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Visit Tutorials below:
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Power of Digital Root
The following interaction program determine the power digital root of your input x^y. Your input must be positive integer number of any digit up to 32,759 digits for base x and 28 digits for exponent y. Let define dr[x] as digital root of x and dp[x, y] as power of digital root of Power of digital root dr[z] =dp[x, y] = dp[ dr[x], y] For instance, we know that 225 = 15^2. Then, we can say that the digital root of 225 is equal to power digital root of 15 with exponent 2. In turn, this equal to the power of 6^2=36 and the digital root of 36 is 9. In formula, it becomes dr[225] = dr[15^2] = dp[15, 2] = dp[ dr[15], 2] = dp[6, 2] = dr[36] = 9
The power of digital root has regular interesting pattern. The base has cycle length 9 and the exponent has cycle length 6. The power of digit root 1 is always 1, thus it has constant pattern whatever the power value is. Digit root 2 has repeating long pattern of 2-4-8-7-5-1. Digit root 3, 6 and 9 has constant pattern of 9 for any value of the exponent except 1. Digit root 4 has repeating pattern of 4-7-1 while digit root 7 has repeating pattern of 7-4-1. Digit root 8 has repeating short pattern of 8-1. [Compare this pattern to Division of Digital Root]
Looking the pattern of digital root below, the pattern of the base is repeating every 9 cycle length and the pattern of the exponent is repeating every 6 cycle length. This pattern can be formulated as the following property: dr[z] = dp[x, y] = dp[ x+9, y+6] for x=1,2,3,..., y=2,3,4,... For example, we can add the base with 9 and the exponent with 6 and still get the same digital root dr[225] = dr[15^2] = dp[15, 2] = dp[15+9, 2+6] = dp[24, 8] = dr[110075314176]=9. In fact, we can write add only the base value with 9 dp[15, 2] = dp[24, 2] = dp[33, 2] = dp[42, 2] = 9. Or, we can add only the exponent with 6 dp[15, 2] = dp[15, 8] = dp[15, 14] = dp[15, 20] = 9.
Practically, the pattern of digital root power cannot be produced using simple formula The following code and explanation come from the contribution of Robert Sawyer ("r.e.s.") a math hobbyist from the USA. The power of digital root of x^y can be computed without computing any powers, as follows: Let dr(x) be the digital root of x, i.e. dr(x) = 1 + (x-1) mod 9. Then Therefore, a function f(x,y) can be defined recursively to compute dr(x^y) without evaluating any powers: f(x,y) In Excel Macro, the codes are shown below. Function RecursiveDigitalPower(x As Integer, y As Integer) As Integer Another way is to compute the power of digital root iteratively. A power of a number Function IterativeDigitalPower (x_, y_) Still another better way to use the nice property of power digital root dr[z] = dp[x, y] = dp[x+9, y+6] as explained above. The code below is the best we got so far that it can accommodate the base up to 32,766 digits and exponent up to 28 digits. Function DigitalPower(x, y) One easy way to tested the function above is to place the following in a cell =DigitalPower(CONCATENATE(REPT(9,32765),"8"), CONCATENATE(REPT(9,27),"9")) which should then display 8 (i.e., the digital root of 8^999...9 with 28 9s). Changing the last digit of x from "8" to "9" should of course change the display to 9. The 999...98 with 32765 9s is chosen because it has a digital root of 8, and 8^n has a digital root that simply alternates between 1 and 8 when n is changed to n+1 or n-1, making it easy to verify that all the digits of y are being processed correctly; i.e., changing the last digit of y from "9" to "8" in the above line should change the display from 8 to 1. You can download the MS Excel companion of this article here Historical Note:I hope this historical note will prevent you from doing the same mistake. In my earlier attempt, I used simple iterative Macro to compute the power of digital root and it was my mistake to deduce that the power of digital roots has complexity pattern as shown below:
On the left side, it is almost regular but getting to the right (higher number of exponent) the pattern is irregular. The boundary between irregular and regular pattern is a curve with two asymptotes in the bases and in the power. The part of irregular pattern is really complex that one cannot predict what will be the next pattern. It is quite amazing that such random pattern can be produced using simple deterministic formula of digital root The apparent complex pattern above may be generated due to inaccuracy of
round off error as suggested by the comment below.
If you are interested in complex pattern of digital root, check the next section on digital root of prime number. Hi Kardi, I have a general interest in complexity and was intrigued by the
These tutorial is copyrighted. Preferable reference for this tutorial is Teknomo, Kardi (2005). Digital Root. http:\\people.revoledu.com\kardi\tutorial\DigitSum\ |
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. In this case, we call x as base and y as exponent. Then, based on the pattern of power digital root, we can say these properties:
because of Mod function in most programming language usually cannot take very large number. One alternative is to use recursive or iterative method (macro) to produce such pattern above. [
for 
and the product of digital root has a nice property of 
. Therefore, we can write the power of digital root dr[z] = dr[x^y] as an iterative function.
