{"id":3154,"date":"2015-03-14T21:23:13","date_gmt":"2015-03-14T21:23:13","guid":{"rendered":"http:\/\/www.lifeandnews.com\/articles\/?p=3154"},"modified":"2016-08-08T05:20:46","modified_gmt":"2016-08-08T05:20:46","slug":"pi-day-is-silly-but-%cf%80-itself-is-fascinating-and-universal","status":"publish","type":"post","link":"https:\/\/www.lifeandnews.com\/articles\/pi-day-is-silly-but-%cf%80-itself-is-fascinating-and-universal\/","title":{"rendered":"Pi Day is silly, but \u03c0 itself is fascinating and universal"},"content":{"rendered":"

By Daniel Ullman<\/a>, George Washington University<\/a><\/em><\/p>\n

Math students everywhere will be eating pies in class this week in celebration of what is known as Pi Day, the 14th day of the 3rd month.<\/p>\n

\"\"<\/a>
Larry Shaw initiated the first Pi Day in 1988.<\/span>
\nRon Hipschman<\/a>, CC BY-SA<\/a><\/span><\/figcaption><\/figure>\n

The symbol \u03c0 (pronounced pa\u026a in English) is the sixteenth letter of the Greek alphabet and is used in mathematics to stand for a real number of special significance. When \u03c0 is written in decimal notation, it begins 3.14, suggesting the date 3\/14. In fact, the decimal expansion of \u03c0 begins 3.1415, so this year\u2019s Pi Day, whose date we can abbreviate as 3\/14\/15, is said to be of special significance, a once-per-century coincidence. (Yet we might anticipate a similar claim next year on 3\/14\/16, since 3.1416 is a closer approximation to \u03c0 than is 3.1415.)<\/p>\n

Besides a reason to enjoy baked goods while feeling mathematically in-the-know, just what is \u03c0 anyway?<\/p>\n

\"\"<\/a>
A circle\u2019s measurements define \u03c0.<\/span>
\nRobotics Academy<\/a><\/span><\/figcaption><\/figure>\n

It\u2019s defined to be the ratio between the circumference of a circle and the diameter of that circle. This ratio is the same for any size circle, so it\u2019s intrinsically attached to the idea of circularity. The circle is a fundamental shape, so it\u2019s natural to wonder about this fundamental ratio. People have been doing so going back at least to the ancient Babylonians<\/a>.<\/p>\n

\"\"<\/a>
The hexagon\u2019s perimeter is shorter than the circle\u2019s, while the square\u2019s is longer.<\/span>
\nCC BY<\/a><\/span><\/figcaption><\/figure>\n

You can see that \u03c0 is greater than 3 if you look at a hexagon inscribed within a circle. The perimeter of the hexagon is shorter than the circumference of the circle, and yet the ratio of the hexagon\u2019s perimeter to the circle\u2019s diameter is 3. And you can see that \u03c0 is less than 4 if you look at the square that circumscribes a circle. The square\u2019s perimeter is longer than the circle\u2019s circumference, and yet the ratio of this perimeter to the diameter of the circle is 4. So \u03c0 is somewhere in there between 3 and 4. OK, but what number is<\/em> it?<\/p>\n

A little experimentation with a measuring tape and a dinner plate suggests that \u03c0 might be 22\/7, a number whose decimal expansion begins 3.14. But it turns out that 22\/7 is approximately 3.1429, while even 2,250 years ago Archimedes<\/a> knew that \u03c0 is approximately 3.1416. The fraction 355\/113 is much closer to \u03c0 but still not exactly equal to it.<\/p>\n

Fractionally closer?<\/h2>\n

So this raises the question: is there some other fraction out there that equals \u03c0, not merely approximately but exactly? The answer is no. In 1761, Swiss mathematician Johann Lambert<\/a> proved that no fraction exactly equals \u03c0. This implies that its decimal expansion is never-ending, with no repeated pattern.<\/p>\n

The German mathematician Ferdinand Lindemann<\/a> proved in 1882 that \u03c0 is in fact transcendental<\/em>, which means that it does not solve any polynomial equation with integer coefficients. This implies in some sense that there isn\u2019t ever going to be a simple way of describing \u03c0 arithmetically. Nowadays, machines can compute trillions of decimal digits of \u03c0, but that in no way helps us understand what \u03c0 is exactly. It\u2019s easiest just to say that, to be exact, \u03c0 is equal to \u2026 \u03c0.<\/p>\n

No one knows whether each of the ten digits \u2013 0 through 9 \u2013 appears with equal frequency in the decimal expansion of \u03c0, as we would expect if the digits of \u03c0 were produced by a random digit generator. This illustrates that a strikingly elementary question can be out of reach of modern mathematics. Perhaps in a century mankind will know the answer to this question, but it\u2019s not even clear at this time how to attack it effectively.<\/p>\n

\"\"<\/a>
You could measure circumference and diameter of these pies to get \u03c0.<\/span>
\nDennis Wilkinson<\/a>, CC BY-NC-SA<\/a><\/span><\/figcaption><\/figure>\n

Everything\u2019s coming up \u03c0<\/h2>\n

What is astonishing about \u03c0 is that it appears in many different mathematical contexts and across all mathematical areas. It turns out that \u03c0 is the ratio of the area of a circle to the area of the square built on the radius of the circle. That seems like a coincidence, because \u03c0 was defined to be a different ratio. But the two ratios are the same. \u03c0 is also the ratio of the surface area of a sphere to the area of the square built on the diameter of the square. And what about the ratio of the volume of sphere to the volume of the cube built on the sphere\u2019s diameter? That\u2019s \u03c0\/6.<\/p>\n

The area under the bell-shaped curve y=1\/(1+x\u00b2) is \u03c0. But this curve isn\u2019t actually the well-known and universal bell-shaped curve seen in statistics that has the formula y=e-x\u00b2<\/sup>. The area under that curve is the square root of \u03c0! If you drop a pin of length one centimeter on a sheet of lined paper with lines spaced at centimeter intervals, the probability that the pin crosses one of the lines is 2\/\u03c0. If you choose two whole numbers at random, the probability that they will have no common factor is 6\/\u03c0\u00b2.<\/p>\n

There are thousands of formulas for \u03c0 of one sort or another, although it isn\u2019t clear whether any of them will satisfy the desire to know what \u03c0 is exactly. One such formula is<\/p>\n

\"\"<\/a>
Ramanujan\u2019s equation for \u03c0.<\/span>
\nAuthor provided<\/span><\/span><\/figcaption><\/figure>\n

where the sigma symbol indicates that one must plug in all the whole numbers in place of the symbol \u201ck\u201d in the subsequent formula and add up the resulting infinitely-many fractions. What is remarkable about this expression is that it was discovered by the legendary Indian genius Srinivasan Ramanujan<\/a> in 1914, working alone. No one knows how Ramanujan came up with this amazing formula. Moreover, his formula wasn\u2019t even shown to be correct until 1985 \u2013 and that demonstration used high-speed computers to which Ramanujan had no access.<\/p>\n

\u03c0 is beyond universal<\/h2>\n

The number \u03c0 is a universal constant that is ubiquitous across mathematics. In fact, it is an understatement to call it \u201cuniversal,\u201d because \u03c0 lives not only in this universe but in any conceivable universe. It existed even prior to the Big Bang. It is permanent and unchanging.<\/p>\n

\"\"<\/a>
Math enthusiasts need to cut loose sometimes too.<\/span>
\nVancouver Island University<\/a>, CC BY-NC-ND<\/a><\/span><\/figcaption><\/figure>\n

That\u2019s why the celebration of Pi Day seems so silly. The Gregorian calendar, the decimal system, the Greek alphabet, and pies are relatively modern, human-made inventions, chosen arbitrarily among many equivalent choices. Of course a mood-boosting piece of lemon meringue could be just what many math lovers need in the middle of March at the end of a long winter. But there\u2019s an element of absurdity to celebrating \u03c0 by noting its connections with these ephemera, which have themselves no connection to \u03c0 at all, just as absurd as it would be to celebrate Earth Day by eating foods that start with the letter \u201cE.\u201d<\/p>\n

\"The<\/p>\n

This article was originally published on The Conversation<\/a>.
\nRead the original article<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"

By Daniel Ullman, George Washington University Math students everywhere will be eating pies in class this week in celebration of what is known as Pi Day, the 14th day of the 3rd month. Larry Shaw initiated the first Pi Day in 1988. Ron Hipschman, CC BY-SA The symbol \u03c0 (pronounced pa\u026a in English) is the […]<\/p>\n","protected":false},"author":40,"featured_media":5245,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[7],"tags":[892,1002,885,891,860,665,1001],"_links":{"self":[{"href":"https:\/\/www.lifeandnews.com\/articles\/wp-json\/wp\/v2\/posts\/3154"}],"collection":[{"href":"https:\/\/www.lifeandnews.com\/articles\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.lifeandnews.com\/articles\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.lifeandnews.com\/articles\/wp-json\/wp\/v2\/users\/40"}],"replies":[{"embeddable":true,"href":"https:\/\/www.lifeandnews.com\/articles\/wp-json\/wp\/v2\/comments?post=3154"}],"version-history":[{"count":3,"href":"https:\/\/www.lifeandnews.com\/articles\/wp-json\/wp\/v2\/posts\/3154\/revisions"}],"predecessor-version":[{"id":5246,"href":"https:\/\/www.lifeandnews.com\/articles\/wp-json\/wp\/v2\/posts\/3154\/revisions\/5246"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.lifeandnews.com\/articles\/wp-json\/wp\/v2\/media\/5245"}],"wp:attachment":[{"href":"https:\/\/www.lifeandnews.com\/articles\/wp-json\/wp\/v2\/media?parent=3154"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.lifeandnews.com\/articles\/wp-json\/wp\/v2\/categories?post=3154"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.lifeandnews.com\/articles\/wp-json\/wp\/v2\/tags?post=3154"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}