Volume 11, 1999

University of Bialystok

Copyright (c) 1999 Association of Mizar Users

**Freek Wiedijk**- University of Nijmegen
- Written while a guest of the Institute of Mathematics of the University of Bia{\l}ystok.

- We prove the irrationality of square roots of prime numbers and of the number $e$. In order to be able to prove the last, a proof is given that {\tt number\_e = exp(1)} as defined in the Mizar library, that is that $$\lim_{n\rightarrow\infty} (1+{1\over n})^n = \sum_{k=0}^\infty {1\over k!}$$

- Square Roots of Primes are Irrational
- A proof that $e = e$
- The Number $e$ is Irrational

- [1]
Grzegorz Bancerek.
The fundamental properties of natural numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [2]
Grzegorz Bancerek.
The ordinal numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [3]
Grzegorz Bancerek and Krzysztof Hryniewiecki.
Segments of natural numbers and finite sequences.
*Journal of Formalized Mathematics*, 1, 1989. - [4]
Czeslaw Bylinski.
Functions and their basic properties.
*Journal of Formalized Mathematics*, 1, 1989. - [5]
Czeslaw Bylinski.
Finite sequences and tuples of elements of a non-empty sets.
*Journal of Formalized Mathematics*, 2, 1990. - [6]
Czeslaw Bylinski.
The sum and product of finite sequences of real numbers.
*Journal of Formalized Mathematics*, 2, 1990. - [7]
Marek Chmur.
The lattice of natural numbers and the sublattice of it. The set of prime numbers.
*Journal of Formalized Mathematics*, 3, 1991. - [8]
Andrzej Kondracki.
Basic properties of rational numbers.
*Journal of Formalized Mathematics*, 2, 1990. - [9]
Jaroslaw Kotowicz.
Convergent sequences and the limit of sequences.
*Journal of Formalized Mathematics*, 1, 1989. - [10]
Jaroslaw Kotowicz.
Monotone real sequences. Subsequences.
*Journal of Formalized Mathematics*, 1, 1989. - [11]
Jaroslaw Kotowicz.
Real sequences and basic operations on them.
*Journal of Formalized Mathematics*, 1, 1989. - [12]
Jaroslaw Kotowicz.
Functions and finite sequences of real numbers.
*Journal of Formalized Mathematics*, 5, 1993. - [13]
Rafal Kwiatek.
Factorial and Newton coefficients.
*Journal of Formalized Mathematics*, 2, 1990. - [14]
Rafal Kwiatek and Grzegorz Zwara.
The divisibility of integers and integer relatively primes.
*Journal of Formalized Mathematics*, 2, 1990. - [15]
Jan Popiolek.
Some properties of functions modul and signum.
*Journal of Formalized Mathematics*, 1, 1989. - [16]
Konrad Raczkowski and Andrzej Nedzusiak.
Real exponents and logarithms.
*Journal of Formalized Mathematics*, 2, 1990. - [17]
Konrad Raczkowski and Andrzej Nedzusiak.
Series.
*Journal of Formalized Mathematics*, 3, 1991. - [18]
Andrzej Trybulec.
Subsets of real numbers.
*Journal of Formalized Mathematics*, Addenda, 2003. - [19]
Andrzej Trybulec and Czeslaw Bylinski.
Some properties of real numbers operations: min, max, square, and square root.
*Journal of Formalized Mathematics*, 1, 1989. - [20]
Michal J. Trybulec.
Integers.
*Journal of Formalized Mathematics*, 2, 1990. - [21]
Zinaida Trybulec.
Properties of subsets.
*Journal of Formalized Mathematics*, 1, 1989. - [22]
Yuguang Yang and Yasunari Shidama.
Trigonometric functions and existence of circle ratio.
*Journal of Formalized Mathematics*, 10, 1998.

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