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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 »
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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 »
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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.
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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:
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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.
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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:
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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 »
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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 »
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:
i2 = j2 = k2 = i j k = -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 + 3i + 0.342j – 2k is a quaternion. More »