mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a Gregory number, named after James Gregory, is a

real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...

of the form: :

G_x = \sum_^\infty (-1)^i \frac

where ''x'' is any rational number greater or equal to 1. Considering the

power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''a_n'' represents the coefficient of the ''n''th term and ''c'' is a co ...

expansion for

arctangent In mathematics, the inverse trigonometric functions (occasionally also called ''antitrigonometric'', ''cyclometric'', or ''arcus'' functions) are the inverse functions of the trigonometric functions, under suitably restricted domains. Specific ...

, we have :

G_x = \arctan\frac.

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

\frac=\arctan 1

is a Gregory number.

Properties

G_=-(G_x)

\tan(G_x)= \frac

References

Sets of real numbers {{numtheory-stub

Properties

See also

References