Value of e, Base of Natural Logarithms

1. e to One Hundred Decimal Places

e = 2.71828 18284 59045 23536 02874 71352 66249 77572 47093 69995 95749 66967 62772 40766 30353 54759 45713 82178 52516 64274

Source: Encyclopedic Dictionary of Mathematics, The Mathematical Society of Japan (Cambridge, Mass.: MIT Press, 1980);

cited in Maor, E., e: The Story of a Number, Princeton NJ: Princeton Univ. Press, 1994, p. 211.
2. Another representation of e, the base of natural logarithms, is as exponentiation involving only a few digits, namely:


= 2.718290995

This is due to Brewster, G.W. The Mathematical Gazette, 25-263, 49, Feb. 1941;

It was cited in Gaither, C. C. and Cavazos-Gaither, A. E.,
Mathematically Speaking, A Dictionary of Quotations, 1998.

For clarity. Note that the exponent for 2 is 5/2 raised to the 2/5.
3. The online calculator at

gives 1.638721270700128 for the sq root of e, and 1.280125490215755 for the sq root of sq root of e if you input this e approximation: 2.7182818284590452.

4. The fourth root of e, e^0.25, from a TI-81 hand calculator is about 1.284025417 .

5. In email to the NCTM-L mailing list Han Sah commented on a property of e, that Martin Gardner wrote of in the context of "six major discoveries of 1974" (Gardner, M., Scientific American, April 1975), indicating that the article was was an April fool's joke and the specific statement of interest here, the integer nature of an expression below (following from the fact that it was said by Gardner to terminate in an infinite number of nines), was false.

If e(pi*sqrt[163]) isn't actually integer, it has an interesting property that Ramanujan knew about.

The "first 12 digits after the decimal point are 9's" (Sah,, accessed 10/31/03) but not the 13-th.
6. A word mnemonic for e follows.

Take the portrait of Andrew Jackson. There are two things about it. He was the seventh president of the United States. He was elected in 1828. He was installed in office in 1828. If you cut it on a diagonal you get a 45-90-45 triangle.

2.718281828459045 is accurate to a large number of digits (see 1.).

3/16/2009 Version
©2009 Allen Klinger