Friday, March 14, 2014

5 Common Pi Myths


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 16.1527897 pixels, which is approximately π * π + π + π. This is the second post, for π-day; for the first, go to http://greatmst.blogspot.com/2014/03/pi-month-pi-day-post-1.html.

In this post, I will list 5 common myths about π, and explain why they're wrong.


1. Pi is wrong

Actually, π is defined as the ratio between the circumference of a circle and its diameter. There really isn't anything about this ratio that could possibly be wrong, because a ratio is a ratio - just like an orange is an orange or a piano is a piano. You can't really say that oranges are wrong, and you can't say that a ratio is wrong either. So π can't be wrong.

One might reword the statement to say that the commonly accepted value for π is wrong. This is possible, but is not the case, because the formulas commonly used to derive the digits of π have been proven to be accurate, and those formulas are the only methods we have for finding digits of π.

This myth likely came as a misinterpretation of an article written by a mathematician named Robert Palais that was titled "Pi is Wrong!", published as an opinion article in the journal called "The Mathematical Intelligencer." In this article, Palais does not actually argue that π itself is wrong; he argues that π was the wrong choice for the circle constant, and that a constant equal to the ratio between the circumference and the radius should have been used instead. This constant is now known as τ (tau). As I argued in my last π-post, it's actually the other way around - τ is the "wrong" constant, not π.

2. Pi cannot be exactly defined

I can't remember where, but I did find a news article that claimed that π cannot be perfectly defined. In reality, this is not the case; π can, and has been, defined. In fact, the exact definition for π is the ratio between the circumference of a circle and its diameter. That's the exact definition.

Where did this myth come from? Probably from the fact that when expressed in decimal, the digits of π never end and never repeat, because π is irrational, and cannot be expressed as a fraction. The decimal value for π is not the same thing as its definition, however, so π can still be defined.

3. Pi is infinity

One of my friends tried to argue that π is infinity. Apparently there are others who think that as well. In reality, π is a number smaller than 4 and larger than 3, and isn't anywhere near infinity. This myth, like so many others, comes from the fact that the decimal expansion of π never ends.

4. Archimedes used area to approximated the value of pi

A very commonly held belief is that Archimedes approximated π by finding the area of two polygons, one which was larger than a unit circle and the other which was smaller, and then averaging those areas. In reality, Archimedes found the perimeters of the two polygons, not the areas, and averaged them. This myth doesn't matter too much, because either way the idea is the same: Archimedes used two polygons to find an approximation for π.

5. The Bible used a value of pi = 3.0

This is a surprisingly common belief among math geeks. Sometimes this argument is used to prove that the Bible is wrong. The myth originates in the passage that says "And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and its height was five cubits: and a line of thirty cubits did compass it about." It would seem that the value for π being used was 30/10, or 3. If a perfect value for π were used, then the line that compassed the molten sea would be 31 cubits, not 30.

Actually, there are multiple explanations for this. One is that the diameter was measured from brim to brim, but the circumference was measured around the vessel itself, and since the brim of the vessel protruded from the rest of the vessel, the apparent π-value was not actually a π-value at all. Another explanation is that the numbers were rounded to the nearest whole number, so that if the diameter of the vessel had been 9.6 (rounded to 10) cubits, the circumference would be about 30.1 (rounded to 30) cubits.

The best explanation, I think, involves a look at the numerical values for words in the Hebrew language. As one student explained in his essay "The History of Mathematics," the Hebrew for "line" consists of the Hebrew letters Kuf Vov Heh, which are assigned the values of 100, 6, and 5, respectively. In Hebrew math, these numbers would be added up, so that the word's value would be 111. But the letter Heh is unnecessary, because it isn't pronounced; without that extra letter, the word's value would be 106. Wilson says that if you take the ratio of the first word to the second, and multiply it by 3, you get a value of π = 3.1415094..., which is far more accurate than any previous approximation for π up to that point.

It seems like this is just a coincidence, especially since we had to do those multiplications and divisions to arrive at that value for π. But I think there's more to this than Wilson described. Look at the phrase "line of 30 cubits did compass it about." Remember that the Hebrew word for "line" can be replaced by the ratio 111/106, as Wilson explained. Insert that into the sentence, and you get: "111/106 of 30 cubits did compass it about." Notice the part that says "111/106 of 30." That's the same as saying "111/106 times 30," which is approximately 31.415. This is almost the exact circumference of a circular vessel of diameter 10. Now is that a coincidence? I think not.

Even if the Bible did use a stunted value for π, it would not prove that the Bible is wrong. The whole idea of the Bible is that it tells about the history of man's relationship with God; to put this in another way, it is not a math textbook.

1 comment:

  1. Interesting post, but the snow background is a bit annoying, in my opinion.

    ReplyDelete