Why is 1111^2=1234321?

Question

And why is it true more generally?
As in 11111111^2=123456787654321.

Answer 1

It's pretty obvious if you write out the multiplication:

......1111
....x1111
-----------
......1111
....1111
..1111
1111
----------
1234321

Algebraically 1111 x 1111 = 1111 x 1 + 1111 x 10 + 1111 x 100 + 1111 x 1000 = 1 x 1 + (1+1) x 10 + (1+1+1) x 100 ....

Each row is 1111 and each row shifts one to the left.

The nice symmetry would break down after 111111111, because then you'd have to start carrying.