Digital Roots Help Check Addition and Multiplication (classes 9-29 thru 10-5)

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.

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.

USING DIGITAL ROOTS TO VERIFY ADDITION (this is for 1st through 5th graders)

Suppose we are adding two numbers: 345 + 789.
· The Digital Root of 789 is 6.
· The Digital Root of 345 is 3
· Now add 3 + 6 = 9
· This implies that the Digital Root of our final answer obtained after adding 345 + 789 will be 9.
· Let’s verify that 345 + 789 = 1134
· The Digital Root of 1134=1+1+3+4=9 This equals the sum obtained at by adding the digital roots of the original two numbers above.

USING DIGITAL ROOTS TO VERIFY MULTIPLICATION (this is for 3rd through 5th graders)

Suppose we are multiplying two numbers: 12 x 8

· The Digital Root of 12 is 3.
· The Digital Root of 8 is 8.
· Now multiply 3x8=24; whose digital root is 6.
· This implies that the Digital Root of our final answer obtained after multiplying 12 by 8 will be 6.
· Let’s verify that 12x8=96
· The Digital Root of 96 is 9+6=15 and 1+5=6 This equals the product obtained at by multiplying the digital roots of the original two numbers above.

Fill in the multiplication table in the pdf file. However, instead of filling each box in with the products of the two numbers on the horizontal and vertical axis, use the DIGITAL ROOTS.

*now, look for magnificent patterns in the digital roots.

USE THE WORKSHEETS IN THE PDF FILE TO PRACTICE USING DIGITAL ROOTS TO VERIFY ADDITION AND MULTIPLICATION