Gregory number

In mathematics, a Gregory number, named after James Gregory, is a real number of the form:

G_{x} = \sum_{i = 0}^{\infty} (- 1)^{i} \frac{1}{(2 i + 1) x^{2 i + 1}}

where x is any rational number greater or equal to 1. Considering the power series expansion for arctangent, we have

G_{x} = \arctan \frac{1}{x} .

Setting x = 1 gives the well-known Leibniz formula for pi. Thus, in particular,

\frac{π}{4} = \arctan 1

is a Gregory number.

Properties

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v t e Real numbers
Definitions	Absolute difference Cantor–Dedekind axiom Completeness Construction Real line Tarski axiomatization
Notable subsets	Cantor set Gregory number Irrational number Normal number Rational number Vitali set
Related topics	0.999... Decidability of first-order theories Extended real number line Rational zeta series Real coordinate space