Monday, September 14, 2009

Vedic Mathematics Lesson 19: Digital Roots

The digital root of a number is the single-digit number that one gets by adding up the digits of the number. Why is the digital root of a number important? In this lesson, we will learn about how to calculate the digital roots of numbers. Then, we will also explore how they are useful.

You can find all the previous posts about Vedic Mathematics below:

Multiplication Special Case 3
Vertically And Crosswise I
Vertically And Crosswise II
Squaring, Cubing, Etc.
Subtraction
Division By The Nikhilam Method I
Division By The Nikhilam Method II
Division By The Nikhilam Method III
Division By The Paravartya Method

The simplest way to calculate the digital root of a number is to take the definition of the digital root literally: one can simply add up the digits of the number. If the sum is not a single digit, take the sum of the digits of this sum again. Keep repeating this process until the sum becomes a single digit. This method always works and is entirely fool-proof. But it can be quite tiresome when you start with a large number with big digits. It can also be error-prone.

To illustrate this simple method let us try to find the digital root of 45769486. When we add up the digits, we first get 4 + 5 + 7 + 6 + 9 + 4 + 8 + 6 = 49. Since 49 is not a single-digit number, we repeat the process on 49 once again. We get 4 + 9 = 13. 13 is not a single-digit number either, so we are forced to repeat the process one more time to get 1+ 3 = 4. 4 is the digital root of 45769486.

The second way to calculate the digital root is to find the remainder of the division of the number by 9. Recalling how to divide by 9 from this lesson, we calculate the digital root as below:

9 1
4576948|6
0496215|43
001233
-------------
5085493|49
5085498|4

Once again, we find the digital root to be 4, just as we found out using the brute-force addition method.

What happens when the remainder after division by 9 turns out to be zero? For instance, what is the digital root of 45769482? We find that the division by 9 gives us a remainder of 0 as below:

9 1
4576948|2
0496215|43
001233
--------------
5085493|45
5085498|0

When we perform the brute-force addition of the digits, we find the following:

4 + 5 + 7 + 6 + 9 + 4 + 8 + 2 = 45
4 + 5 = 9

So, the digital root of 45769482 is 9. The remainder after division by 9 is 0. So, we can conclude that if the division by 9 gives us a remainder of 0, the digital root is actually 9. Remember that if a number N divided by 9 gives us a quotient of Q and a remainder of 0 (i.e. Q x 9 = N), then we could also say that N divided by 9 gives us a quotient of Q - 1 and a remainder of 9 (i.e. (Q - 1) * 9 + 9 = N).

This equivalence between 0 and 9 can be exploited even further in finding the digital root. One obvious way would be to consider all 9's in the original number to be equivalent to 0 and not include them in the brute-force addition or the division by 9. Let us take our original number, 45769486, once again. Setting 9 = 0 in the number and then adding up the digits gives us 4 + 5 + 7 + 6 + 4 + 8 + 6 (we skipped the 9 because it is considered equivalent to zero) = 40, and 4 + 0 = 4. We find that we got the original answer once again.

We can verify that division by 9 gives us equivalent results. First we perform the division by 9 substituting 0 for 9.

9 1
4576048|6
0496226|34
001222
--------------
5084494|40
5084498|4

This gives us the same digital root, 4, as before. Next, we perform the division by 9 after throwing out the 9 from the number entirely.

9 1
457648|6
049626|34
00122
-------------
508494|40
508498|4

This also gives us the same digital root as before, 4.

We are now ready to go one step further and make a more radical simplification. We formalize this simplification as "casting out the nines". Essentially, just like we threw out the 9 (cast out the 9) in the original number to make the calculation of the digital root simple, we will cast out any numbers that add up to 9 in the original number. Let us take our original number once again: 45769486.

We see that 4 + 5 = 9. Let us cast out 4 and 5 in addition to any 9's in the number. We are left with 76486. Using the brute-force method on this number, we find the digital root to be 7 + 6 + 4 + 8 + 6 = 31, 3 + 1 = 4. Once again, we are left with our original digital root of 4. We will find that this result holds true when we use the division by 9 method also:

9 1
7648|6
0437|25
011
----------
8195|31
8198|4

We are once again left with a remainder of 4, giving us a digital root of 4 once again.

The method of casting out 9's makes finding the digital root quite easy for lots of numbers. In many numbers, it is easy to find individual digits that add up to 9. Throwing them out makes the original number much shorter, making the brute-force addition much simpler.

A simple extension of this method involves the casting out of not just 9's, but also all multiples of 9. Let us see this in action as below, by trying to find the digital root of 45769486.

First we throw out 4 and 5 because they add up to 9. We are left with 769486. Next we throw out 9, because it obviously adds up to 9! We are left with 76486. Now, we notice that 4 + 8 + 6 = 18, which is a multiple of 9. Throwing out those three digits leaves us with 76. It is trivially simple to then add up 7 and 6 to get 13, which is then simplified to 4, the original digital root.

Unfortunately, while identifying digits which add up to 9 is simple, identifying digits that add up to multiples of 9 is not as easy. So, this simplification may not save us much time. But there is one final simplification that we have not dealt with that will make the task of finding digital roots even simpler. It is a subtle extension of the principle of casting out the 9's, and it works as follows:

• We add up the digits of the number one by one.
• Any time the sum of the digits equals or exceeds 9, we cast out the 9 and retain only the part that is in excess of 9.
• What we are left with at the end is the digital root of the number. If what we are left with is 0, the digital root is 9.

That is all there is to it! To convince ourselves, let us apply this method to 45769486. We can choose to identify and throw out 9's and numbers that add up to 9 or multiples of 9. But we don't have to. It is all automatically taken care of using the above method. Let us walk through the steps one by one:

• 4 + 5 = 9. Since the sum equals or exceeds 9, we cast out the 9 and are left with 9 - 9 = 0.
• 0 + 7 = 7
• 7 + 6 = 13. Since this exceeds 9, we cast out the 9 from it. We are left with 13 - 9 = 4.
• 4 + 9 = 13. Once again, this exceeds 9, so cast out the 9 to get 13 - 9 = 4 (we can use the fact that the sum does not change when we add 9 to just skip the 9 if we choose to).
• 4 + 4 = 8
• 8 + 8 = 16. Cast out the 9 from this to get 16 - 9 = 7.
• 7 + 6 = 13. Cast out the 9 from this to get 13 - 9 = 4.

We are done with all the digits and we are left with 4. We see that we have once again calculated the digital root, but this time, we did not have to do brute-force addition multiple times or perform a division by 9. All we did was simple single-digit additions and a few subtractions of 9.

Casting out the 9's whenever possible simplifies the calculation of the digital root immensely, so that it is possible to do it very fast mentally without having to keep track of big sums that keep growing. We are guaranteed that the largest sum we ever have to deal with using this method is 17 (and we have to deal with 17 only if we choose not to cast out the 9 directly instead of adding it to the sum and then casting it out. Otherwise, the largest sum we need to deal with is 16).

Now that we have explored different methods of calculating the digital roots, and figured out a trivially simple method involving a few single digit additions and simple subtractions, we ask ourselves why we need to do it in the first place. What is the use of calculating a digital root?

Let us take a few simple applications of digital roots first. As anybody who has gone through elementary school can tell you, the rule for figuring out whether a number is divisible by 3 is that a number is divisible by 3 if the sum of the digits of that number is divisible by 3. A simpler way of expressing this rule (since to figure out whether the sum is divisible by 3 might involve calculating more sums) is to say that the number is divisible by 3 if the digital root of that number is divisible by 3. That is a very powerful application of the concept of digital roots.

Similarly, the rule for divisibility by 9 is that the sum of the digits of the number should be divisible by 9 for the number to be divisible by 9. The simpler way of expressing this is to say that the number is divisible by 9 if the digital root of the number is 9.

But there is more to a digital root than just that. Consider the following properties of a digital root: The digital root of the sum of two numbers is the sum of the digital roots of the two numbers. In other words, if D() is a function to find the digital root of a number, then D(n1 + n2) = D(n1) + D(n2).

The obvious application of this property is in verifying one's arithmetic to see if it passes the "smell-check". In this case, the smell is the digital root. If the sum you calculated has a digital root that is not the same as the sum of the digital roots of the 2 numbers you added up, your answer is obviously wrong. Obviously, if the digital roots do match, that does not guarantee that the sum is correct, but at least, it has passed the "smell-test".

Let us apply this property to the following sum:

437285437
975984648
------------

We calculate the digital sum of the first number as 7. We calculate the digital sum of the second number as 6. The sum of the digital roots is 13, and expressing it as a digital root gives us 4. Now let us calculate the sum of the numbers. We get 1413270085. To do a smell-check on this answer, let us find its digital root: we get 4. We can not guarantee that the answer is correct because it may be missing a zero or 9 somewhere or might have digits interchanged. But if we had gotten a digital root other than 4, we could guarantee that the sum was wrong and we would have to redo it to correct some mistake.

We can easily see that this property holds true for subtraction also. That is, D(n1 - n2) = D(n1) - D(n2). Here, we have to be a little careful in the application though. It is possible for the digital root of n1 to be lower than that of n2. In that case, D(n1) - D(n2) will turn out to be a negative number. However, the expression on the left hand side of that equation, D(n1 - n2) is always positive, and is restricted to being between 1 and 9 (at least in the base 10 arithmetic we are using here). To derive an equivalent positive digital root from a negative sum, if we do get one on the right hand side, we add 9 to the number. That is all there is to it, but it has to be kept in mind when using this method for verification of the result of a subtraction.

Let us verify this using the example below:

86958678
-32984878
-------------

The digital root of the first number is 3. The digital root of the second number is 4. We see that D(n1) - D(n2) = -1. Following the rule above about converting negative numbers into equivalent digital roots, we add 9 to this answer to get 8. Indeed, the difference is 53973800, whose digital root is 8.

The property is particularly useful in the verification of multiplication problems. As you guessed, D(n1 x n2) = D(n1) x D(n2). Let us verify this by applying the property on the following problem:

474387 x 390989

The digital root of the first number is 6. The digital root of the second number is 2. The product of the digital roots is 12, which can be simplified to 3. We find that the product of the two numbers is 185480098743. Its digital root is indeed 3 as predicted by the property above.

In addition to being useful for verifying the results of multiplication problems, this property can also be used to solve various puzzles that are commonly posed in IQ tests, and sometimes, in recruitment tests also. The question usually goes something like this: what is the digital root of the 987654321st power of 123456789? In fact, I frequently bamboozle job applicants to my department with this question or something similar!

It may not be this exact problem, but you get the basic idea. Essentially, you are asked to find the digital root of a number you can never hope to calculate. Even if you had enough time to do the calculation, you would require reams of paper to hold the answer. The trick to solving the problem is knowing that the examiner does not expect you to compute the answer to the exponentiation problem at all. And in fact, you don't have to.

Consider this: 123456789^987654321 is actually equal to 123456789 x 123456789 x 123456789 . . . 987654321 times. Now, the digital root of 123456789 is 9. So, the digital root of 123456789 x 123456789 is 9 x 9 = 81, which is once again 9. In fact, it is simple to establish that multiplying even three 123456789's with each other does not change the digital root of the answer. It is always 9. Hence, the answer to the problem is 9.

What if the examiner is not quite so obliging and throws in something like 7^777? The trick here is to identify a pattern of digital roots that you can then exploit to get your answer. We notice that the digital root of 7 is obviously 7 itself. The digital root of 7 x 7 is 4. The digital root of 7 x 7 x 7 is 1 (4 x 7 = 28, which is the same as 1 in digital root terms). The digital root of 7^4 then works out to 7 once again. Thus we see the following pattern:

Number -- Digital Root
7^1 ------- 7
7^2 ------- 4
7^3 ------- 1
7^4 ------- 7
7^5 ------- 4
7^6 ------- 1

and so on and so forth. Now, it is simply a matter of predicting what the digital root will be if we extend the pattern all the way to the 777th power of 7. Since the digital roots repeat after very 3 terms, we find the modulus of 777 with respect to 3 and find that it is 0. A modulus of 0 is equivalent to a modulus of 3 since the divisor is 3. We then look up the pattern for what the digital root is when the power is 3, and we immediately derive the correct answer: 1.

The pattern of digital roots with successive powers for various digital roots is as below:

• If the digital root of the original number is 1, then the digital root remains 1 for all higher powers of that number.
• If the digital root of the original number is 2, then the digital roots for higher powers follow the pattern 2, 4, 8, 7, 5, 1, 2, . . . The pattern repeats after every 6 powers.
• If the digital root of the original number is 3, then the digital roots for higher powers follows the pattern 3, 9 , 9 , 9, . . . The digital root stays at 9 for all higher powers of that number.
• If the digital root of the original number is 4, then the digital roots for the higher powers follow the pattern 4, 7, 1, 4, . . . The pattern repeats after every 3 powers.
• If the digital root of the original number is 5, then the digital roots for the higher powers follow the pattern 5, 7, 8, 4, 2, 1, 5, . . . The pattern repeats after every 6 powers.
• If the digital root of the original number is 6, then the digital roots for the higher powers follow the pattern 6, 9, 9, . . . The digital root stays at 9 for all higher powers of that number.
• If the digital root of the original number is 7, then the digital roots for the higher powers follow the pattern 7, 4, 1, 7, . . . The pattern repeats after every 3 powers.
• If the digital root of the original number is 8, then the digital roots for the higher powers follow the pattern 8, 1, 8, . . . The pattern repeats after every 2 powers.
• If the digital root of the original number is 9, then the digital roots of all higher powers of that number are also 9.

That is the power of digital roots! Hope you found this lesson and the discussion on the properties of digital roots useful. As always, it is important to practice the procedure for quickly calculating digital roots.

When I was young, I used to calculate the digital root of every phone number I encountered on my way from home to school and back (I had no idea about Vedic Mathematics at that time, it was just that I was fascinated with digital roots for some reason, and I had independently figured out the procedure explained in this lesson). It was difficult to calculate digital roots before I lost sight of the phone numbers initially as my school bus drove by, but after some practice, I could look at a number and tell its digital root out loud in a second or less. If I can do it, anybody can! Good luck!!

ankita said...

thanks a lot...! It was really helpful...!

I had a query though... Can we use digital roots in case of division?

D{n1/n2} = D{n1}/ D{n2}
Does that exist?????

Blogannath said...

Yes, we can use it for division also, but not directly and as easily as for addition, subtraction and multiplication. Take 14/7 for instance. The digital root of 14 is 5 and that of 7 is 7. But 5 is not divisible by 7. The trick is to keep adding 9 to the digital root of the numerator until it becomes divisible by the digital root of the denominator. Then, the digital root of the whole number answer of that division is the digital root of the initial division answer also.

So, take 364/14 as an example. The digital root of the numerator is 4. The digital root of the denominator is 5. Since 4 is not divisible by 5 directly, keep adding 9 to 4 repeatedly until you get a number divisible by 5. In this case, we add 9 four times to 4 get 40, which is divisible by 5. The quotient of 40/5 is 8, whose digital root is 8. And the quotient of 364/14 is 26, whose digital root is also 8.

So, to answer your question, D(n1/n2) = D((D(n1) + a*9)/D(n2)), where a is some whole number such that D(n1) + a*9 is divisible by D(n2).

sreedhar said...

For division, Digital Roots work as expected in most cases, only thing is we must define DR of a fraction in the natural manner, e.g. D(7/8) = D(0.875) = DR(8+7+5) = 2.

We need to be careful only when

D(n2) is 3, 6 or 9 && D(n1) is none of these. In this case, it is akin to division by zero (i.e., not defined).

D(n2) is 3, 6 or 9 && D(n1) is also 3, 6 or 9. Then multiple answers are possible.

Dr. Kardi Teknomo has a nice tutorial on Digital Root, showing the division table (color coded).

Regards,
Sreedhar

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