Notes

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] = 2n, n (n + 1) and (n + 1)!/n all yield 1. Simple choices for a[n] yield many standard transcendental numbers: n!: - 1; n!2: BesselI[0, 2] - 1; n 2n: Log[2]; n2: π2/6; (3n - 1)(3n - 2): π 3/9; 3 - 16n + 16n2: π/8; n n!: ExpIntegralEi[1] - EulerGamma. (See also page 902.)



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