Chapter 4: Systems Based on Numbers

Section 5: Mathematical Constants

Egyptian fractions

Following the ancient Egyptian number system, rational numbers can be represented by sums of reciprocals, as in 3/7==1/3+1/11+1/231. With suitable distinct integers a[n] one can represent any number by Sum[1/a[n], {n, ∞}]. The representation is not unique; a[n]=2^n, n (n+1) and (n+1)!/n all yield 1. Simple choices for a[n] yield many standard transcendental numbers: n!: E-1; n!^2: BesselI[0,2]-1; n 2^n: Log[2]; n^2: Pi^2/6; (3n-1)(3n-2): Pi Sqrt[3]/9; 3-16n+16n^2: Pi/8; n n!: ExpIntegralEi[1] - EulerGamma. (See also page 902.)

From Stephen Wolfram: A New Kind of Science [citation]