# A Theorem About 1 and 0     4.60 (5 votes) Loading...

Well, I don’t know if this is known already, so if it is, please comment the name of the theorem. If it isn’t, then someone put a comment on what it should be called. A number with all of its digits less than 2 (which means only 0 and 1) and has less than 10 digits, multiplied with a number with the same characteristics has a product reverse of that of another set of two numbers with the same digits as the first set but has its digits reversed. Example below.
111111111 x 111111101 = 12345677876543211
111111111 x 101111111 = 11234567877654321

Thanks to Kirk Bamba for this submission!

### 7 thoughts on “A Theorem About 1 and 0”

1. Doesn’t always work:

11101 * 110101 = 1222231201
11101 * 101011 = 1121323111

I guess that’s why it’s listed under “math fail” then =).

1. Nope. If you construct the 2nd line correctly:
10111 x 101011 = 1021322221
it works (you have to reverse both factors).

On the other hand, this is not really a theorem, if you do the multiplication with pen and paper you’ll see that the result is no surprise.

1. Then that’s my own fail haha. I guess it wasn’t clear because after reading I thought it had meant to reverse only the second number multiplied. Perhaps if the example used something other than “11111111”

2. Agreed. This is not a theorem at all. To be called a ‘theorem’, such a statement must be proven rigorously, which this is not. Your observation is simply one of numerous number patterns that exist in the base 10 system. For clarification, please refer to some of the more famous number patterns: Fibonacci Sequence, Pascal’s Triangle, Pythagorean Triples, The Rule of 3, Multiples of 11 (which mimics your observation), etc.

1. . . . Or, more to your point, the base 2 number system.

3. You can call it a theorem but no one else will ever use it, because it’s just too simple and it doesn’t extend to higher level theorems.
You can prove it in a minute if you know how multiplication works.
To find 110101…1 * 11101…1
You do:
110101……………….1
110101……………….1
110101……………….1
110101………………1

+ 110101…………….1
—————————————
=122213………………………….1

If you reverse both numbers, you simply just rotate the block above by 180 degrees. And there’s no number greater than 9 in the sum of each column.
So, of course the result will simply be reversed.
It can be extended to a more general case. 1301*2111=2746411, 1031*1112=1146472.
But it’s not useful.

1. 