So, without further ado, I present to you digital root extraction.
Step 1: What Is a Digital Root?
A better demonstration would be to take a number (such as 179) and demonstrate. So, the digital root of 179 is something like this:
179 -> 1+7+9 = 17 -> 1+7 = 8
So, the digital root of 179 is 8.
Another, more interesting way to look at a digital root is that it is a modulo 9 operator. In other words, if you were to divide the number by 9, the integer remainder and the digital root are the same thing. So, digital roots can be rewritten as mod9(Number).
Now that we know what digital roots are, let's go ahead and find them.
Step 2: Finding Digital Roots
One thing the more clever people in the audience may have noticed, though, is that, since the digital root is equivalent to taking the number modulo 9, and modulo 9 is 0, we can ignore any number or numbers which add up to 9 or a multiple of 9.
So, for a number like 4529, the astute observer will see that 4+5 =9, so 4 and 5 can be ignored, and by the same reasoning, 9 can be ignored. Much quicker than dividing by 9 or recursively adding digits.
Quickly, it should be noted that the modulus definition is slightly off. That is, for a number greater than or equal to 9, the mod9 operation returns a digital root of 0, whereas it should be 9. So, for numbers such as 36, the digital root is clearly 9, even though the modulus would suggest it is 0. This is not particularly difficult, just keep it in mind when taking digital roots.
So, for practice, find the digital roots of the following numbers:
1) 47 (Digital Root: 2)
2) 365 (Digital Root: 5)
3) 1,852 (Digital Root: 7)
4) 136,257 (Digital Root: 6)
5) 234,181,360 (Digital Root: 1)
In the next step, we will talk about what applications digital roots have.
Step 3: Applications of Digital Roots
In the now dark and forgotten age before the creation of the calculator, digital roots were used by some (such as bank tellers) to check the accuracy of their computations. When 2 numbers are added, multiplied, or subtracted, the digital root of the result should be the digital root of the sum, product, or difference of the operands.
So, for example, consider the subtraction problem 342-173, which is 169. The digital root of 342, using the techniques from step 2, is 9, while the digital root of 173 is 2. Therefore the digital root of the difference should be the difference of the digital roots (or 7). Using techniques from step 2, we see that it is.
So, in summary, you can use digital roots as a primitive way to check the accuracy of subtraction, multiplication, and addition operations, which can be useful for any mental mathematicians out there.
In the next step, we will prove why digital roots work on multiplication, addition, and subtraction in this way.
Step 4: Prove It!
From earlier, we mentioned that the digital root is equivalent to taking a number modulo 9. From properties of the modulus operator, it can be shown that:
mod9(a)+mod9(b) = mod9(a+b)
mod9(a)-mod9(b) = mod9(a-b)
mod9(a)*mod9(b) = mod9(a*b)
Thus, we have an understanding of why digital roots can be applied to addition, subtraction, and multiplication. (They can also be applied to division, but that case requires some special treatment.)
Step 5: Summary