Digital Root

11/23/05

Check http://digicc.com/fido out.

Whoah!

This is one of my favorite number tricks. I'll explain how it works.

It works by using a 'digital root'. A digital root is the sum of the digits in a number, and you keep summing them until you get a single digit. The digital root, since it is a digit, only takes on the values 0 through 9. For example,

• digital root(102) = 3, because 1+0+2 = 3.
• digital root(67) = 4, because 6+7 = 13 and 1+3 = 4
• digital root(563) = 5, because 5+6+3 = 14, and 1+4 = 5
• digital root(82) = 1, because 8+2 = 10, and 1+0 = 1.

The digit that you circled, and the one the program reveals, is:

This works because the digital root of your original number minus a jumbling of your original number, always gives a digital root of 9. This is because if a*10^3+b*10^2+c*10+d is your original number, and your jumbled number (without loss of generality) is d*10^3+c*10^2+a*10+b, then

This can also be written as

to make it more obvious that it is a multiple of 9. Multiples of 9 always have digital roots of 9. With other variations of jumbling your number, you'll also get digital roots of 9.

For example, say your number is 9741, and you jumble it to get 7194, and subtract the two to get 2547 (notice this has a digital root of 9). You circle the 5 and type in 247.

9-digital root(247) = 9-4 = 5, so the computer knows you circled the 5.

The jumbling at the end it asks you to do is just to mystify some more, because digital root(247) = digital root(274), etc., because the digital root is just a type of sum.

Also, the program says to not circle a zero because things like 2250 and 2259 have the same digital root (of 9), so by the formula above, it can't tell if what you circled is a 0 or a 9. So if your number is 2250 and you circle 0 and enter in 225, the program will think you circled 9.

The digital root has many applications. I use it as a quick mental check for arithmetic. For example, if I write that 13*15 = 185, I can do a basic check using the digital root.

digital root(13)*digital root(15) = 4*6 = 24

digital root(24) = 6

But, digital root(185) = 5

Because the digital roots are not equal, the original equation of 13*15 = 185 is wrong.

However, be warned: if digital root(left side of equation) is not equal to digital root(right side of equation), then you made a mistake, but if digital root(left side of equation) = digital root(right side of equation), this does not mean your original equation is correct. This is because different numbers can have the same digital root. It is a good and quick basic check.