By Hojae Lee, a Berkeley Math Circle assistant, and Laura Pierson, a high-school student at The College Preparatory School, in Oakland, Calif.
It’s time to celebrate Π Day! And 3/14/15 is a special Π Day, as the date forms the first five digits of Π (3.1415), and at 9:26:53, the date and time will be the first ten digits.
Most of us have heard of the famous circle constant, slightly more than three. It arises in the simplest and most natural way: the ratio of the circumference of a circle to its diameter. It has been studied for more than 2,000 years. The ancient Greek mathematician Archimedes even correctly calculated its value to three decimal places by inscribing a regular 96-sided polygon in a circle.
Yet this seemingly simple number has many strange and fascinating properties. For instance, what happens when you alternately add and subtract the reciprocals of the odd numbers? Surprisingly, the sum is exactly Π/4:
1 – 1/3 + 1/5 – 1/7 + … = Π/4
Intuitively, you would never expect that a sum of rational numbers would be connected to the geometry of circles. The sum of the reciprocals of the squares of positive integers is also connected to p:
1/12 + 1/22 + 1/32 + … = Π2/6.
Take another seemingly unrelated problem, known as “Buffon’s needle.” Suppose you drop a one-inch long needle on a surface that has equally spaced lines drawn on it, each one inch apart. What is the probability that the needle will land on one of the lines? Once again, the result is unexpectedly connected to Π: the probability is 2/Π.
Π is also connected to the imaginary number i, the square root of negative one. The complex numbers (sums of ordinary real numbers and multiples of the imaginary i) have an elegant geometric representation as points in a plane, and it turns out raising a number to an imaginary power rotates it about the origin of this plane. Among other things, this leads to Euler’s famous formula
eiΠ = -1,
where e = 2.71828… is Euler’s constant.
This is what is great about math. You can take seemingly simple or unrelated ideas, and yet they turn out to be connected in surprising and beautiful ways. The world of mathematics is much stranger and more interesting than it might seem. For many Bay Area kids, the Berkeley Math Circle has been a window into this world.
At the Berkeley Math Circle, students in elementary, middle, and high school meet on a weekly basis to explore such beauties of mathematics. It is the second Math Circle that started in the United States in 1988 and that is modeled after Eastern European programs; hundreds of others have started since then to foster many great mathematicians.
There are now more than 400 students in the Berkeley Math Circle (meeting every Tuesday of the academic year, 6-8 p.m. in Evans Hall), each with a unique background in math. Nevertheless, there is an axiom that applies to all Math Circle students: “We have an unending enthusiasm and love for math!”