Digital Root

By Kardi Teknomo, PhD.

<Contents | Previous | Next >

Motivating Example

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 write is 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 is mean.

What is Digital Root?

Think of any positive integer number, for example 179. Add all the digits of that number (1 + 7 + 9 = 17). Repeat the addition of the digits until it is only single digit (thus, 17 -> 1 + 7 = 8). The last single digit is called digital root of the number.

Thus, digital root of 179 is 8.

Digital root or digit sum of a positive integer number is a single digit obtained from the iterative summation of the digits in the number. For our base-10 decimal system, the single digit is one of number 1 to 9.

The number of transformation from the number until digital root is called additive persistence . For number 179 we have additive persistence of 2 because it requires 2 transformations from 179 -> 17 -> 8.

Digital root of a number can be computed using the following formula

If you like iterative programming of digital root below is the function. For some computer or programming language, iterative programming more appropriate because it can handle very large number and also can give the additive persistence value. For example if you use = 25^25 and use MS Excel (or even Visual Basic), you cannot get the digital root using the formula above but you can get the digital root using the function below [See also: Power of Digital Root]

Function DigitalRoot(inputNum, Optional AdditivePersistence As Integer) As Variant 

'#################################################### 

' This function return digital root using iteration 

' copyright (c) 2005 by Kardi Teknomo 

' see more tutorial in http://people.revoledu.com/kardi/ 

'#################################################### 

Dim s As Variant, sum As Long 

Dim g As Integer 

Dim i As Integer 

Dim num As String 

If Not IsNumeric(inputNum) Or inputNum = 0 Then Exit Function 

s = Val(inputNum) 

Do While Len(Str(s)) - 1 > 1 

    sum = 0 

    g = Len(Str(s)) - 1 

    For i = 1 To g 

       sum = sum + Val(Mid(s, i, 1)) 

    Next 

    AdditivePersistence = AdditivePersistence + 1 

    s = sum 

Loop 

DigitalRoot = s 

End Function 

 

<Contents | Previous | Next >

The PDF file of this tutorial is also available for download here.

This tutorial is copyrighted.

Preferable reference for this tutorial is

Teknomo, Kardi. Digital Root. http:\\people.revoledu.com\kardi\ tutorial\