Wonders of arithmetics!

11111111 * 11111111 = 123456787654321

The square of a number with eight 1's produces the sequence of all numbers from 1 to 8 and backwards.

You won't be able to verify this on a normal pocket calculator because it doesn't give you more than 8 digits. And you probably don't have the patience to do the math yourself.

On a PC, you can verify this by using software that gives you arbitrary numerical precision. With Excel, it's very simple: do the calculation and choose the 'Number' cell output format.