# Definition:Champernowne Constant

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## Definition

The **Champernowne constant** is the real number whose decimal expansion is formed by concatenating the positive integers in ascending order:

- $C_{10} = 0 \cdotp 12345 \, 67891 \, 01112 \, 13141 \, 51617 \, 18192 \, 02122 \ldots$

This sequence is A033307 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

## Also known as

The **Champernowne constant** is also known as **Mahler's number** for Kurt Mahler, who proved it transcendental in $1937$.

Some sources report it as **Champernowne's number**.

## Also see

- Results about
**the Champernowne constant**can be found**here**.

## Source of Name

This entry was named for David Gawen Champernowne.

## Historical Note

The **Champernowne constant** was invented by David Gawen Champernowne as an example of a real number which can be demonstrated to be normal with respect to base $10$.

He achieved this in $1933$ while still an undergraduate.

## Sources

- 1933: D.G. Champernowne:
*The Construction of Decimals Normal in the Scale of Ten*(*J. London Math. Soc.***Vol. 8**: pp. 254 – 260) - 1983: François Le Lionnais and Jean Brette:
*Les Nombres Remarquables*... (previous) ... (next): $0,1234567891011 \ldots$ - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**Champernowne's number** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**Champernowne's number**