One of the most popular queries on Wolfram|Alpha is for definite integrals. So we’re especially excited to announce that Step-by-step solutions for these are now available! The general method used to find the steps for definite integrals is to tap into the already existing “Show steps” functionality for indefinite integration, and then to use the fundamental theorem of calculus. More »

Every four years for more than a century there’s been an International Congress of Mathematicians (ICM) held somewhere in the world. In 1900 it was where David Hilbert announced his famous collection of math problems—and it’s remained the top single periodic gathering for the world’s research mathematicians.

This year the ICM is in Seoul, and I’m going to it today. I went to the ICM once before—in Kyoto in 1990. *Mathematica* was only two years old then, and mathematicians were just getting used to it. Plenty already used it extensively—but at the ICM there were also quite a few who said, “I do *pure* mathematics. How can *Mathematica* possibly help me?”

It’s that time of year—the torrential downpour of spring showers is paving the way for leafy trees and blooming flowers. While some may love getting caught in the rain, many of us are looking eagerly at the promise of sunshine and bonfires to come.

So what can you do to pass the time during the soggy days ahead?

Well, Wolfram|Alpha can help you figure out how best to stay dry! When you have no other choice but to venture out into the wet world around you, what can you do to minimize your chances of getting soaked? Should you walk or run that 100 yards to the car? Does it really make that big of a difference?

Well, if you walk a leisurely 1 mph, you will acquire 204.8 cubic centimeters of rainwater. Ugh. That’s like someone throwing a baseball-sized water balloon at you. More »

Every year, members of some of the biggest and most influential mathematics associations get together to dedicate the month of April to math awareness. The initiative was set in motion in 1986 by President Reagan, who said, “To help encourage the study and utilization of mathematics, it is appropriate that all Americans be reminded of the importance of this basic branch of science to our daily lives.”

More »

Today is the birthday of two famous physicists, though that is not how they are commonly remembered. More »

Last week the weather here was pretty bizarre. Overnight, it went from 66°F and outdoor soccer matches to 28°F and a blanket of snow and ice. You know what else is pretty bizarre? Some of the things people can—and do—ask Wolfram|Alpha. So in case, like us, you’re stuck inside for a few more weeks of winter and in need of inspiration, read on for a few examples of some of the more… unique types of queries that you, too, can ask Wolfram|Alpha. More »

Whether math is your favorite subject or the bane of your existence, we can think of at least one day on which you might look forward to math class. Every March 14, many teachers take it upon themselves to indulge students’ sugar cravings with a variety of pies, but not before forcing them into some kind of plate-measuring, digit-memorizing, or Pi-ku-writing event (yes, these are real things). More »

The start of the XXII Olympic Winter Games means one thing for me: at least six hours a day of watching people ski down treacherous slopes, do crazy 720-degree spins with their snowboards, and perform triple toe loop jumps. Whether or not you’re spending every waking moment watching athletically superior individuals accomplish seemingly impossible feats, you can take this opportunity to explore some of Wolfram|Alpha’s math and physics calculators. Now Wolfram|Alpha can add a fun science and math spin to the Olympics. More »

We are happy to announce the *Mathematica* Summer Camp 2014! This camp, for advanced high school students entering grades 11 or 12, will be held at Bentley University in Waltham, Massachusetts July 6–18. If you are ready for two weeks of coding fun, apply now on our website. Students who attend the camp have a unique opportunity to work one-on-one with Wolfram mentors in order to build their very own project in *Mathematica*. More »

As the winter term kicks into gear, you might start hoping you had an ODE-solving pet monkey as the math and physics problem sets start piling up. Now, we do not offer ODE-solving primates at the moment, but we can help you with your differential equations problem sets. Wolfram|Alpha can solve a plethora of ODEs, each using multiple methods. More »

Wolfram|Alpha has been adding more step-by-step functionality to accommodate the needs of students at various levels of education. Now with Wolfram Problem Generator and Step-by-step solutions, students essentially have their own private tutor to help them better understand their homework and advance their knowledge. More »

Wolfram|Alpha Pro has become well known for its ability to show not only the answer to a question, but also the Step-by-step solution to show how to find the answer. Since its first release, we have developed new features and content for Step-by-step solutions. We’ve added hints and the ability to walk through problems one step at a time, and we’ve added support to show multiple methods for solving problems whenever possible. More »

Even our biggest Wolfram|Alpha fans may have missed some of the stories we’ve shared this year—but here’s your chance to catch up! Without further ado, our 10 most popular Wolfram|Alpha Blog posts from 2013 More »

It wasn’t all that long ago that I was in high school—and I can still remember that looming dread of mid-terms that comes around this time of year. Getting a head start on studying may seem impossible, but Wolfram|Alpha has plenty of ways to make it a little easier! More »

I was a raging sugar-holic as a kid. (Let’s face it, who wasn’t?) So, naturally, Halloween was a glorious, much-anticipated, high-energy free-for-all. My brother and I used to return from trick-or-treating dragging heavy pumpkin buckets and overstuffed pillowcases behind us—the candy wrappers would make that *crinkle-squish* sound as we dumped out our riches to sort, trade, construct giant candy pyramids… and then devour. More »

We’re excited to introduce some brand new features to our step-by-step functionality! Wolfram|Alpha can now guide you through factoring polynomials and completing the square, in addition to being updated to include FOIL and the binomial expansion theorem. Let’s take a look. More »

By now, most of you students are likely getting into the thick of the academic year, preparing for the first wave of exams and projects and presentations to come your way… But don’t freak out just yet! Here’s a list of Wolfram’s most recent apps and programs that might help make your life a little easier. After all, it never hurts to have a few powerful resources on your side. More »

We are proud to announce Wolfram Problem Generator, a website where students decide which topic they want to practice and we provide the questions and solutions. This is an exciting new way to help students with their classes: previously, students provided their own practice questions and Wolfram|Alpha helped them find answers with Step-by-step solutions. Students can now ask Wolfram|Alpha for help with practice and homework questions and can do practice problems with Wolfram Problem Generator. More »

Did you know that October 11 is World Egg Day?

Like most people, you probably go to the grocery store and eventually end up in the dairy aisle, where, unless you’re a vegetarian or vegan, you probably pick out a dozen eggs and place them into your cart without a second thought. They’re pretty much a staple food—from savory breakfasts to the sweet wonders of baking. More »

A lot of cool things happened this summer on Wolfram|Alpha and the Wolfram|Alpha Blog. And just wait—we have even better stuff planned for the coming months! But in case you missed it, here’s a quick recap of some of our best posts from this summer. More »

As a physics major, I sometimes find myself solving interesting problems for fun. However, I have never been very quick at doing simple math in my head, so I often resort to using computers to do tedious calculations. This keeps me interested in the answer to the problem and not focused on the details of the calculations, which can be very boring. Computers are much faster at doing calculations than I am, and Wolfram|Alpha is no exception: for instance, arctan(3^4^3)/pi. More »

As part of our ongoing plan to expand Wolfram|Alpha’s numerical method functionality to more kinds of algorithms, we recently addressed solving differential equations. There are a number of different numerical methods available for calculating solutions, the most common of which are the Runge–Kutta methods. This family of algorithms can be used to approximate the solutions of ordinary differential equations. More »

We’re pleased to announce that computerbasedmath.org is partnering with UNICEF for the third Computer-Based Math™ Education Summit. More »

*(This is the third post in a three-part series about electrostatic and magnetostatic problems involving sharp edges.)*

In the first blog post of this series, we looked at magnetic field configurations of piecewise straight wires. In the second post, we discussed charged cubes and orbits of test particles in their electric field. Today we will look at magnetic systems, concretely, mainly at a rectangular bar magnet with uniform magnetization. More »

As we continue to expand the functionality of Wolfram|Alpha, we want to include not only the symbolic and exact results, but also allow you the option to explore the numerical approximations for solving mathematical problems such as differential equations and integrals. These methods, both simple and complex, continue to underpin many of our modern day calculations. More »

Last year we greatly expanded our step-by-step functionality for mathematical problems in Wolfram|Alpha. These tools can be a great aid for students to understand the methods of solving integrals and equations symbolically. But what if we are not looking for a symbolic result? What if we need a numerical approximation? For example, we might be looking at an integral or differential equation that cannot be solved in a closed form, or we might just want to find where an equation intercepts the *x *axis. More »

In my last blog post, I discussed how to construct closed-form trigonometric formulas for sketches of people’s faces. Using similar techniques, Wolfram|Alpha has recently added a collection of hundreds of such closed-form curves for faces, shapes, animals, logos, and signatures. More »

Recently, we’ve been showcasing some new math features in Wolfram|Alpha, particularly those relevant to primary and secondary school students. Our idea is that when using Wolfram|Alpha, learning math can be a fun experiment. We’d like you to think of Wolfram|Alpha as your own infinitely patient robot, which you can use to explore mathematical ideas, test your knowledge, and generally answer any specific math question you have. More »

Here at Wolfram Research and at Wolfram|Alpha we love mathematics and computations. Our favorite topics are algorithms, followed by formulas and equations. More »

Mathematics has many faces. It deals with diverse objects such as integers, points and lines, equations, graphs, categories, thousands of different spaces (from R^{3} to Hilbert, Banach, Fréchet, …), and so on. Mathematics can be constructive or just prove the existence of certain structures. Wolfram|Alpha has made a good fraction of computable constructive mathematics freely available to everyone: from line through (2,3) and (4, 5) to Fréchet derivative of (integrate exp(-f(x)^2) dx from -inf to inf) wrt f(y) to fractional derivative of ln(z). More »

A century ago, Srinivasa Ramanujan and G. H. Hardy started a famous correspondence about mathematics so amazing that Hardy described it as “scarcely possible to believe.” On May 1, 1913, Ramanujan was given a permanent position at the University of Cambridge. Five years and a day later, he became a Fellow of the Royal Society, then the most prestigious scientific group in the world. In 1919 Ramanujan was deathly ill while on a long ride back to India, from February 27 to March 13 on the steamship Nagoya. More »

Some common questions from the many student users of Wolfram|Alpha include “Isn’t cbrt(-8) = -2?” and “Why doesn’t the plot of the cube root include the negative part?” The answers are that -2 is just one of the three cube roots of -8, and that *Mathematica*, the computational engine of Wolfram|Alpha, has always chosen the principal root, which is complex valued. More generally, odd roots of negative numbers are typically assumed to be complex. You can see this in the output of (-8)^(1/3). More »

In our previous post about expanding Step-by-step solutions, we introduced a revamped equation solver. I’m proud to say that it has now been extended to solve systems of linear equations. In addition, you have four different methods to choose from when looking for a solution! These methods are elimination, substitution, Gaussian elimination, and Cramer’s rule. Let’s look at *x* + *y* = 5, *x* – *y* = 1 to see all four methods in action. More »

As a continuation of our new math content blog series, I’d like to talk about an exciting new Step-by-step feature. Previously I talked about differential equations, but today I’d like to look toward the other end of the spectrum: basic arithmetic. Wolfram|Alpha can now help you work out long addition, subtraction, multiplication, and division with hints and steps! Let’s go ahead and look at some examples. More »

The Wolfram|Alpha math team adds new and exciting content to Wolfram|Alpha on a daily basis! In fact, over the past few months we’ve added a wide range of features and we will be introducing them in a blog series here. Lately, we’ve made an effort to make Wolfram|Alpha a powerful learning tool for those learning arithmetic! If you are either teaching or learning addition, multiplication, or basic math word problems, Wolfram|Alpha can help you. More »

To celebrate Halloween, last year we discussed what you can do with 1,818 pounds of pumpkin. It was a popular blog post, and it put an awful lot of smiles on peoples’ faces. An entire lamina (filled shape) of smiles, in fact. More »

Wolfram|Alpha answers millions of queries every day. For instance:

What do the following two math problems have in common?

- If I have 12 apples, and Jane has 7, and then Jane gives 2 apples to me, how many more apples do I have than Jane?
- (12 + 2) – (7 – 2)

Answer: two things, actually. More »

In my last blog post, we looked at various examples of electrostatic potentials and magnetostatic fields. We ended with a rectangular current loop. Electrostatic and magnetostatic potentials for squares, cubes, and cuboids typically contain only elementary functions, but the expressions themselves are often quite large compared with simple systems with radial symmetry. In the following, we will discuss some 3D charge configurations that have sharp edges. More »

Step-by-step solutions, one of the most popular features for mathematics in Wolfram|Alpha, has just received a dramatic expansion in its functionality! With our new interface, you now have the ability to walk through all of our Step-by-step solutions at your own pace, revealing only one step at a time. Some of our programs will offer to guide you with hints when walking through solutions. And for common math problems, we can even show multiple ways to find the solutions. More »

While Wolfram|Alpha can do many sorts of computations, mathematical calculations are one of its particular specialties. In fact, using the power of *Mathematica*‘s computational capabilities under the hood, Wolfram|Alpha can do many things, ranging from the very simple to the fiendishly complicated, with mathematical functions. More »

In a previous post, I discussed Wolfram|Alpha’s facility with probability distributions, such as the distribution of probability among the possible totals that can be shown with a pair of fair dice: More »

*(This is the first post in a three-part series about electrostatic and magnetostatic problems involving sharp edges.)*

*Mathematica* can do a lot of different computations. Easy and complicated ones, numeric and symbolic ones, applied and theoretical ones, small and large ones. All by carrying out a *Mathematica* program.

Wolfram|Alpha too carries out a lot of computations (actually, tens of millions every day), all specified through free-form inputs, not *Mathematica* programs. More »

A study of mathematical probability typically begins by considering a random situation with a low number of possible outcomes, like a coin flip or a toss of a die. Wolfram|Alpha has long been able to compute probabilities involving coins and dice. More »

Imagine you are building a roller coaster. You need to create a curved shape for the track, which will be designed on a computer before being built out of metal. You want a curve that is fun to travel along, which means you want a lot of sharp curves—but not too sharp, unless you want your amusement park guests to get sick or pass out.

A similar consideration is faced by engineers building a railroad or a highway: you want the path of the road to have curves that are not too sharp—in this case, to prevent the cars or trains from having to slow down (reducing efficiency) or even to wreck. The reason, of course, is that when you travel along a strongly curved section of track or road, you feel an acceleration. The higher the curvature, the stronger the acceleration, all other things being equal. However, the acceleration you feel depends on how fast you are going along the track (the faster you go, the greater the acceleration), while curvature is a property intrinsic to the track itself.

More »

It’s usually in precalculus class that students are first exposed to the more exotic and subtle aspects of functions on the real line. This first exposure comes through studying limits and discontinuities.

Most functions that we see every day, from the parabolic arc of a thrown ball to the exponential growth of money in a bank account, are “continuous.” That is, they don’t change their value suddenly. Thought of another way, a continuous function is one you can graph without having to lift your pen up from the paper.

Conversely, a discontinuity of a function is a point where the value of the function experiences a sudden change. In technical language, a discontinuity of a function from reals to reals is a point where either the left- or right-hand limit does not exist, or where these limits exist but aren’t both equal to the value of the function at this point.

Wolfram|Alpha now has the ability to find and analyze the discontinuities of most functions of real numbers. When it comes to such functions, there are three main kinds of discontinuities.

An “infinite” discontinuity is a point where the function increases to infinity and/or decreases to negative infinity (i.e., where it has a vertical asymptote). 1/x is the standard example:

One of the features of calculus is the ability to determine the arc length or surface area of a curve or surface. An arc length is the length of the curve if it were “rectified,” or pulled out into a straight line. You can also think of it as the distance you would travel if you went from one point to another along a curve, rather than directly along a straight line between the points.

To see why this is useful, think of how much cable you would need to hang a suspension bridge. The shape in which a cable hangs by itself is called a “catenary,” but with a flat weight like a roadway hanging from it, it takes the shape of a more familiar curve: a parabola.

Wolfram|Alpha has been steadily growing since its initial release nearly three years ago, and this growth is directed, in part, by the queries it receives. For example, the Wolfram Education Portal was created largely in response to the obvious demand for Wolfram|Alpha in the classroom. As a more specific example, we’ve recently enabled Wolfram|Alpha to respond to domain and range queries for real functions.

The domain of a real function is the set of real numbers that can be plugged in so that the function returns a real value. If, for example, we wish to evaluate f(*x*) = √(*x* + 2) / (*x* – 1), then we should ensure that *x* + 2 > = 0 and *x* – 1 ≠ 0:

Wolfram|Alpha has become well-known for its ability to perform step-by-step math in a variety of areas. Today we’re pleased to introduce a new member to this family: step-by-step differential equations. Differential equations are fundamental to many fields, with applications such as describing spring-mass systems and circuits and modeling control systems.

From basic separable equations to solving with Laplace transforms, Wolfram|Alpha is a great way to guide yourself through a tough differential equation problem. Let’s take a look at some examples.

Wolfram|Alpha can show the steps to solve simple differential equations as well as slightly more complicated ones like this one:

What rule gives the integer sequence 3, 10, 17, 18, 7, …? Wolfram|Alpha can easily find that this sequence comes from a simple cubic polynomial, -*x*^{3} + 6*x*^{2} – 4*x* + 2.

A different sequence, 1, 1, 3, 7, 22, 82, 333, 1448, … can be identified as the sequence of the polyhexes. After that, the input sequence of the polyhexes recovers the above sequence.

More »

Finding the tangents and normals of a mathematical function or relation is one of the most common exercises in any calculus course. In this post, I’ll show you the newest functionality in Wolfram|Alpha for discovering and investigating them.

The simplest example of a tangent is the “tangent line” to a one-dimensional curve in the plane. Graphically, the tangent line is a line that “just touches” the curve at some point, so that if it were moved just slightly, this one point of contact would become two.

If you ask Wolfram|Alpha for the tangent line of a specific function and point, it gives it in both graphical and algebraic/numerical form:

What do your alarm clock, thermostat, coffeemaker, car radio, anti-lock brakes—and almost every other electrical and mechanical device you encounter in your daily life—all have in common? They are all examples of “control systems,” one of the most ubiquitous yet unseen modern technologies. A control system is any system or device that controls or regulates the behavior of another system. Using various kinds of sensors and actuators, these systems automatically control most common appliances, industrial processes, and even your body’s own biological processes!

Take your home’s humble thermostat. The temperature of your home depends on many factors, especially how long and how recently the home’s furnace was on. With a thermostat installed, the reverse is also true: the state of the furnace depends on the temperature of the house (it comes on if the temperature is too low, and turns off if the temperature is too high). There is a closed loop of causation formed between the home’s temperature and the state of the furnace. By design, the thermostat creates a kind of closed loop called a “negative feedback loop,” which tends to stabilize the temperature around a desired value. Most control systems are like this: sensors feed information back into the system, which is then used to decide on an action. More »

The hyperlink has been one of the most powerful tools of the information age. Links make it easier to navigate the complex web of information online by combining the information itself with the method for retrieving it. Clicking a link means “tell me more about this thing,” which naturally lends itself to “surfing.”

At Wolfram|Alpha, we strive to integrate and leverage technologies to create the most powerful computational capabilities and user experiences possible. In Wolfram|Alpha, the output comes in the form of a “report.” If you want to know more about something in the output of an Wolfram|Alpha query, clicking it as a link will generate another such report. Though we’ve had links in Wolfram|Alpha for a while, we’ve recently taken them to the next (computable) level: Wolfram|Alpha now computes links dynamically based on the output generated by your query.

Clicking a link basically feeds the plaintext of that link back into Wolfram|Alpha, creating new output with new links. Thus the navigational ability of the world wide web and the computational ability of Wolfram|Alpha are now intertwined and can feed off each other. You can now surf Wolfram|Alpha like you can surf the Internet. More »

Permutations are among the most basic elements of discrete mathematics. They are used to represent discrete groups of transformations, and in particular play a key role in group theory, the mathematical study of symmetry. Permutations and groups are important in many aspects of life. We all live on a giant sphere (the Earth) whose rotations are described by the group SO(3) (the special orthogonal group in 3 dimensions). On the micro-scale, the Hungarian-American physicist Eugene Wigner (November 17, 1902–January 1, 1995), who received a share of the Nobel Prize in Physics in 1963, discovered the “electron permutation group”, one of many applications of permutation groups to quantum mechanics.

Permutations deserve a full treatment in Wolfram|Alpha. Since *Mathematica* 8 provides new functionality to work with permutations using both list and cyclic representations, we now have a powerful new way of working with permutations using natural language.

Let’s first define permutations, then discuss how to work with them in Wolfram|Alpha. A permutation of a set *X* is a bijective (one-to-one and onto) mapping of *X* to itself. There is a convenient way of specifying a permutation *α* of a finite set of *n* elements: write down the numbers 1, 2, …, *n* in a row and write down their images under *α* in a row beneath:

As most Wolfram|Alpha blog readers know, the engine behind the Wolfram|Alpha computational knowledge engine is Wolfram Research’s powerful mathematics and computation software, *Mathematica*. Ironically, while Wolfram|Alpha contains thousands of datasets on diverse and sundry subject areas, until very recently, its computable knowledge of the *Mathematica* language itself has been somewhat limited. More »

One fall evening in 1843, a man walked past the Brougham Bridge along the Royal Canal in Dublin, Ireland. Suddenly, he felt a flash of insight so strong he was compelled to etch his thoughts into the rock on the side of the bridge. This is what he wrote:

i^{2}=j^{2}=k^{2}=ijk= -1

The man was mathematician William Rowan Hamilton, and the insight was of a number system that could represent forces and motions in three-dimensional space. Hamilton called his numbers “quaternions”, because each has four parts: a real number part, and three other parts labeled with *i*, *j*, and *k*, each of which is also a real number. For example, 2 + 3*i* + 0.342*j* – 2*k* is a quaternion. More »

Today we are releasing our Wolfram Pre-Algebra Course Assistant App for iOS, expanding our planned series of Course Assistant Apps built using Wolfram|Alpha technology.

Game theory is a rich branch of mathematics that deals with the analysis of games, where, mathematically speaking, a “game” can be defined as a conflict involving gains and losses between two or more opponents who follow formal rules.

Mathematical games can be very simple, such as the game of chicken (which is not recommended in practice):

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.

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 »

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)

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:

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 »

Today we are pleased to announce the Wolfram Tides Calculator and Wolfram Fractals Reference Apps for iPhone, iPod touch, and iPad. These are the first two in our series of reference apps that utilize Wolfram|Alpha technology to shed light on some fascinating subjects. Like our series of Wolfram Course Assistant Apps, the Wolfram Reference Apps are each designed with an optimized interface and specialized keyboards to enhance usability for mobile users.

The Wolfram Tides Calculator will become your go-to guide for tide information. Calculate the present tide or today’s high and low tides, do historical computations, or plan your vacation using the tide forecast. The app can automatically detect your current location or provide data from around the world.

For example, with the Tides Calculator you can enter a future date and location to see the tide forecast for a specific date. Here is the prediction for tides in Miami on July 4:

More »

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:

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 »

Plotting functions in the Cartesian plane is such a simple task with Wolfram|Alpha: just enter the function you are looking to graph, and within seconds you will have a beautiful result. If you are feeling daring, enter a multivariate function, and the result will be a 3D Cartesian graph. Wolfram|Alpha is certainly not limited to Cartesian plotting; we have the functionality to make number lines, 2D and 3D polar plots, 2D and 3D parametric plots, 2D and 3D contour plots, implicit plots, log plots, log-linear plots, matrix plots, surface of revolution plots, region plots, list plots, pie charts, histograms, and more. Furthermore, in Wolfram|Alpha we can generate specialized plots for illustrating asymptotes, cusps, maxima, minima, inflection points, saddle points, solutions of ordinary differential equations, poles, eigenvalues, series expansions, definite integrals, 2D inequalities, interpolating polynomials, least-squares best fits, and more. Let’s take a look at the plotting functionality in Wolfram|Alpha, some of which is newly improved!

We will start simple with 2D Cartesian plots.

Here we plot sin(√7*x*)+19cos(*x*) for *x* between -20 and 20.

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:

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

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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.

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

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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.

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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:

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/*z*^{n} 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 *z* – *z*_0. As an example, let’s look at the Laurent expansion of 1/Sin[*z*] at *z* == 2 Pi:

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:

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

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.

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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”.

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 3*x*<7*x*^2+2”? This plot will show that the solutions to this inequality are all real numbers between negative and positive infinity.

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Consider packing circles inside a circular container, or less abstractly, placing cookie dough on a cookie sheet. In the case of cookies, which expand to be a roughly circular shape, you don’t want them so close that they run into each other. At the same time, you don’t want them too far apart, because that would mean fewer cookies.

One of the latest features of Wolfram|Alpha is the ability to get information about packing circles into circles.

For instance, suppose you have a circular baking sheet with a diameter of 12 inches, and you want to make 20 cookies. You can ask Wolfram|Alpha “pack 20 circles in a diameter 12 inch circle”; not only does it give you a diagram of the densest packing, but also the largest radius of the circular cookies on the 12-inch baking sheet.

Or you might know the size of the cookies and want to know how many can fit? One way to get the answer would be “pack r=1 circles in a diameter 12 circle”.

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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:

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.

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.

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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.

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 »

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:

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 »

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:

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 »

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.

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 »

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.

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 »

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.

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 »

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.

“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”:

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.

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

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

Wolfram|Alpha computes things. While the use of computations to predict the outcomes of scientific experiments, natural processes, and mathematical operations is by no means new (it has become a ubiquitous tool over the last few hundred years), the ease of use and accessibility of a large, powerful, and ever-expanding collection of such computations provided by Wolfram|Alpha is.

Virtually all known processes occur in such a way that certain functionals that describe them become extremal. Typically this happens with the action for time dependent processes and quantities such as the free energy for static configurations. The equations describing the extremality condition of a functional are frequently low-order ordinary and/or partial differential equations and their solutions. For example, for a pendulum: Frechet derivative of Integrate[x'[t]^2/2 – Cos[x[t]], {t, -inf, inf}] wrt x[tau]. Unfortunately, if one uses a sufficiently realistic physical model that incorporates all potentially relevant variables (including things like friction, temperature dependence, deformation, and so forth), the resulting equations typically become complicated—so much so that in most cases, no exact closed-form solution can be found, meaning the equations must be solved using numerical techniques. A simple example is provided by free fall from large heights:

On the other hand, some systems, such as the force of a simple spring, can be described by formulas involving simple low-order polynomial or rational relations between the relevant problem variables (in this case, Hooke’s law, *F* = *k x*):

Over the last 200+ years, mathematicians and physicists have found a large, fascinating, and insightful world of phenomena that can be described exactly using these so-called special functions (also commonly known as “the special functions of mathematical physics”), the class of functions that describe phenomena between being difficult and complicated. It includes a few hundred members, and can be viewed as an extension of the so-called elementary functions such as exp(z), log(z), the trigonometric functions, their inverses, and related functions.

Special functions turn up in diverse areas ranging from the spherical pendulum in mechanics to inequivalent representations in quantum field theory, and most of them are solutions of first- or second-order ordinary differential equations. Textbooks often contain simple formulas that correspond to a simplified version of a general physical system—sometimes even without explicitly stating the implicit simplifying assumptions! However, it is often possible to give a more precise and correct result in terms of special functions. For instance, many physics textbooks offer a simple formula for the inductance of a circular coil with a small radius:

While Wolfram|Alpha knows (and allows you to compute with) this simple formula, it also knows the correct general result. In fact, if you just ask Wolfram|Alpha for inductance circular coil, you will be simultaneously presented with two calculators: the one you know from your electromagnetics textbook (small-radius approximation) and the fully correct one. And not only can you compute the results both ways (and see that the results do differ slightly for the chosen parameters, but that the difference can get arbitrarily large), you can also click on the second “Show formula” link (near the bottom of the page on the right side) to see the exact result—which, as can be seen, contains two sorts of special functions, denoted E(m) and K(m) and known as elliptic integrals: More »

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.

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 »

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.

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 »

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:

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 »

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!

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”.

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 »

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:

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 »

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 to euros or computing a numerical value for the 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 , the “Greek” keyboard accessed by one press of the *left-shift* key , and the “symbol” keyboard accessed by a second press of the *left-shift* key.

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 key on the math keyboard. The same is true for other symbols common in math, such as and . Specifying geometric shapes, such as a triangle, is straightforward as well.

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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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.

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

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

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 »

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 »

*(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 3*x*^{2}+*x*–7=4*x*, Wolfram|Alpha can break down the steps for you if you click the “Show steps” button in the Result pod.

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 »

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.

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 »

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 »

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.

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.

*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.*

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 »

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”:

Get the latest Flash Player.

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

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.

Get the latest Flash Player.

See Part 1 here.

Every aspect of Wolfram|Alpha has been thought through in great detail. Its logo is no exception.

As a tip of the hat to the vast and powerful computational engine that powers Wolfram|Alpha, a natural place to start brainstorming for an appropriate logo was in *Mathematica* itself. And this is where I, geometry enthusiast and the developer of the `PolyhedronData` computational data collection, came into the picture.

As many of you may know, *Mathematica*‘s logo is a three-dimensional polyhedron affectionately called “Spikey.” In its original (Version 1) form, Spikey consisted of the spiked solid obtained from an icosahedron (the regular 20-faced solid that is one of the five Platonic solids) with regular tetrahedra (triangular pyramids) affixed to its faces.

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 »

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

Remember your old friend pi?

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.

To give an amusing example, every school child has at one time or another written a report on the moon, and they probably included the wrong figure for how far the moon is from the earth. Why wrong? Because the distance from the earth to the moon is not constant: it changes by as much as a mile a minute. If you ask Wolfram|Alpha the distance to the moon, it tells you not only the conventionally quoted average distance, but also the actual distance *right now*, which can at times be well over ten thousand miles off the average. The actual distance is a figure that can be arrived at only by computation based on the moon’s known orbital parameters. It’s rocket science, if you will.

More »

As people might imagine, I’m pretty busy right now getting Wolfram|Alpha ready for launch. But yesterday afternoon I took a few hours out to give an early preview of Wolfram|Alpha at Harvard.

There were lots of interesting questions and comments, particularly about the broader intellectual context of Wolfram|Alpha.

There’s really a very long and rich history behind the kinds of things we’re doing with Wolfram|Alpha.

And in fact, a little while ago my staff took some notes of mine and assembled a timeline about the history of “The Quest for Computable Knowledge.” I think it makes interesting reading; there’s quite a diverse collection of elements, some very well known, some not.

I’ve always been particularly struck by Gottfried Leibniz’s role. He really had pretty much the whole idea of Wolfram|Alpha—300 years ago.

More »