Friday, March 14, 2014

Should Tau Replace Pi?


The digits of π, organized in a very new way

Happy π-day! And happy π-month! Today's month and day - that is, March 14 or 3.14 - includes the first 3 digits of π. And today's month and year - March 2014 or 3.14 - also includes the first 3 digits of π. We won't have another double-day for π for the next 100 years, so enjoy this one!

For the special occasion, I'm posting two π-related posts - one for π-month, and the other for π-day. In both posts, I'm setting the font size to approximately π * π + π + π. This is the first post, for π-month; to see the second, go to http://greatmst.blogspot.com/2014/03/pi-month-pi-day-post-2-5-common-pi-myths.html.

In this post, I am including an essay I wrote about whether π or τ is the more superior constant. This was written for people who know very little about math, so the basic idea should be easy to understand even for people who are not mathematically inclined.

Should Tau Replace Pi?
A constant is any number or value that never changes. Many mathematical constants begin as definitions, and the actual values are later determined based on those definitions. Then the constants are assigned special characters or symbols to distinguish them from each other, so that mathematicians can refer to them more easily and consistently. One of the most famous constants is represented by the Greek letter π, pronounced "pie." π is the number of times that the diameter of a circle goes into its circumference, or in more mathematical terms, it is the ratio between the circumference and diameter of a circle. All geometric circles are the same shape – circular – so the distance across the circle in relation to the distance around the circle is always the same, so it is a constant. The value for the so-called "circle constant" happens to be slightly more than 3.14 in decimal, but the exact value is impossible to expand in a numeral system because π is irrational, meaning it is not a fraction (Kaiser 1975). As a result, when represented in a numeral system, its digits continue forever without repeating.
Because the decimal expansion of π never ends or repeats, it can be used to test the limits of a person's memory. No matter how many digits a person memorizes, there are always more, and he can continue memorizing as many as he needs to. This property has made the number π very popular, even to the point where people eat pie on March 14 (or 3.14) and celebrate it as Pi Day. π also has many applications in the real world, including finding the amount of water needed to fill a cylindrical container or estimating the width of a 5-pound spherical bowling ball.
However, in an article titled "Pi is Wrong!" which was published in the math journal "The Mathematical Intelligencer," math professor Bob Palais argues that π is wrong. He explains that he does not mean that the numerical value for π is wrong, or that π is not the ratio between the circumference and diameter of a circle, but that there is a better mathematical constant that should have been defined as the circle constant instead of π. This constant, he writes, is equal to 2π (Palais 7).
Palais' main argument for why π is wrong involves the radian unit of angle measurement, which is the most commonly used unit of angle measurement in mathematics. To understand this system, it can be compared to the degree unit of angle measurement. A degree is 1/360th of a full turn, so that a full turn is equal to 360 degrees, a half turn is equal to 180 degrees, and a quarter turn is equal to 90 degrees. A radian, on the other hand, describes the angle subtended by an arc of length 1 and radius 1, so that an angle measured in radians is equal to the length of the arc that subtends it. A whole turn would therefore have a radian measurement of 2π, which is the circumference of a unit circle. Palais argues that if π were twice as large as it currently is, this would work out much better; a whole turn (or 360 degrees) would equal π radians, a half turn would equal π/2 radians, a quarter turn would equal π/4 radians, etc. Because of this, the circle constant should have been twice as large (Palais 7). Another mathematician named Michael Hartl named this new constant after the Greek letter τ, pronounced "tau," so that 360 degrees would equal τ radians, 180 degrees would equal τ/2 radians, etc. (Hartl).
If τ replaced π, the definition of the circle constant would be a lot more consistent with the rest of mathematics. Instead of being defined as the ratio between the circumference and diameter of a circle, it would be defined as the ratio between the circumference and radius of a circle. This is better because circles in math are usually defined by radius, not diameter. τ would also simplify many equations and formulas. For example, the equation for the circumference of a circle would become C = τr instead of C = 2πr, and in trigonometry, the period of the sine and cosine functions would change from 2π to τ. As Hartl points out, τ would make mathematics easier for children as a result (Hartl). After Hartl largely advertised this new number, a group of people now known as the "tauists" formed and are now pushing a conversion from π to τ.
However, I believe that π should not be replaced. π is being very widely used in many areas of mathematics, and it would be a bad idea to switch to a new constant. Too many math textbooks use π; almost every calculator uses it as well. If π were replaced, thousands of textbooks and calculators would be outdated, resulting in a lot of confusion. Not only that, but many people are opposed to the idea of changing, and with a good reason – although there are benefits to using τ instead of π, there many more downsides, and many times when τ does not work as well as π.
Probably the formula where π outdoes τ the most is A = πr2, which is the formula for the area of a circle with radius r. In terms of τ, the formula would be A = r2τ/2, which is more complicated than the original formula. Although this is only one of the many formulas that use π and there are many more in which τ would work better, I have found in my math studies that the area of a circle is a very important formula in mathematics; it is therefore important to keep it simple. In fact, the formula can even be used in everyday situations as common as deciding whether a medium $8 pizza is a better value than a large $12 pizza, which would involve dividing the price of each pizza by its area, and choosing the pizza with the smaller value. In this situation, the circumference would be useless, since a pizza with twice the circumference does not have twice the area, because area grows faster than circumference (Tent). π in this case is the preferable constant to use, and since situations such as these are common, τ should not replace π.
Even if circumference and area were used exactly as often as each other, π would still be a simpler choice. If τ replaced π, the formula for the circumference of a circle would no longer require multiplication by 2, but the formula for the area of a circle would require division by 2. Division by 2 is a more complicated task to perform than multiplication by 2, so with respect to these two formulas, π is the simpler constant to use. Since mathematicians usually prefer simpler numbers and formulas to more complicated ones, π is better than τ.
Furthermore, the definition of the circle constant as being the ratio between the circumference and diameter  is more natural than the definition being the ratio between the circumference and radius. Although using the radius may be more consistent with the rest of mathematics, defining the constant in terms of diameter is actually more practical for use in real-world applications, because radius cannot be directly measured in a circle, but diameter can. However, an even better definition for the circle constant would be the ratio between the area of a circle and the square of its radius, because as I have found, area tends to be used more often than circumference. In fact, this definition was used by many of the earliest mathematicians (Blatner 16). This definition yields π, not τ, which is another reason that τ should not replace π.
To me, the most convincing argument in favor of τ is that it would make the measurement of angles in radians more logical, so that 360 degrees would be equal to τ, instead of 2π. Although this is a valid argument, it is a weak one, because it assumes that a complete angle is 360 degrees, which is not necessarily correct; in fact, 180 degrees could actually be the more desirable unit. This is because an angle between two intersecting lines can never be greater than 180 degrees, so that any angle greater than 180 degrees can never really exist in the real world. As a result of this, most math problems do not require the use of angles greater than 180 degrees, and even when they do, the problem can usually be restated to use an angle that is less than or equal to 180 degrees. Because of this property of angles, I find it useful in math to think of 180 degrees as a whole unit equal to π radians rather than a half unit equal to τ/2 radians. π is better than τ in this case.
There are some cases, though, when it still may be slightly more advantageous to use τ. It may seem logical to use both constants, switching from π to τ or vice versa when needed. However, the awkwardness of switching back and forth between two different circle constants would be too confusing and the gain would be very slight. After all, π is exactly τ/2, and most good mathematicians find multiplication by 2 very trivial. Not only that, but putting buttons for both π and τ on a calculator would take up too much space and would likely never be done, so every time somebody needed to evaluate a math statement that used τ he would have to convert to π. If we used τ at all, we would have to use it entirely. We don't have to worry about losing π, though – the majority of mathematicians have just been ignoring τ, or disagreeing that it is any better than π (Life). So even while the tauists are pushing to change mathematics, π will live on.
Works Cited
Blatner, David. The Joy of Pi. Walker Publishing Company, Inc. New York, 1997.
Hartl, Michael. "The Tau Manifesto." N.p. 14 Mar 2013. Web. 22 Nov 2013. <http://tauday.com/tau-manifesto>.
Kaiser, Jocelyn. "Pieces of Pi." Science 283.5410 (1999): 1975. Print.
"Life of Pi in No Danger." The Telegraph Calcutta. 30 Jun 2011. Web. 4 Dec 2013.
Palais, Bob. "π is Wrong!" The Mathematical Intelligencer. 23.3 (2001): 7-8. Print.


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