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Viewing: Blog Posts Tagged with: History of Mathematics, Most Recent at Top [Help]
Results 1 - 5 of 5
1. Is Tau Better Than Pi? Irrational Arguments

Happy Tau Day, the most exciting math holiday you’ve yet to discover! Today, June 28th is 6/28, which contains in order the first three digits of tau (τ), the rival of math’s most popular irrational number, pi (π). In 2001, Bob Palais wrote an article for The Mathematical Investigator called ,“π is wrong!” In it, he insists that the choice of using π in our mathematical formulas for hundreds of years is no good. He argues that the use of τ would simplify many formulas and its derivation is much more intuitive. (Notice that the symbol resembles that for pi, but with one "leg" instead of two.) The significance of our beloved irrational number π is that it is equal to the ratio of the circumference of any circle to its diameter--in notation, π = C/d. However, the most defining characteristic of a circle is not its diameter but its radius. A circle is defined as the collection of points on a plane that are exactly the same distance, its radius, from a point, its center. Palais argues that intuition should direct us to the use of a more elegant Circle Constant, tau, where τ is the ratio of the circumference of a circle to its radius--in notation, τ = C/r. Self-described “notorious mathematical propagandist” Michael Hartl takes the argument even further in his now-famous “The Tau Manifesto,” which he published on Tau Day of 2010, exactly one year ago. He demonstrates with many adapted formulas that the factor of 2 is unnecessary if we incorporate it into the ratio itself. For instance, the periods of basic trigonometric functions f(x) = sin(x), and f(x) = cos(x), are in both cases 2π. Why not change them to tau instead? Palais and Hartl each list numerous other examples from calculus and physics, in which the factor of 2 is rendered obsolete by replacing 2π with τ. The really intuitive part is revealed if you think of angle measure. How things are done now with π, a half turn of the circle is π radians, and a full turn is 2π radians. Should we adopt τ instead, τ radians would be a full turn, τ/2 radians a half turn, τ/4 radians a quarter turn, and so on. There are, of course, instances where π appears un-doubled. For instance, the formula for area of a circle: A = πr2. Hartl shows, in a mathematically sophisticated way, that the replacement of π by τ even in this instance is the more sound choice, since it is analogous to similar formulas in physics. An article in today’s BBC News paints the issue as a violent conflict, with pi detractors up in arms over a lifetime of educational betrayal, which seems to this mathematician something of a manufactured controversy. (I can imagine you'd be upset if you are the sort of mathematician that has memorized pi to the nth digit. If you are one of these folks, here's the start for your new parlor trick: reciting tau, 6.283185307...)
Is it worthwhile to switch to tau use, an

2 Comments on Is Tau Better Than Pi? Irrational Arguments, last added: 6/29/2011
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2. Free Math Apps for the iPhone and iPod Touch!

Last week we offered the Android users among you a selection of free math-related apps. If you’re an iPhone user like I am, you will be pleased to know that there are equivalent apps for your device!

Many of the specific apps for Android are not available on iPhone, but that’s not to say that there is any shortage of math-related iPhone apps. For instance, Andie Graph is not available for iPhone, nor is Graph Lite. However:

There are TONS of graphing calculator apps that are available for free:
Or if you don’t mind spending $1.99, check out the very nice-looking Graphing Calculator from Appcylon LLC: Algebra 1 Lite is a different mini-interactive textbook: students read some examples and then try out some practice problems. The app also keeps track of student progress. Here’s an example of a quiz question:


Math Genius. (Also available on Android). Practice basic arithmetic skills.
Cool math resources include (free) Digital Protractor, ($0.99) Slide Rule, and (free) Abacus.

Another app I recommen

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3. Q-What? Why Do the Quadrants Go Counterclockwise?

The Cartesian Plane
In the new national Common Core Learning Standards, mathematics students in 5th grade begin to explore graphing in the first quadrant (where the values of each coordinate are positive), and in 6th grade expand to all four quadrants of the plane. The appearance of the coordinate plane at this early time in the curriculum emphasizes its importance for the study of mathematics. The plane is the brain-child of René Descartes, prominent 17th-century French philosopher and mathematician. (That’s why we also call it the Cartesian plane.)

 Two perpendicular lines, the x-axis and the y-axis, split the plane into four infinite quarter-planes, which are called quadrants. As mentioned earlier, the first quadrant has positive values for both its x and y coordinates. Quadrants are numbered using Roman numerals, so we label it I. Then we label the other quadrants, naming them with Roman numerals II, III, and IV, in counterclockwise order. As shown in the graph, QII has negative x-values, but positive y-values. QIII has negative x-values and negative y-values. QIV has positive x-values, but negative y-values. Even as, more and more frequently, time presents itself digitally, some students may wonder why the quadrants are not arranged clockwise. Some may be initially confused. For others, even if it makes sense at first, a clockwise orientation may feel more natural, and the draw of the familiar may provide an opportunity for error.

Teachers should expect to be questioned about this: “Why not clockwise?” The stock answer is, “That’s the rule.” Take the time to delve a little deeper. This response may sound to a young person like, “because I say so.” Dissatisfaction with explanations and confusion about them go hand in hand. A solid understanding of how to label the quadrants should enable your students to discuss graphing more efficiently. Graphing on the coordinate plane is a skill that will be immensely valuable throughout high school math and well into college courses, including and beyond multi-variable calculus and electro physics. The purpose of this convention and most conventions in general, is to avoid ambiguity. (And ambiguity can lead to chaos! Imagine if we didn’t have the convention of driving on only one side of the road.)
Physicists in particular use the right-hand rule to navigate in three dimension

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4. Big Nothing: The History of Zero

While rocking out to Patti Smith, in celebration of her victory winning the National Book Award, I rediscovered her tribute, “Radio Baghdad.” The song celebrates the Iraqi city’s rich cultural and intellectual history, and as a refrain she specifically mentions its involvement in the invention of zero: “We created the zero/But we mean nothing to you.”

Smith honors Baghdad’s intellectual contribution to the establishment of zero as a number. Zero deserves her praise for its usefulness as a placeholder (as in the number 306), for its role as the additive identity element (if you add zero to any number, you get that number—in symbols, n + 0 = n for any number n), and for its contribution to the development of calculus. As the late writer David Foster Wallace elegantly claimed, “The invention of calculus was shocking because for a long time it had simply been presumed that you couldn't divide by zero.” Zero is a game-changer, a distinct value, and the barrier between positive and negative.

The richly informative book 100 Greatest Science Inventions of All Time tells the story of Al-Khwarizmi. In 810 A.D., this famous Baghdad mathematician convinced a group of fellow scholars that zero must be a number by demonstrating that zero behaves like a number when subject to common operations. Not only did Al-Khwarizmi thus effectively demonstrate zero as a number, but he also established himself as the founder of algebra. 
I love this story because I think it eloquently demonstrates the following dispositi

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5. Girls, Math, and Image


Remember Winnie Cooper from The Wonder Years, Kevin Arnold’s first love and ultimate crush? Danica McKellar, the actress whom we all wished to end up with our hero, Fred Savage, grew up to be a mathematician! She graduated summa cum laude from UCLA with a degree in mathematics and even co-authored a theorem that bears her name. As an actress, she made the experience of falling in love as a kid relatable to millions. Now she aims to guide a new generation through a different part of growing up: middle school math class. She’s written three books, Math Doesn’t Suck: How to Survive Middle School Math without Losing Your Mind or Breaking a Nail, Kiss My Math: Showing Pre-Algebra Who’s Boss, and Hot X: Algebra Exposed!
Going from child actor to math book author is rare enough, and McKellar has gotten a lot of attention, notably with a racy photo shoot in Maxim magazine (accompanied by an article about her books and the coolness of math). In an interview with Salon.com, she offers, “Most of the images that girls are getting of what is attractive are so limiting. And that's what I want to fight against.” That’s great!

I think there’s a lot of potential for good here, but is it appropriate for educators to conflate attractiveness with math ability? And what exactly is she advocating, a bait and switch? She seems to be going on the assumption that math isn’t cool enough on its own, but we can coerce students into liking it, like offering them chocolate-covered vitamins. The books have increasingly racy titles. I also have a gripe that the books argue, “If you are good at math, then you are attractive.” (Conceding there is no accounting for taste, I’ll argue that there is no shortage of counterexamples here.)

I’d like to continue this argument mathematically by offering the following two statements. (Please note that the arguments are, in general, centered on girls.)

A: You are good at math.
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