When you roll dice, all numbers have the same probability to show up (assuming that the dice aren’t loaded in any way):

However, the leading digits of numbers in very large accumulated datasets—for example, the amount you pay for each household bill over the course of a year—follow a very different pattern. In such cases it is much more likely that a given number will start with one, with decreasing probability for each higher digit up to nine. This statistical phenomenon is called Benford’s law. 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 »

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!

When we launched Wolfram|Alpha in May 2009, it already contained trillions of pieces of information—the result of nearly five years of sustained data-gathering, on top of more than two decades of formula and algorithm development in Mathematica. Since then, we’ve successfully released a new build of Wolfram|Alpha’s codebase each week, incorporating not only hundreds of minor behind-the-scenes enhancements and bug fixes, but also a steady stream of major new features and datasets.

We’ve highlighted some of these new additions in this blog, but many more have entered the system with little fanfare. As we near the end of 2009, we wanted to look back at seven months of new Wolfram|Alpha features and functionality.

More »

There’s new data flowing into Wolfram|Alpha every second. And we’re always working very hard to develop the core code and data for the system. In fact, internally, we have a complete new version of the system that’s built every day. But before we release this version for general use, we do extensive validation and testing.

In addition to real-time data updates, we’ve made a few changes to Wolfram|Alpha since its launch three weeks ago. But today, as one step in our ongoing, long-term development process, we’ve just made live the first broad updates to the core code and data of Wolfram|Alpha. More »