# Definition:Wallis's Number

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## Definition

**Wallis's number** is the real root of the cubic:

- $x^3 - 2 x - 5 = 0$

Its approximate value is:

- $2 \cdotp 09455 \, 1 \ldots$

This sequence is A007493 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

## Also known as

The name of this number is also presented as **Wallis' number**.

## Source of Name

This entry was named for John Wallis.

## Historical Note

The cubic $x^3 - 2 x - 5 = 0$ was used by John Wallis as a example to demonstrate how Newton's Method could be used to solve certain equations numerically.

The real root of this equation has since become known as Wallis's number.

It has subsequently been used as a test for a number of different methods of approximation.

It is known to over $4000$ digits.

## Sources

- April 1984: Fred Gruenberger:
*How to handle numbers with thousands of digits, and why one might want to*(*Scientific American***Vol. 250**,*no. 4*: 19 – 26)

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $2 \cdotp 094 \, 551 \ldots$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $2 \cdotp 09455 \, 1 \ldots$