The Saturday (Simple) Question

admin — Sun, 07 Feb 2010 19:03:34 +0000

Question : What is meant by ‘Casting out nines’ ?

Answer : Casting out the nine is a method of checking answers to multiplication, addition and subtraction.
Casting out nines was known to the Roman bishop Hippolytos as early as the third century. It was employed by Twelfth-century Hindu mathematicians.
It is based on the idea of finding the sum of the digits in a number and then adding the digits in the resulting sum, etc., until a one digit number results. 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 check number.
For example,
8216 x 4215 = 34630440, this is how you proceed.

4 + 2 + 1 + 5 = 12, casting out the nines leaves 3 as check number.

8 + 2 + 1 + 6 = 17, casting out the nines leaves 8 as check number.

3 + 4 + 6 + 3 + 0 + 4 + 4 + 0 = 24, casting the nines leaves 6 as check number.

3 x 8 = 24, 2 + 4 = 6 which was the check number of the original product. So the multiplication is correct.
Casting out nines is not in wide use today, mainly because of the widespread use of calculators and computers.

Casting out Nines

admin — Fri, 05 Feb 2010 13:53:28 +0000

There are many little tricks in mathematics can make solving problems much easier. Casting out the 9s is one of the tricks can help make sure your answer is right much easier. Many people check the digit sum of the answers to make sure that their answers are correct. Casting out the 9s can make checking the answers and getting digit sums much easier.
Let’s start with Digit Sums. It is possible to take a number regardless of how big it is and transform it into a single digit. You can start with the number 15. In order to get it down to just one digit, you would add the two numbers together which would equal 6. This means that the digit sum of 15 is 6. It is also possible to do this with larger numbers such as 547. You would begin by adding 5 and 4 together which would be 9 then you would add 9 and 7 together which would be 16. However, you still got two digits so you would need to add 1 and 6 together which would be 7. With a bigger number such as 72546 you would add 7 and 2 which is 9 then 9 and 5 which is 14. 14 and 4 would equal 18 and finally 18 and 6 would equal 24. Finally, 2 plus 4 equals 6 so the digit sum of 72546 equals 6. This can be rather frustrating with larger numbers.

Thankfully, there is another way to shorten it down and make it easier.
Okay, now you’re probably wondering where casting out the 9s comes into play in this situation. When it comes to digit sums, it is possible to regard 9 as a 0. This basically means that you can cast out the 9s when working with digit sums. Then you would add the remaining digits together to get the digit sum. Let’s work with the numbers mentioned above. Let’s look at 547. 5 plus 4 is equal to 9. We can quickly eliminate these two numbers since they are equal to 9 when added together. Therefore, we are left with 7. This is the same answer we got above.

We just got with much easier and quicker when casting out the 9s. Okay, now let’s look at 72546 which was rather complicated to break down and get the digit sum earlier. 7 plus 2 is equal to 9 which can be casted out. 5 and 4 is also equaled to 9 and can also be casted out. This leaves us with 6 which is the same sum digit that we received using the more complicated method above.

Casting Out Nines » Casting out Nines

The Saturday (Simple) Question

Casting out Nines