10

The number 10 (ten) is the basis for the Decimal system of notation. In this system, each ``decimal place'' consists of a Digit 0-9 arranged such that each Digit is multiplied by a Power of 10, decreasing from left to right, and with a decimal place indicating the s place. For example, the number 1234.56 specifies

$\begin{displaymath} 1\times 10^3+2\times 10^2+3\times 10^1+4\times 10^0+5\times 10^{-1}+6\times 10^{-2}. \end{displaymath}$

The decimal places to the left of the decimal point are 1, 10, 100, 1000, 10000, 100000, 1000000, 10000000, 100000000, ... (Sloane's A011557), called one, ten, Hundred, Thousand, ten thousand, hundred thousand, Million, 10 million, 100 million, and so on. The names of subsequent decimal places for Large Numbers differ depending on country.

Any Power of 10 which can be written as the Product of two numbers not containing 0s must be of the form $2^n\cdot 5^n=10^n$ for an Integer such that neither nor contains any Zeros. The largest known such number is

$\begin{displaymath} 10^{33}=2^{33}\cdot 5^{33} = 8,589,934,592\cdot 116,415,321,826,934,814,453,125. \end{displaymath}$

A complete list of known such numbers is

$\displaystyle 10^1$	$\textstyle =$	$\displaystyle 2^1\cdot 5^1$
$\displaystyle 10^2$	$\textstyle =$	$\displaystyle 2^2\cdot 5^2$
$\displaystyle 10^3$	$\textstyle =$	$\displaystyle 2^3\cdot 5^3$
$\displaystyle 10^4$	$\textstyle =$	$\displaystyle 2^4\cdot 5^4$
$\displaystyle 10^5$	$\textstyle =$	$\displaystyle 2^5\cdot 5^5$
$\displaystyle 10^6$	$\textstyle =$	$\displaystyle 2^6\cdot 5^6$
$\displaystyle 10^7$	$\textstyle =$	$\displaystyle 2^7\cdot 5^7$
$\displaystyle 10^9$	$\textstyle =$	$\displaystyle 2^9\cdot 5^9$
$\displaystyle 10^{18}$	$\textstyle =$	$\displaystyle 2^{18}\cdot 5^{18}$
$\displaystyle 10^{33}$	$\textstyle =$	$\displaystyle 2^{33}\cdot 5^{33}$

(Madachy 1979). Since all Powers of 2 with exponents $86<n\leq 4.6\times 10^7$ contain at least one Zero (Cook), no other Power of ten less than 46 million can be written as the Product of two numbers not containing 0s.

See also Billion, Decimal, Hundred, Large Number, Milliard, Million, Thousand, Trillion, Zero

References

Madachy, J. S. Madachy's Mathematical Recreations. New York: Dover, pp. 127-128, 1979.

Pickover, C. A. Keys to Infinity. New York: Wiley, p. 135, 1995.

Sloane, N. J. A. Sequence A011557 in ``The On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html.