What is the concept of casting out nines?

Answer 1

Its an old arithmetic trick used when adding two large numbers. You sum the first numbers digits, then the second numbers, ... etc, then take the remainder when you divide that sum by 9. If the sum of the total's digits does not yield the same remainder, you made a mistake. For example:

213+756+1148=2417
2+1+3=6
7+5+6=18/9 = 2 r 0
1+1+4+8=14/9=1 r 5
Adding remainders, we have 11/9=1 r 2. But the sum of our digits in 2+4+1+7=14/9=1 r 5. So something is wrong. Turns out the 4 should be a 1, 2117. So its just an error correction method.

Steve

Answer 2

Casting out nines is the name of technique for checking arithmetic.
It depends for its use on the idea of the digital sum of a number.
The digital sum of any positive integer (or whole number) is gotten by
adding up all the digits of the number. If the result has more than
one digit, repeat this, until the result is a one-digit number. That
digit is the digital sum of the starting positive integer.

Example:

9974 -> 9+9+7+4 = 29 -> 2+9 = 11 -> 1+1 = 2,

so 2 is the digital sum of 9974.

Let's write s(9974) = 2.

If you are familiar with modular arithmetic, the digital sum of a
number is the smallest nonnegative representative of its congruence
class modulo 9.

Now the important facts about digital sums and arithmetic are that:

s(a+b) = s(s(a)+s(b)),
s(a*b) = s(s(a)*s(b)).

We use this to check addition and multiplication as follows:

9974 + 2348 ?=? 12422.
s(s(9974)+s(2348)) = s(2+8) = s(10) = 1,
s(12422) = 2.

This means that the sum given is incorrect.

9974*2348 ?=? 23418952.
s(s(9974)*s(2348)) = s(2*8) = s(16) = 7,
s(23418952) = 7.

This means that the product given is likely to be correct.

This kind of checking will find many errors, but not all! An
interchange of two digits (23418952 vs. 23419852) will not be
detected, and replacing a 9 by a 0 or vice versa will not be detected.

To check subtraction, use the fact that a - b = c means a = b + c.
To check division, use the fact that a/b = c means a = b*c.
To deal with zero, you can define s(0) = 0.
To deal with negative numbers, you can define s(-a) = 9 - s(a).

Answer 3

it is a way to verify addition

you take the numbers you are adding up
and add them to each other - like
36 is = to 9 - so thats a zero
19 is a 10 - so that is a 1 (minus the nine)

36 + 19 = 55

3+6=9 so that's a zero
1+9=10 so thats a one

our adding numbers = 1
our answer 55 which is 5+5=10 cast out the nines = 1

1=1 our addition is correct

Answer 4

Casting out nines is the name of technique for checking arithmetic.
It depends for its use on the idea of the digital sum of a number.
The digital sum of any positive integer (or whole number) is gotten by
adding up all the digits of the number. If the result has more than
one digit, repeat this, until the result is a one-digit number. That
digit is the digital sum of the starting positive integer.

Answer 5

back in the olden olden days people would cast lots. like throwing dice and gambling. the roman soldiers did it gambling for Jesus' robe and stuff when he was hung on the cross.

Answer 6

Here's a good explanation.