Significant of Digital Root

By Kardi Teknomo, PhD.

<Previous | Next | Contents>

To understand the significant of digital root pattern, in this section, you will learn some applications of digital root. The first example is just for fun and the second example is the practical application of digital root.

Guess a Digit

Consider to play this guess-a-digit game with one of your friend

1. Ask your friend to write a positive integer number and write it down so that you can see this number (for example, the number she wrote was 1329351)
2. Ask her to take out one digit of any arbitrary location from that number so that you will guess later without telling you which digit she chose. Let her write this digit and hide it from you (suppose she choose to take out the fourth digit, that is 9, but she did not tell you about it)
3. Ask her to sum the remaining digit and report to you only the sum. (she counted the sum as 1 + 3 +2 + 3 + 5 + 1 = 15, and tell you that the sum is 15)
4. Then you did simple calculation and you guess the digit she hide was 9

“How you did that?” she exclaimed on the accuracy of your guess. The secret lies on digital root.

The sum of the number 1329351 is 1 + 3 + 2 + 9 + 3 + 5 + 1 = 24 and the digital root of 24 is 2+4 = 6, while the sum she report to you was 15 with digital root of 1+5 = 6. Since the subtraction of two digital roots was 6 - 6 = 9 then you guess. Hey, how could 6 – 6 is 9? Read on this article and you will understand what it means.

Fibonacci Sequence

Fibonacci number is a sequence of numbers obtained by adding two numbers in the sequence recursively. We start with F(1) = 1 and F(2) =1 and we get the next Fibonacci number by adding the last two numbers:

F(1) = 1
F(2) = 1
F(3) = F(2) + F(1) = 1 + 1 = 2
F(4) = F(3) + F(2) = 2 + 1 = 3
F(5) = F(4) + F(3) = 3 + 2 = 5
F(6) = F(4) + F(3) = 5 + 3 = 8
and so on

This addition may be very simple problem but when the numbers are quite large, the addition of two numbers may produce computation error in the computer.

For example,

F(72) = 498454011879264
F(73) = 806515533049393
F(74) = F(73) + F(74) = 1304969544928657

But when you use spreadsheet like Microsoft Excel, for example, you will get

F(74)   1304969544928660

You may not aware of this inaccuracy just by looking at numbers. This kind of inaccuracy is due to round off error and can be detected easily using the pattern of digital root.

The digital root pattern of Fibonacci sequence has 24 digits length cycle: 1-1-2-3-5-8-4-3-7-1-8-9-8-8-7-6-4-1-5-6-2-8-1-9.  After 24 digits, the pattern of digital root repeats itself (see figure below). You can download the MS Excel companion of this article here 

Knowing this cyclical pattern, we shall expect the pattern repeat itself for large Fibonacci number. However, starting from F(74) the pattern of digital root does not repeat itself.

For example,

Digital root of F(72) = 9
Digital root of F(73) = 1
Digital root of F(74) supposed to be 1

But when you use spreadsheet like Microsoft Excel, for example, you will get

F(74)   1304969544928660 (digital root = 4)

After 9-1 pattern in F(72) and F(73), the digital root should be 1 in F(74) but the Excel shows that the digital root of F(74) = 4 which breaks the cyclical pattern of digital root. Thus, digital root is a great help to detect the round off error that Microsoft Excel does.

Similar problem may happen in any programming languages aside from MS Excel. It may also happen in any arithmetic operation such as addition, subtraction, multiplication, division and power. Using the pattern of digital root, we can detect the inaccuracy of computation.

<Previous | Next | Contents>

These tutorial is copyrighted.

Preferable reference for this tutorial is

Teknomo, Kardi (2005). Digital Root. http:\\people.revoledu.com\kardi\tutorial\DigitSum\