Introduction: Mental Math: Digital Root Extraction

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.



babai9name. (author)2016-06-25

423.62 - 269.21 ÷ (11.9 % of 78) = ?

what is the digital root of the equation ... plz describe step by step

RakeshJ4 (author)2015-09-06

4558 digital root is 4 and for 86 it is 5 so

4558/86=4/5 but answer is 53 with digital root as 8. am i wrong somewhere or this shortcut cannot be applied everytime??

PardhasaradhiV (author)RakeshJ42016-05-04

you don't whole concept first learn full concept digital root method 4/5=0.8 and 5+3=8 both => 0.8 = 8 is same value
neglect points
neglect Negatives
Eg: -5 is value .subtract high value of digits that means 9-5=4

pantasangam (author)2015-01-31

this is amazing. i never realized the benefit of digital root. nobody ever taught us about it in the entire school life. thank you so much. greetings from nepal. namaste. :)

CJHayosh (author)2014-12-13

I love digital roots. How about this? If you subtract any 2 unique integers with the same digital root, the answer is always evenly divisible by 9. Which is easy to prove by expanding on modulo(a)-modulo(b)=modulo(a-b).

mluft (author)2009-08-29

so, i am a 4th grade teacher and i'm teaching my students about digital root. i'm teaching them that it is a tool to use to check computation. i have a parent all up in my face saying it is too abstract for 4th graders and it is some new-fangled crap i'm teaching that is going to screw up her kid's understanding of math for life. i've searched and searched for something, anything to help her understand, but no luck. any ideas?

purduecer (author)mluft2009-09-02

In case you didn't know already, you can find out more about the digital root here. Digital roots are a topic that is indeed fairly abstract in math, but as you mentioned, they can be a quick but useful way to check hand computations. If I were you, and trying to explain to the parent, I would mention simply that you're teaching your kids a helpful shortcut for finding the remainder, and remind the parent that when they add all the digits together to see if a number is divisible by 9, they are in fact already taking the digital root.

Hope that helps, let me know.

foobear (author)2008-10-27

Thank you, this is wonderful. I always wondered what the term "casting out 9s" meant. Please show us how to do the division one too.

Perhaps you could team up with this guy: math museum and maybe this guy:
advanced subject explained with ascii art

And one day they could make the perfect math book that would actually imbue the average math phobe with a true intuition for math! A sort of "math pill" if you will.

purduecer (author)foobear2008-10-27

haha, thanks for the praise (nice secret squirrel icon, btw). I will continue to post math instructables as I discover them, and in the meantime will try and bring interesting material to the Instructables fan. --Purduecer

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