# Article

# Article

- Mathematics
- Analysis (calculus)
- e (mathematics)

# e (mathematics)

Article By:

**Lowan, Arnold N. **Formerly, Department of Physics, Yeshiva University, New York, New York.

**Bochner, Salomon **Formerly, Department of Mathematics, Rice University, Houston, Texas.

Last reviewed:2012

DOI:https://doi.org/10.1036/1097-8542.208500

**The number e usually defined as the limit approached by the expression**

*n*approaches infinity. If the given expression is expanded by the binomial theorem, and if one uses the theorem that the limit of the quotient of two polynomials of equal degree as the variable tends to infinity is equal to the ratio of the coefficients of highest degree, one obtains the expansion in relation (1).

*e*is larger than 2; it may be easily shown that it is smaller than 3. It can also be shown by elementary methods that

*e*is irrational; that is, it cannot be represented as the quotient of two integers. Furthermore,

*e*is transcendental; it does not satisfy any algebraic equation with integral coefficients. The transcendentality of

*e*was proved by the French mathematician C. Hermite in 1873; the proof constitutes an important milestone in the history of mathematics.

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