proposed future investment we were met with, we would go directly to Wolfram|Alpha to compute the cash flows. We even went as far to show screenshots, such as the one below, of inputs and outputs in our final case presentation last week.

Net present value calculation in Wolfram|Alpha

I’ve met other students on my campus who have found Wolfram|Alpha to be helpful in their courses. A few months ago while studying in the library, I walked by a table of freshman students all using Wolfram|Alpha on their laptops. I decided to stop and chat with them because I knew one of the girls. They explained how they were using Wolfram|Alpha to model functions and check portions of their math homework. All three girls are enrolled in Calculus III, and not exactly overjoyed about the fact of future— and most likely harder—math classes. More »

May 4, 2010– 9

As we all know by now, Wolfram|Alpha is a computational knowledge engine. That means not only should it be able to do computations on a wide variety of topics, but also that it needs detailed knowledge of the names and salient properties of a wide variety of entities that are commonly encountered in human inquiry and discourse.

This is obvious in the case of classes of objects that fit neatly into curated data collections, such as mathematical surfaces (e.g., Möbius strip), countries of the world (e.g., New Zealand), chemicals (e.g., caffeine), and so forth.

What is perhaps slightly less obvious is just how much knowledge needs to be encoded to have a reasonable “understanding” of almost any named result in math and the sciences. For example, most people (including non-mathematicians) have heard of Fermat’s last theorem and therefore would rightly expect Wolfram|Alpha to be able to say something sensible about it.  And as one of my other hats involves writing the online encyclopedia of math known as MathWorld, which is hosted and sponsored by Wolfram Research, putting this information into Wolfram|Alpha naturally fell to me. So, for the past several months, I have been attempting to gradually build up Wolfram|Alpha’s knowledge base on named results in math and physics.

The screenshot below shows what Wolfram|Alpha now returns for Fermat’s last theorem:

Fermet's last theorem in Wolfram|Alpha

As you can see, Wolfram|Alpha begins by giving you the standard name for the result in question, followed by a clearly worded (or at least as clearly worded as could be managed in the marginal space available ;) ) plain English statement of the result. Next, at least in cases where it is possible to do so, a mathematically precise “formal statement” of the result is given. This is followed by any common alternate names the result might have, a listing of historical information, and finally an enumeration of prizes associated with it (where relevant). More »

March 26, 2010– 5

Exciting new math features have arrived in Wolfram|Alpha! Our programmers have spent the past two months developing new capabilities in optimization, probability, number theory, and a host of other mathematical disciplines. Searching for elusive extrema? Look no further! Just feed your function(s) into Wolfram|Alpha and ask for their maxima, minima, or both. You can find global maxima and minima, optimize a function subject to constraints, or simply hunt for local extrema.

We’ve also added support for a wide variety of combinatorics and probability queries. Counting combinations and generating binomial coefficients has been simplified with syntax like 30 choose 18. Want to spend less time crunching numbers and more time practicing your poker face? You can ask directly for the probability of a full house or other common hands, as well as the probabilities of various outcomes when you play Powerball, roll two 12-sided dice, or repeat any sequence of trials with a 20% chance 4 times.

The pursuit of primes has never been so simple. Imagine yourself walking the streets of an infinite city in search of “prime real estate.” You can find the nearest one simply by requesting (for example) the prime closest to 100854; alternatively, you could scope out the entire neighborhood by asking Wolfram|Alpha to list primes between 100,000 and 101,000. Would you prefer the greatest prime number with 10 digits, or will you be satisfied with any random prime between 100,000,000 and 200,000,000? The aspiring real estate agent—er, number theoretician—can also tinker with quantities like the sum of the first hundred primes or the product of primes between 900 and 1000. If your explorations take you to the realm of the composites (the addresses of houses with “sub-prime” mortgages, perhaps), you can identify numbers with shared factors by querying Wolfram|Alpha for, say, multiples of 5, 17, 21.

Other additions have brought everything from Archimedes’ axiom to semiaxes and square pyramid syntax into our body of computable knowledge and functions. Wolfram|Alpha grows daily, so stay tuned to this blog for further updates. Better yet, apply to become a Wolfram|Alpha tester for privileged access to the newest features before they go public!

March 19, 2010– 5

Steven Strogatz, a professor of applied mathematics at Cornell University, is currently blogging for The New York Times about issues “from the basics of math to the baffling”. It’s been a fascinating series, starting with preschool math and progressing through subtraction, division, complex numbers, and more. As Wolfram|Alpha is such a powerful tool for working with mathematical concepts, we thought it’d be fun to show how to use it to explore some of the topics in Strogatz’s blog.

First up is Strogatz’s post on “Finding Your Roots”. For a brief introduction to Wolfram|Alpha’s ability to find roots, try “root of 4x+2”.
Wolfram|Alpha solves "root of 4x+2"

Here we found the one and only root of 4x+2, but what if there is more than one root? Not a problem for Wolfram|Alpha—try “4x^2 + 3x – 4”. More »

March 12, 2010– 7

In my blog post last month, I wrote about Valentine’s Day in Wolfram|Alpha. Strangely, we received a number of comments indicating that the computational power of Wolfram|Alpha was not always sufficient to melt the hearts of some non-mathematically inclined sweethearts of the world. But not to fear; I have decided to persist undeterred in spite of that disappointing and surprising news, now that we’re on the verge of another holiday (and a more inherently mathematical one).

The holiday in question is Pi Day. As with a large number of other holidays, simply typing its name (in this case, “pi day”) into Wolfram|Alpha gives you basic calendrical information about it:

Wolfram|Alpha's results for "pi day"

Now, because Wolfram|Alpha users are both intelligent and discriminating, all of you have I’m sure already noticed that when the digits in the date 3/14 (March 14 in the United States style for dates—a bit more about this later) are run together with a decimal place between, the result is 3.14. And that that decimal expansion is connected with a certain famous mathematical constant given by the ratio of the circumference of a circle to its diameter. And that little fact explains why Pi Day is celebrated on the 14th of each March. More »

February 5, 2010– 16

Version 1.1 of the Wolfram|Alpha App for the iPhone & iPod is now available in the App Store. The new version includes a number of new features that continue to improve the app’s unique mobile Wolfram|Alpha experience. Perhaps its most iconic feature, the specialized keyboards that greet you when you first open the Wolfram|Alpha App, have been painstakingly constructed to ease the burden of entering queries, whether you’re converting from pounds Pounds key to euros Euro key or computing a numerical value for the Weierstrass p-function Weierstrass p-function. Our goal in creating these keyboards was to form families of characters that naturally occur together both in common use and in traditional mathematical applications. We also wanted mathematical expressions to look and feel natural to enhance usability and understanding. Version 1.1 has four specialized keyboards: the default keyboard, the “math” keyboard accessed by the right-shift key Integral Infinity key , the “Greek” keyboard accessed by one press of the left-shift key Right-shift key on the math keyboard, and the “symbol” keyboard accessed by a second press of the left-shift key.

The default keyboard in the Wolfram|Alpha App for the iPhone and iPod touch The math keyboard in the Wolfram|Alpha App for the iPhone and iPod touch The Greek keyboard in the Wolfram|Alpha App for the iPhone and iPod touch The symbol keyboard in the Wolfram|Alpha App for the iPhone and iPod touch

To determine the optimal keyboard layout, we scoured Wolfram|Alpha’s server logs for the most commonly entered phrases that have characters with meaning in Wolfram|Alpha. Given that Wolfram|Alpha is built on Mathematica, one of its core strengths is advanced mathematics. True to form most of the commonly typed characters are related to math. For example, you would generally type the word “integrate” to compute an integral on the Wolfram|Alpha website. In the Wolfram|Alpha App you could simply type the Integral key key on the math keyboard. The same is true for other symbols common in math, such as Theta key and Pi key. Specifying geometric shapes, such as a triangle, is straightforward as well.

Entering an integral into the Wolfram|Alpha App for the iPhone and iPod touch Using Greek characters in the Wolfram|Alpha App for the iPhone and iPod touch More »

January 22, 2010– 11

Is it cheating to use Wolfram|Alpha for math homework? That was the presentation topic of Conrad Wolfram, Wolfram Research’s Director of Strategic Development, at the TEDx Brussels conference at the European Parliament. Conrad shares his viewpoint in this thought-provoking (and often entertaining) video.

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January 14, 2010– 30

Need a tutor for solving equations? Solving equations is just one of hundreds of mathematical tasks that can be done using Wolfram|Alpha. Wolfram|Alpha can solve equations from middle school level all the way through college level and beyond. So next time you are stumped on an equation, consult Wolfram|Alpha for a little help.

Let’s start with the simpler stuff. Wolfram|Alpha can easily solve linear and quadratic equations, and even allows you to view a step-by-step solution of each problem.

Wolfram|Alpha solves 2x+1=3 for x

What if the roots of the equation are complex? No worries; Wolfram|Alpha has no trouble solving equations over the complex plane.

Wolfram|Alpha solves 3x^2+x+1=0 for x

Wolfram|Alpha can also solve cubic and quartic equations in terms of radicals.

Wolfram|Alpha solves Wolfram|Alpha solves x^3-3x^2-3x+11=0 for x

Of course, some solutions are too large or cannot be represented in terms of radicals; Wolfram|Alpha will then return numerical solutions with a “More digits” button. More »

December 29, 2009– 3

Prior to releasing Wolfram|Alpha into the world this past May, we launched the Wolfram|Alpha Blog. Since our welcome message on April 28, we’ve made 133 additional posts covering Wolfram|Alpha news, team member introductions, and “how-to’s” in a wide variety of areas, including finance, nutrition, chemistry, astronomy, math, travel, and even solving crossword puzzles.

As 2009 draws to a close we thought we’d reach into the archives to share with you some of this year’s most popular blog posts.

April

Rack ’n’ Roll

Take a peek at our system administration team hard at work on one of the
many pre-launch projects.
Continue reading…

May

The Secret Behind the Computational Engine in Wolfram|Alpha

Although it’s tempting to think of Wolfram|Alpha as a place to look up facts, that’s only part of the story. The thing that truly sets Wolfram|Alpha apart is that it is able to do sophisticated computations for you, both pure computations involving numbers or formulas you enter, and computations applied automatically to data called up from its repositories.

Why does computation matter? Because computation is what turns generic information into specific answers. Continue reading…

Live, from Champaign!

Wolfram|Alpha just went live for the very first time, running all clusters.

This first run at testing Wolfram|Alpha in the real world is off to an auspicious start, although not surprisingly, we’re still working on some kinks, especially around logging.

While we’re still in the early stages of this long-term project, it is really gratifying to finally have the opportunity to invite you to participate in this project with us. Continue reading…

June

Wolfram|Alpha Q&A Webcast

Stephen Wolfram shared the latest news and updates about Wolfram|Alpha and answered several users’ questions in a live webcast yesterday.

If you missed it, you can watch the recording here. Continue reading… More »

December 21, 2009– 7

We’re really catching the holiday spirit here at Wolfram|Alpha.

We recently announced our special holiday sale for the Wolfram|Alpha app. Now we are launching our first-ever Wolfram|Alpha “Holiday Tweet-a-Day” contest.

Happy holidays from Wolfram|Alpha!

Here’s how it works.

From tomorrow, Tuesday, December 22, through Saturday, January 2, we’ll use Twitter to give away a gift a day. Be the first to retweet our “Holiday Tweet-a-Day” tweet and you get the prize! You can double your chances to win by following and playing along with Wolfram Research.

Start following us today so you don’t miss your chance to win with our Wolfram|Alpha “Holiday Tweet-a-Day” contest.

December 1, 2009– 186

(January 15, 2014 Update: Step-by-step solutions has been updated! Learn more.)

Have you ever given up working on a math problem because you couldn’t figure out the next step? Wolfram|Alpha can guide you step by step through the process of solving many mathematical problems, from solving a simple quadratic equation to taking the integral of a complex function.

When trying to find the roots of 3x2+x–7=4x, Wolfram|Alpha can break down the steps for you if you click the “Show steps” button in the Result pod.
Wolfram|Alpha finds the roots of 3x2+x–7=4x

As you can see, Wolfram|Alpha can find the roots of quadratic equations. Wolfram|Alpha shows how to solve this equation by completing the square and then solving for x. Of course, there are other ways to solve this problem! More »

November 11, 2009– 37

Having announced the Wolfram|Alpha API just over three weeks ago, I am pleased to share in announcing Microsoft’s Bing decision engine as one of our first API customers.

Starting today, Wolfram|Alpha’s knowledge, computed from expertly curated data, will enrich Bing’s results in select areas across nutrition, health, and advanced mathematics. Wolfram|Alpha provides immediate, unbiased, and individualized information, making it distinctly different from what has traditionally been found through web search. By using Wolfram|Alpha, Bing recognizes the complementary benefits of bringing computational knowledge to the forefront of the search experience.

By using our API, Bing will be able to seamlessly access the tens of thousands of algorithms and trillions of pieces of data from Wolfram|Alpha, and directly incorporate the computations in its search results.

Microsoft’s initiative and interest in Wolfram|Alpha began earlier this year. In fact, there is an interesting story that circulates within our walls around some of our early discussions with Microsoft.

Highlighting examples of Wolfram|Alpha to the most senior executives at Microsoft, Stephen Wolfram entered the query “2^2^2^2^2”. Upon seeing the result, Bill Gates interrupted to say, “What, is that right?”

A profound silence fell over the entire room.

Stephen replied, “We do mathematics!”

Amused, Stephen, Bill, and the other executives dissected the calculation and determined that the result was, indeed, correct. Microsoft continues to pepper us with questions to this day, reflecting its continued enthusiasm in Wolfram|Alpha.

We applaud Microsoft’s vision and foresight in augmenting their search with Wolfram|Alpha, and we look forward to a fulfilling and productive partnership.

October 21, 2009– 2

When we were preparing for Wolfram|Alpha Homework Day, a tweet from @mwarntzen caught our attention: “just learned how to use an abacus while messing around on Wolfram|Alpha.” It brought smiles to our faces to think about this ancient tool being explored with our modern-day technology, and to think about how learning tools have evolved.

The abacus was developed as a counting tool long before the time of calculators. More modern versions of the abacus are wooden frames with rows of beads used for counting. Query “abacus” in the computation bar, and Wolfram|Alpha will return an abacus page (as shown below). You can enter a number, and Wolfram|Alpha will show you how the number would appear on a modern Chinese abacus. More »

September 8, 2009– 12

college-is-hard

We know college is hard. So we’re highlighting examples of how Wolfram|Alpha can make subjects and concepts a bit easier to learn. Wolfram|Alpha is a free computational knowledge engine that can help you tackle everything from calculus, to computing the number of pages for a double-spaced 1000-word essay, to comparing the flash points of methane, butane, and octane, to figuring just how much money it’s going to cost you to drive home to do your laundry. Check out a quick introduction to Wolfram|Alpha from its creator, Stephen Wolfram.

We want to help you take full advantage of this resource. Over the next term, we’ll be highlighting helpful computations and information here on the blog, and even providing ways you can get involved with our company. (Would you like to be a part of the Wolfram|Alpha Team on your campus? Stay tuned to find out how you can be involved.) For this post we selected several of our favorite examples to help you start thinking about how you can use Wolfram|Alpha in your courses, and in your always-changing college life. More »

September 3, 2009– 16

We use this blog to provide helpful tips on using Wolfram|Alpha. So when a relevant screencast caught our eye on Twitter—”Wolfram|Alpha for Calculus Students,” produced by Robert Talbert, PhD, an associate professor of mathematics and computing science at Franklin College—we wanted share it with you. We think his straightforward video is a great demonstration of just how valuable Wolfram|Alpha is for students. In the screencast, Professor Talbert discusses the concept of Wolfram|Alpha, and illustrates how it solves problems such as factoring or expanding expressions, solving quadratic equations, and more.

Wolfram|Alpha for calculus students screen cast

The screencast covers just a few of the ways educators and students are using Wolfram|Alpha. Are you an instructor who has found innovative ways to incorporate Wolfram|Alpha into your lesson plans? Or are you a student using Wolfram|Alpha to assist in your studies? You can join others having these conversations on the Wolfram|Alpha Community site.

June 29, 2009– 29

Stephen Wolfram recently received an award for his contributions to computer science. The following is a slightly edited transcript of the speech he gave on that occasion. (The audio version of the original speech is here.)

I want to talk about a big topic here today: the quest for computable knowledge. It’s a topic that spans a lot of history, and that I’ve personally spent a long time working on. I want to talk about the history. I want to talk about my own efforts in this direction. And I want to talk about what I think the future holds. More »

June 25, 2009– 7

He’s developing some of the most popular frameworks in Wolfram|Alpha. She’s on the front lines of handling and managing all of your feedback. Meet them both in Part 3 of our video series, “A Moment with the Wolfram|Alpha Developers”:

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Other interviews with Wolfram|Alpha team members can be found in Part 1 and Part 2 of this video series.

June 7, 2009– 18

Members of the Wolfram|Alpha development team give insight on what goes into building a system like Wolfram|Alpha and how exciting it is to be a part of the project.

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See Part 1 here.

In the first 24 hours of our launch weekend, we received nearly 10,000 messages forwarded from the feedback forms on the bottom of each Wolfram|Alpha page. The compliments have been very gratifying.

The feedback has been insightful and entertaining. You’ve offered lots of suggestions, from additional domains and analysis to computations that have gone awry. We thought you might enjoy seeing some of the feedback we’ve received. More »

May 13, 2009– 1

Some of you have asked whether you’ll be able to use Wolfram|Alpha for challenging math. Of course!

Remember your old friend pi?

Wolfram|Alpha computers pi to 100 digits

More »