Is there a name for this pattern? I'm not a mathematician, but I was calculating multiplication of some numbers and I saw a pattern emerging. What is this phenomenon called? And does it happen in other cases?
    6 *     9 =         54
   66 *    99 =       6534
  666 *   999 =     665334
 6666 *  9999 =   66653334
66666 * 99999 = 6666533334

6,6    * 9,9    = 65,34
6,66   * 9,99   = 66,5334
6,666  * 9,999  = 66,653334
6,6666 * 9,9999 = 66,66533334

Any ideas? Looks like 6's and 3's are introduced.
 A: You are multiplying $2/3 \times (10 - 10^{-k})$ by $10 - 10^{-k}$, and so you obtain
$2/3 \times (10 - 2 \times 10^{1-k} + 10^{-2k})$.  The initial $6$'s come from 
$2/3 \times 10 = 6.666\ldots$.  The $5$ and $3$'s come from subtracting 
$4/3 \times 10^{1-k} = 13.333\ldots \times 10^{-k}$.
A: I don’t know if it has a name, but here’s a way to see what is happening.
$$\begin{align}
65,34 +0,66&=66 & & = 6,6\times(9,9+0,1)=6,6\times10\\
66,534 +0,066&=66,6 & & = 6,66\times(9,99+0,01)=6,66\times10\\
66,6534 +0,0066&=66,66 & & =6,666\times(9,999+0,001)=6,666\times10\\
66,66534 +0,00066&=66,666 & & =6,6666\times(9,9999+0,0001))=6,6666\times10
\end{align}$$
Similar patterns would occur with repeated digits other than $6$, but the repeated $9$s are more special.
A: The reason for that is you are multiplying by something close to 10.  9; 9.9; 9.99; 9.999 ... converges to 10 so you are just observing convergence. and 0.666666666666666
it is much better to see it as
0.6 * 0.9 = 6 * 0.1 * (1-0.1) 
and
0.66 * 0.99  = 6 * (0.1 + 0.01) * (1 - 0.01)
and
0.666   * 0.999   = 6 * (0.1+0.01+0.001) * (1- 0.001 )
etc.
basically the product of two brackets converges to 1/9  *  1 so you are observing something converging to 6/9. more and more digits are becoming exact.
