Digital Roots of Sequences: Circular and Graphic Designs

The digital root of a number is the number obtained by adding all the digits, then adding the digits of that number, and then continuing until a single-digit number is reached.

Digital roots were known to the Roman bishop Hippolytos as early as the third century. It was employed by Twelfth-century Hindu mathematicians as a method of checking answers to multiplication, division, addition and subtraction.

For example, the digital root of 65,536 is 7, because 6 + 5 + 5 + 3 + 6 = 25 and 2 + 5 = 7.

The children will remember that two weeks ago we created fractals from Pascal’s triangles by coloring in the multiples of 3 and used the digital root test of adding the digits and if the sum was a multiple of 3, then the original number was also a multiple of 3.

This addition of the digits in a number is further simplified by first discarding or casting away any digits whose sum is 9. The remainder is set down, in each case as the digital root.

For example, the digital root of 972,632 is 2 because we cast out the nines or numbers who's sum is nine like the 7 + 2 and the 6 + 3. The remainder of 2 is the digital root.  

When casting out nines, if there is a remainder of zero, the digital root is 9.  The digital root of 927 is 9 because we cast out the 9 and the 7 + 2 leaving a remainder of zero.

FIND THE DIGITAL ROOT OF:

Number Calculation            Digital Root

3 3                   3

24 2 + 4 = 6             6

93 9 + 3 = 12 → 1 + 2 = 3 3

126 1 + 2 + 6 = 9        9

146 1 + 4 + 6 = 11 → 1 + 1 = 2 2

389,257 3 + 8 + 9 + 2 + 5 + 7 = 34 → 3 + 4= 7 7

96,871,565,493,528,698 9 + 6 + 8 + 7 + 1 + 5 + 6 + 5 + 4 + 9 + 3 + 5 + 2 + 8 + 6 + 9 + 8 = 101 →  1 + 0 + 1 = 2 2

Shortcut called "Casting Away Nines."

If we take the number 4,569,512,597,853, losing both 9s gives you 45,651,257,853. Then you can do the same with numbers that add to 9. So in the number we have now, we can also lose 4&5, 6&3, 1&8, 2&7 which leaves 555. Now we can find the digital root of 555 quite easily: 5+5+5=15, then 1+5=6. That's nice - no big additions to do to get the digital root of 4 569 512 597 853 to be 6!

This week I had the children list the digital roots of the multiples of the numbers from 1-12. They found amazing patterns of numbers from:

Multiples of 1: 1,2,3,4,5,6,7,8,9, repeat   1,2,3,4,5,6,7,8,9

Multiples of 2: 2,4,6,8,1,3,5,7,9, repeat   2,4,6,8,1,3,5,7,9

Multiples of 3: 3,6,9, repeat  3,6,9,

Multiples of 4: 4,8,3,7,2,6,1,5,9 repeat 4,8,3,7,2,6,1,5,9

Multiples of 5: 5,1,6,2,7,3,8,4,9 repeat 5,1,6,2,7,3,8,4,9

Multiples of 6: 6,3,9, repeat 6,3,9

Multiples of 7: 7,5,3,1,8,6,4,2,9 repeat 7,5,3,1,8,6,4,2,9

Multiples of 8: 8,7,6,5,4,3,2,1,9 repeat 8,7,6,5,4,3,2,1,9

Multiples of 9: 9, repeat 9

Multiples of 10: 1,2,3,4,5,6,7,8,9, repeat 1,2,3,4,5,6,7,8,9

Multiples of 11: 2,4,6,8,1,3,5,7,9, repeat 2,4,6,8,1,3,5,7,9

Multiples of 12: 3,6,9, repeat  3,6,9,

Square Numbers: 2,4,8,7,5,1 repeat 2,4,8,7,5,1

Powers of 3: 3,9, repeat 9

Powers of 4: 4,7,1, repeat 4,7,1

Powers of 5: 1,4,9,7,7,9,4,1,9, repeat 1,4,9,7,7,9,4,1,9,

Then I had the children graph the sequences on a circle graph with 9 points to create amazing designs. They noticed that the Multiples of 1, 8, and 10 were identical, so were Multiples of 2,7, and 11, so were Multiples of 3,6, and 9 and even Powers of 4, so were Multiples of 4 and 5, and Multiples of 9 and Powers of 3 were very close. Square Numbers had a pattern of a repeat of 6 digits and Powers of 5 was also a repeat of 9 digits but a little strange.

Talk about strange behavior, we then graphed the digital roots on a square grid but going in a direction of right and then taking a 90 degree turn to the right in a continuous clockwise fashion. These graphs made the most mundane sequence come alive.