Digital Roots

2013 May 23 06:56 PM MDT | Math⁷ Programming¹⁸ PHP⁴ Python⁴ [2013]³

What happens to the digital root as the numbers increase?

 n | dr(n)
---+------
 0 | 0
 1 | 1
 2 | 2
 3 | 3
 9 | 9
10 | 1
11 | 2
12 | 3
18 | 9
19 | 1

It looks like it is really just a cycle that goes from 1-9 over and over again (except 0). This can be represented using modulus.

What happens if n < 0? Well, I've decided to just use the absolute value of it. That means that dr(-1) = 1.

Here is some PHP code:

<?php
for($i = -11; $i <= 20; ++$i)
	echo "dr($i) = ".(1 + (abs($i) - 1) % 9);

See that? There is no need for looping and recursion (nor iteration) to calculate the digital root. Why take O(≫ 1) time when you can take O(1) time?

Many languages handle negative modulus with truncation, but a special case must be created for some languages that use floored division:

# Python
def dr(i):
  if i:
    return 1 + (abs(i) - 1) % 9
  return 0

for i in range(-11, 20 + 1):
  print "dr(%d) = %d" % (i, dr(i))