111,111,111 x 111,111,111 = 12,345,678,987,654,321
This mathematical equation appeared within the ‘list of facts’ e-mail I received a while back. Whilst many of the others have so far turned out to be popular misconceptions, this one is demonstrably true (it would have been fairly foolish to get it wrong!).
Whilst it might initially seem surprising that 111,111,111 x 111,111,111 = 12,345,678,987,654,321 - perhaps you might even wonder how such a mathematical oddity could exist - when you stop and think about it, the equation is less of a co-incidence than you might think.
Let’s look at the sequence of squaring ever longer strings of 1s:
1 x 1 = 1
11 x 11 = 121
111 x 111 = 12,321
1,111 x 1,111 = 1,234, 321
You can guess the rest…
Why should this be? Well it’s easier to see if you do the maths the high school way:
111 x 111 ----- 111 111 111 ===== 12321
Does that take the mystery out of it?
If you try 1,111,111,1112 then the pattern is slightly messed up, as the middle number is forced to carry (although most calculators won’t show this, so you’ll have to do it on paper). However, if you want to try using base 11 or higher…