Today is a great day. We get to celebrate one of the best numbers in existence: π. π plays a role in many mathematical and scientific applications. Some examples include:
Area of a Circle = π * radius^2
Area of an Ellipse = π * length of semi-major axis * length of semi-minor axis
The functions sin x, cos x, and tan x each have a period of π.
Euler's Formula: e^(i*π) + 1 = 0
Definite integral involving π:
∫(e^(-a*x^2) dx, 0, infinity) = 0.5 * √(π/a)
1 + 1/3 + 1/5 + 1/7 + ... = sum of the fractions of 1/odd integers = π/4
(Convergence might be slow)
Γ(1/2) = √π. (Gamma of 1/2)
Probability: Area under the Normal Curve:
1/√(2 π) * ∫(e^(-x^2/2))
Angular Velocity, general: ω = 2 π f, where f is frequency
Conversion between radians and degrees:
Angle in Radians = Angle in Degrees * π/180
Optics: Double Split Interference - Intensity at an Angle:
I = 4 * I_0 * cos( π * d * sin θ / λ )^2
Also today is Albert Einstein's birthday (March 14). He is primarily responsible for the Theory of Relativity and E = mc^2. I am going to give myself a refresher course.
Here is some information about the Theory of Relativity:
http://en.wikipedia.org/wiki/Theory_of_relativity
This blog is property of Edward Shore. 2014