Uses of Pi
- Ever wondered when you were going to use that annoying π symbol outside of math class? Well then this section is for you.
- Besides knowing that π is Circumference divided by diameter, it's also important to actually be able to use the thing.
How Much Fun Can π Be?
- Pi (π) is one of the most important numbers in Mathematics, and yet it is one of those we know least about. Its mysteries have puzzled some of the best mathematicians throughout history, including Euler and Archimedes.
- We know what you are thinking by now, "Who in the world would want to create a web page about π?" Well, we did. Despite the obvious fact that we must be out of our minds, one can actually have fun with π. We figure that once you get to understand π, you will be able to play with π. Soon we will have you having more fun than you thought you could have with a number.
π is Irrational
A rational number is one that can be expressed as the fraction of two integers. Rational numbers converted into decimal notation always repeat themselves somewhere in their digits. For example, 3 is a rational number as it can be written as 3/1 and in decimal notation it is expressed with an infinite amount of zeros to the right of the decimal point. 1/7 is also a rational number. Its decimal notation is 0.142857142857…, a repetition of six digits. However, the square root of 2 cannot be written as the fraction of two integers and is therefore irrational. For many centuries prior to the actual proof, mathematicians had thought that pi was an irrational number. The first attempt at a proof was by Johaan Heinrich Lambert in 1761. Through a complex method he proved that if x is rational, tan(x) must be irrational. It follows that if tan(x) is rational, x must be irrational. Since tan(pi/4)=1, pi/4 must be irrational; therefore, pi must be irrational. Many people saw Lambert's proof as too simplified an answer for such a complex and long-lived problem. In 1794, however, A. M. Legendre found another proof which backed Lambert up. This new proof also went as far as to prove that π^2 was also irrational.
π is Transcendental
A transcendental number is one that cannot be expressed as a solution of ax^n+bx^(n-1)+...+cx^0=0 where all coefficients are integers and n is finite. For example, x=sqrt(2), which is irrational, can be expressed as x^2-2=0. This shows that the square root of 2 is nontranscendental, or algebraic. It is very easy to prove that a number is not transcendental, but it is extremely difficult to prove that it is transcendental. This feat was finally accomplished for π by Ferdinand von Lindemann in 1882. He based his proof on the works of two other mathematicians: Charles Hermite and Euler. In 1873, Hermite proved that the constant e was transcendental. Combining this with Euler's famous equation e^(i*π)+1=0, Lindemann proved that since e^x+1=0, x is required to be transcendental. Since it was accepted that i was algebraic, π had to be transcendental in order to make i*π transcendental.