Archimedes' constant, or "Pi," is the name given to the ratio of the circumference of a circle to the diameter, but it's actually so much more than that.
Greek mathematician Archimedes is credited with the first theoretical calculation of Pi, which he estimated was between 3 10/71 and 3 1/7 — or 223/71.
Pi is now defined as 3.1415926535... etc
Application: Pi is the key constant in any equation that involves circular or harmonic motion. It's one of the most essential relationships in mathematics.
Source: Math.com
Euler's Number (e): 2.7182...
Euler's number is also known as the exponential growth constant. It is the base for natural logarithms and is found in many areas of mathematics.
Application: In finance, Euler's number is used to determine compound interest, which is extremely vital in understanding the time value of money — the backbone of finance.
Moreover, Euler's number is crucial when describing any decaying relationship - think Carbon 14 dating.
Source: Math Is Fun
Reference:
http://www.businessinsider.com/most-important-numbers-2012-7?IR=T#eulers-number-e-27182-2
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