The term casting out nines comes from the fact that when finding the digital root you can ignore nines. Not only can you ignore nine itself but you can also ignore other numbers that add up to nine.
This allows us to take a shortcut and save ourselves a bit of time by not counting numbers that in the end will have no impact on the result.
We will find the digital root of :
What if we counted the nine?
We get the same result but with more work.
Here are a list of some of the number combinations that add up to nine.
When looking for digits that add up to nine, if you see digits close together that you instantly know add up to nine then ignore them. Don't scan the number too hard looking for digits that add up to nine because as we have seen it does not matter if you include them as they will not affect the result. The more you practice this the easier you will spot the digits that total nine and the faster you will be able to add them up.
Sometimes you may see some digits that have more than one combination of numbers that add up to nine. In these cases just pick one set to ignore and count the rest.
Finding the digital root of we have:
So you can see it does not matter which set of number that add up to nine are ignored the result is the same.
An important point to remember is:
For the the digit root in nines remainder the number 9 is the same as zero
What this means is that you can treat a 9 as zero.
The other term for casting out nines is the nines remainder method because the digital root of a number is in fact the same number you would get as the remainder if you divided the number by nine.
Lets have a look at an example.
For the digit root is:
Now if we divide by we get:
A quick check multiplying by gives:
We see that is less than .
Sum the Sum As You Count
When doing a digital root, especially on larger numbers, there is one more tip you may find useful, that is whenever your total reaches two digits you reduce it to a single figure before continuing on to the next digit.
Lets have a look at the steps to find the digital root of :
We ignore nine and digits adding to nine.
Adding and we get which we sum to .
We add the last to the to get which we sum to get .
In case you were wondering:
the same answer as above.
Now we will have a look at how we can use the digital root when checking our calculations.
The rule for checking multiplication is:
The digital root of the product is equal to the product of the digital roots of the multiplicand and the multiplier.
Lets look at some examples:
numbers: 2 0 8 x 2 3 = 4 7 8 4
digit root: 1 x 5 = 5
For 208 we have 2 + 0 + 8 = 10, then 1 + 0 = 1
For 23 we have 2 + 3 = 5
For 4784 we have 4 + 7 + 8 + 4 = 23, then 2 + 3 = 5
The product of the digit roots of the multiplicand and multiplier is 1 x 5 = 5
The digit root of the product is 5, as both are 5 our answer should be correct.
The rule for checking addition is:
The digital root of the sum is equal to the sum of the digital roots of the numbers added.
Lets look at some examples:
numbers: 2 5 + 2 3 = 4 8
digit root: 7 + 5 3
The sum of the digit roots is 7 + 5 = 12, then 1 + 2 = 3
For the sum 48 the digit sum is 4 + 8 = 12 and the digit root is 1 + 2 = 3
Division is a little different than multiplication and addition in that we multiply digital roots to do the check rather than divide.
The rule for division is:
The digital root of the dividend is equal to the digital root of the quotient multiplied by the digital root of the divisor
To check division we use the following procedure:
- If there is a remainder, subtract it from the dividend first to get the reduced dividend.
- Find the digital root of the dividend or reduced dividend.
- Find the digital root of the divisor.
- Find the digital root of the quotient.
- Multiply the digital roots of the dividend and the quotient.
- Compare the product of the step 5 with result of step 2.
The reduced dividend is actually the number that will be evenly divided by the divisor leaving no remainder.
Here are some examples:
Lets follow the steps above:
- Subtract the remainder from the dividend:
- Find the digital root of the reduced dividend:
- Find the digital root of the divisor:
- Find the digital root of thequotient:
- Multiply the digital roots of the dividend and the divisor:
We can see that the result of step 2 and step 5 are both 3 so our result should be correct.
- The digital root of 1968 is 6 (1 + 8 = 9 so are ignored along with the 9)
- The digital root of the divisor is 8 (the divisor is only one digit so it is already a digit root)
- The digital root of the quotient is 3 ( 2 + 4 + 6 = 12, 1 + 2 = 3)
- Multiplying the digital roots of the dividend and the divisor we get: 8 x 3 = 24, 2 + 4 = 6
This time we see that the result of step 1 and step 4 are the same so our answer is correct.
Note: there was no remainder so there was one less step in the process.
Doing this check is a lot faster to do than it is to explain, with a little practice you will be able to very quickly check your results. It is better that you find your mistakes and correct them than someone else finding them and pointing them out to you.
That is the nines remainder or "casting out nines" method used to check the results of your calculations, there is another method that is a little more involved but is actually a more reliable check called the elevens remainder or "casting out elevens" method.