Today I'm gonna show everyone a trick that can check your arithmetic.
But I think this is something that everyone should know.
I think every school kid should know this.
I think every adult should know this,
and I don't think everyone does know it.
It is called casting out nines.
It's a method to check your arithmetic–
it's what they used in the old days before calculators.
And let's say we're doing something like
2039 plus
172 plus, whatever,
1218 plus 3091.
Let's say we're doing something like that. Let's add these up. I'm just gonna add it up in the traditional way,
I'm not doing anything clever here. So you add up column by column, okay, I reckon it's
6520.
But what if I've done it wrong? What if I've made a mistake?
So here's a check that you can do. So what we're going to do,
we're going to reduce each of these numbers to a single digit, and the way we do that is we're going to add up
the digits in each number. So, 2 plus 3 is 5, plus 9 is 14.
I want to reduce it to a single digit, so I'm just going to do it again, now.
I'm going to take the 14, 1 plus 4. Hope you're happy with that
I'm going to do that with each number. 1 plus 7 plus 2. I reckon that's 10.
Let's reduce that down, so 1 plus 0 is a 1. This is what I get.
So I get this 5 1 3 & 4. Now I'm going to perform the same calculation
using these numbers. So I'm just adding them here. So let's add them together.
So I add them together, okay, 5 plus 1, is a 6, plus a 3 is a 9 plus 4, is a 13.
Actually, I'm gonna reduce that down, if you don't mind, to a single digit, 4.
But now, if you do the same thing with your answer, ok,
6520, add this up is a 13, which gives me a final digit there of 4. And these are
the same and they should be. Now, if they're not the same,
that's how you know your calculation has gone wrong. So this is a way to check your
calculations without having to do the calculation
all over again.
Now
that was a bit laborious the way I did it there.
I can actually show you just a quick way to do it. Just pick a number,
let's say 43. 4 plus 3 equals 7, so 7 would be the final answer.
It's called a digital root. But now, I'm gonna add 9 to 43, gives me 52,
Do the same thing with 52. It's gonna be 5 plus 2, that's also 7.
And that's true. When you add 9
it doesn't change the digital root, so if you look at this original calculation
I did up here, I actually can get rid of nines, so this one at the start,
2039,
Well, this 9 has no effect. And I just get rid of it.
I don't have to consider it. But in the same way, this 9 here which represents 90, has no effect,
That's just like adding 9 over and over again. I can get rid of it. Or in fact,
I can get rid of this 18.
That's like adding 9 twice. That has no effect on the digital root. Same for this 7 and 2.
Any combination that makes 9
can be got rid of, and then you see, it's much easier now. So what I've got now is a 5, a 1, a 3
and a 4. And it's very easy to do, and then you can perform the calculation again. Same thing with this lot here.
Look, the 5 and the 4
cancel out. 1 and 3 left over makes a 4. Very easy to do this arithmetic. It also works for multiplication.
Let's try a multiplication just to see. Let's do
503 times, I'll do something easy,
I'm not trying to show off with how good I am at arithmetic. So, times 15.
What I would do if I saw that is times it by 10 first of all,
so
5030, times it by 5, which is 2515, and then add that up.
So what have I got there?
7545. I reckon that's the answer.
But let's check. Let's do the digital roots of each value.
So that's an 8.
That's a 6, 1 plus 5, that makes a 6. And we're gonna multiply now.
We do the same calculation using the digital roots, 8 times 6, 48.
And 48 will eventually end up being a 3. And now let's check that this is also 3. If it's not,
I've done it wrong.
Let's see. Okay. 5 plus 4, okay, I can get rid of that with no care.
7 plus 5 is a 12, add them together. Hooray!
I'm happy, that ends with a 3, as well. This is what people used to do before calculators.
If you were a scientist or
if you were an accountant, and you needed to check your figures, very important to check your figures,
this is the sort of test that you could do.
So, if you're wrong,
you'll be wrong by some multiple of 9.
Because if you add 9, it doesn't change the digital root, so maybe you're 9 off, or 90 off.
That's possible, true. But this is a check to see if you're right.
Now, you might be thinking why does this work, and if you're thinking that, well done. You're thinking like a mathematician.
Yeah, why does this work? That's a good question to ask.
So,
we saw before adding 9 didn't change the digital root. Adding 9 had no effect.
It's kind of like, if it was Monday, and I add 7 days, it's Monday again.
isn't it? Or, if it was Monday, and I add 17 days,
well, you'd go through 14. That will get you back to Monday, then you would have 3 more days left,
that'll take you to Thursday.
So it's kind of the same idea. Adding nines in this case have no effect, and then anything left is what you have
after a multiple of nine.
What that means is any number
can be written as a multiple of nine plus their digital root. So you pick a number and
you can write it as
some multiple of nine
plus ‘a’ here is the digital root. Now, what happens when I add it to another number? If I add that
to another number, it can be written as
a multiple of 9
plus the digital root. Let's say it's b here. What happens when I do that? We're gonna end up with
a multiple of 9. It's actually 9 times n plus m
plus, these two will get added together, that'll be a plus b.
So, adding these two numbers together has the effect of being a multiple of 9 plus the sum of their digital roots.
And that's why this works. You do a calculations with the original numbers will transfer to the same
calculation with their digital roots. With multiplication, you'll see, we had those two numbers,
let's say 9 n plus a
multiplied by 9 m
plus b.
Then you're going to get
a multiple of 9, which is gonna be 9 times something not very pretty.
So, multiplying two numbers has the effect of being a multiple of 9
plus the product of their digital roots. And of course multiple of nines can be ignored. They just disappear.
... and these, this would be a 9, and
this would then be divided by
1 plus 5 is a 6, that doesn't equal 3, does it? That makes no sense.
We have much further to go in our race for space, so we're gonna leave Sputnik behind. For those who don't know, by the way,
Sputnik was the first satellite ever to orbit the Earth. It was launched by the Russians. We probably should have mentioned that.
October the 4th 1957.
This man knows.