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Numbers Crib Sheet 6 - Special Numbers
Euler's Number (e)
2.718281828459045235360287471352662497757247.....
This number is the base for natural logarithms usually shown by Ln or ex on calculators.
e = (1 + 1/n)n
when n is a very large number (as close to infinity as practicable)
 
e = 1 + 1/1! + 1/2! + 1/3! + 1/4! + 1/5! + ....
 
e = (1 + n)(-n)
when n is a very small number (as close to zero as practicable)
 
e = ∑((2n + 2)/(2n +1)! )
(Sum from n=0 to n=infinity)
Pi (π)
3.141592653589793238462643383279502884197169399...
This number is used extensively in maths, physics and circles.
π = 4/1 - 4/3 + 4/5 - 4/7 + 4/9 - 4/11 + 4/13 - 4/15 ....
 
π = 3 + 4/2x3x4 - 4/4x5x6 + 4/6x7x8 - 4/8x9x10 + 4/10x11x12 - 4/12x13x14 + 4/14x15x16 ....
 
π = ∑( 16(-n) (4/(8n+1) - 2/(8n+4) - 1/(8n+5) - 1/(8n+6) ) )
 

Golden Ratio Φ
1.61803398874989484820458....
Φ = (1+√5)/2
 

Euler-Mascheroni Constant γ
0.5772156649015328606065120900824024310421593359...
γ = ∑(1/n) - Ln(n)
(Sum n=0 to n=very_large_number)
 
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