Name: Casting Out Nines - Numberphile
Uploaded: 2020-03-27T18:12:40.000Z
Duration: 8 min 4 s
Description: Thousands of YouTube videos with English-Chinese subtitles! Now you can learn to understand native speakers, expand your vocabulary, and improve your pronunciation...

Present continuous

Today I'm gonna show everyone a trick that can check your arithmetic.

Modal verbs - A2

But I think this is something that everyone should know.

I think every school kid should know this.

It's a method to check your arithmetic–

it's what they used in the old days before calculators.

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

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

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.

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,

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,

That's like adding 9 twice. That has no effect on the digital root. Same for this 7 and 2.

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.

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

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,

5030, times it by 5, which is 2515, and then add that up.

But let's check. Let's do the digital roots of each value.

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,

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 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.

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.

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,

So it's kind of the same idea. Adding nines in this case have no effect, and then anything left is what you have

can be written as a multiple of nine plus their digital root. So you pick a number and

plus ‘a’ here is the digital root. Now, what happens when I add it to another number? If I add that

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,