Sunday, April 04, 2010

Concerning Circle and the Constant π: Part II

The constant π is a fundamental constant in mathematics. It is also an important constant in science for it arises naturally in so many applications such as in the period of a simple pendulum in mechanics, Maxwell-Boltzmann distribution for gas molecules in thermodynamics, Coulomb's law for electrostatic force in electromagnetic theory, Heisenberg's uncertainty principle in quantum mechanics, Einstein's field equation in general relativity theory.

Since π is such an important constant, I think it should be properly introduced to students, both historically and mathematically. In Malaysian secondary schools, this constant is taught when a typical 14-year-old in Form 2 is learning the properties of circle. In a recent book titled "Essential Mathematics Form 2", published by Longman, the authors wrote, in page 163,
the value of π is an estimated value, which is 22/7 or 3.142
This sentence is very odd from both grammatical/mathematical point of view. I guess the authors probably intend to say: the commonly used approximate value of π is 22/7 or 3.142.

Because of the way these two approximations are represented in textbooks and reference books available in Malaysia, they are usually misunderstood by students and teachers of mathematics as the value of π. My guess is that if you ask this question
Of the two values, which one would you choose to better represent π? (A) 3.142 (B) 3.14159
Most of the students and teachers will choose option (A), which is wrong.

In another local book titled "Focus Goal Additional Mathematics", published by Pelangi, the authors wrote the following in page 151:
The constant π was introduced by a German mathematician Gottfried Wilhem von Leibniz. The value of π is approximately 3.142... The degree of accuracy of π can be determined by the formula: π/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9...
This sentence is problematic for the following reasons:

1. First, Leibniz's name was never associated with the introduction of the notation of π in the history of mathematics. The first person who use π in its present meaning was a Welsh mathematician named William Jones, when he wrote π = 3.14159&c in 1706. But it was the Swiss mathematician Euler who made the notation popular when he used it in 1737.

2. Second, the Nilakantha-Gregory-Leibniz series for π is one of slowest convergence series for π. One would require 1 million terms to compute π correct to 5 decimal places. I would rather use the Zu-ratio 355/113 to obtain a numerical value of π (If you divide 355 by 113, you get 3.141593, this is accurate up to 6 decimal places). Thus it is very inefficient to use the Nilakantha-Gregory-Leibniz series to compute π accurately. However, through a certain manipulation, the series can be made to converge faster.

As the web is full of inaccurate stuffs, it is pretty easy for the uninformed to obtain wrong information and put them in their books.

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