A number which is very close to an Integer. One surprising example involving both *e* and Pi is

(1) |

(2) |

(3) |

(4) |

An interesting near-identity is given by

(5) |

(6) |

(7) |

A whole class of Irrational ``almost integers'' can be found using the theory of
Modular Functions, and a few rather spectacular examples are given by Ramanujan
(1913-14). Such approximations were also studied by Hermite (1859), Kronecker (1863), and Smith
(1965). They can be generated using some amazing (and very deep) properties of the *j*-Function. Some of the
numbers which are closest approximations to Integers are
(sometimes known as the
Ramanujan Constant and which corresponds to the field
which has Class Number 1 and is the
Imaginary quadratic field of maximal discriminant),
,
,
and
, the last three of which have Class Number 2 and are due to Ramanujan (Berndt
1994, Waldschmidt 1988).

The properties of the *j*-Function also give rise to the spectacular identity

(8) |

The list below gives numbers of the form for for which .

Gosper noted that the expression

(9) |

**References**

Berndt, B. C. *Ramanujan's Notebooks, Part IV.* New York: Springer-Verlag, pp. 90-91, 1994.

Hermite, C. ``Sur la théorie des équations modulaires.'' *C. R. Acad. Sci. (Paris)* **48**, 1079-1084 and 1095-1102, 1859.

Hermite, C. ``Sur la théorie des équations modulaires.'' *C. R. Acad. Sci. (Paris)* **49**, 16-24, 110-118, and 141-144, 1859.

Kronecker, L. ``Über die Klassenzahl der aus Werzeln der Einheit gebildeten komplexen Zahlen.'' * Monatsber. K. Preuss. Akad. Wiss. Berlin*, 340-345. 1863.

Le Lionnais, F. *Les nombres remarquables.* Paris: Hermann, 1983.

Ramanujan, S. ``Modular Equations and Approximations to .'' *Quart. J. Pure Appl. Math.* **45**, 350-372, 1913-1914.

Smith, H. J. S. *Report on the Theory of Numbers.* New York: Chelsea, 1965.

Waldschmidt, M. ``Some Transcendental Aspects of Ramanujan's Work.'' In
*Ramanujan Revisited: Proceedings of the Centenary Conference* (Ed. G. E. Andrews, B. C. Berndt, and R. A. Rankin).
New York: Academic Press, pp. 57-76, 1988.

© 1996-9

1999-05-25