Digital Roots

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The digital root of a number is the number obtained by adding all the digits of that number, then adding the digits of that number, and then continuing until a single-digit number is reached. For example, the digital root of 65,536 is 7, because 6 + 5 + 5 + 3 + 6 = 25 and 2 + 5 = 7.

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. The early 19^th Century mathematician Carl Friedrich Gauss made the use of digital roots popular as a method of numerical congruence (although I spoke of the child mathematician Gauss, I did not introduce the term “numerical congruence”).

This property can also be used to show that a number is divisible by 9.

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.

Knowledge of digital roots often enables shortcuts in solving otherwise difficult problems.

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

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

The attached pdf worksheet has 15 separate sequences of numbers and the children were directed to find the digital root of each number and then look for patterns in the digital roots. The second pdf is the answer key and it is amazing to note that there is a pattern in all but one of the sequences.

The younger children should only try a few pages of this exercise at their comfort level. Until they master number sense with me, they should be free to use their fingers, toes or any other method of counting. The 2^nd – 5^th graders should try to push themselves to finish as much of the packet as possible. If they see the pattern they should use it to check their work. The last page of the packet is only for the more advanced student who can look at the multiplication table and record the digital root of each product (the answer to a multiplication problem).

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