Continuing in the tradition of my first 2 Instructables, I present to the interested reader a new mental math technique today. This trick, like the first 2 tricks, is for most intents and purposes simply a parlor trick to astound and amaze your fellow geeks at your next math club meeting. That said, there is an obscure set of circumstances where it could have some practical value.
So, without further ado, I present to you digital root extraction.
Step 1: What Is a Digital Root?
"What is the digital root?", you ask. The digital root is an obscure mathematical operation in which a user recursively adds together all of the digits of a number until only a single remains. This remaining number is known as the digital root.
A better demonstration would be to take a number (such as 179) and demonstrate. So, the digital root of 179 is something like this:
179 -> 1+7+9 = 17 -> 1+7 = 8
So, the digital root of 179 is 8.
Another, more interesting way to look at a digital root is that it is a modulo 9 operator. In other words, if you were to divide the number by 9, the integer remainder and the digital root are the same thing. So, digital roots can be rewritten as mod9(Number).
Now that we know what digital roots are, let's go ahead and find them.
Step 2: Finding Digital Roots
As you probably already guessed, digital roots can simply be found by the use of brute force. That is, simply divide by 9 or recursively add up the digits to find your digital root.
One thing the more clever people in the audience may have noticed, though, is that, since the digital root is equivalent to taking the number modulo 9, and modulo 9 is 0, we can ignore any number or numbers which add up to 9 or a multiple of 9.
So, for a number like 4529, the astute observer will see that 4+5 =9, so 4 and 5 can be ignored, and by the same reasoning, 9 can be ignored. Much quicker than dividing by 9 or recursively adding digits.
Quickly, it should be noted that the modulus definition is slightly off. That is, for a number greater than or equal to 9, the mod9 operation returns a digital root of 0, whereas it should be 9. So, for numbers such as 36, the digital root is clearly 9, even though the modulus would suggest it is 0. This is not particularly difficult, just keep it in mind when taking digital roots.
So, for practice, find the digital roots of the following numbers:
1) 47 (Digital Root: 2)
2) 365 (Digital Root: 5)
3) 1,852 (Digital Root: 7)
4) 136,257 (Digital Root: 6)
5) 234,181,360 (Digital Root: 1)
In the next step, we will talk about what applications digital roots have.
Step 3: Applications of Digital Roots
At this point, those of you still here are probably saying "digital roots are a fun math trick and all, but what's the point to knowing how to use them?
In the now dark and forgotten age before the creation of the calculator, digital roots were used by some (such as bank tellers) to check the accuracy of their computations. When 2 numbers are added, multiplied, or subtracted, the digital root of the result should be the digital root of the sum, product, or difference of the operands.
So, for example, consider the subtraction problem 342-173, which is 169. The digital root of 342, using the techniques from step 2, is 9, while the digital root of 173 is 2. Therefore the digital root of the difference should be the difference of the digital roots (or 7). Using techniques from step 2, we see that it is.
So, in summary, you can use digital roots as a primitive way to check the accuracy of subtraction, multiplication, and addition operations, which can be useful for any mental mathematicians out there.
In the next step, we will prove why digital roots work on multiplication, addition, and subtraction in this way.
Step 4: Prove It!
For most people, knowing how the mathematical technique works is enough to satisfy them. Still, some people aren't happy until they know why it works. So, for those people, consider the following simplified proof.
From earlier, we mentioned that the digital root is equivalent to taking a number modulo 9. From properties of the modulus operator, it can be shown that:
mod9(a)+mod9(b) = mod9(a+b)
mod9(a)-mod9(b) = mod9(a-b)
mod9(a)*mod9(b) = mod9(a*b)
Thus, we have an understanding of why digital roots can be applied to addition, subtraction, and multiplication. (They can also be applied to division, but that case requires some special treatment.)
Step 5: Summary
Digital roots are a fun trick to amuse the nerdiest of your friends, and can be a useful tool in checking peoples math, especially mental math. I hope you enjoyed this math tutorial, and please, Please, PLEASE, rate and comment, and of course let me know if there is something you would like me to try and write an instructable on.