The Magic of Unity

This is to find the multiplication of two numbers each of the form 1111...11 and 11...111 ie. the two numbers consists if 1's only provided each are having the number of digits less than 9.

I would like to introduce a notation aTb which is read as "a times b". This means that the digit "b" is to be written "a" times one after another. For example, 4T5 means 55555 (the digit 5 is written four times continuously)

Let the two numbers be p and q where p consists of m digits and q of n digits and n<=m<=9. On multiplying p and q we get, p * q = 123....(n-1)[(m-n+1)Tn](n-1)(n-2).....4321 

For example, Multiply 1111111 and 111 Let y = 1111111 * 111 

Here m=7 and n=3 

1111111 * 1111 = 12[(7-3+1)T3]21 = 12[5T3]21 = 123333321 

Suppose we want to find the square of 11 

Then square of 11 = 11*11 = 1[(2-2+1)T2]1 = 1[1T2]1 = 121

Here the draw back is that I am not able to extend this formula for more than nine digits :(