Hi everyone! I'm going to show you a little trick today
that will make you look like a mathematical genius.
So you're going to be able to add up a whole bunch of numbers together instantaneously,
and impress all your friends.
So the idea is this: you're going to get a volunteer
and you're going to ask them to pick two numbers.
Now, this works with any two numbers, but for everyone's sake,
you might want to keep these numbers fairly small.
You might, say, keep them under ten, or under twenty--something like that.
So your volunteer picks two numbers.
Now, you can turn away, and you ask your volunteer
to write down those two numbers, one above the other, and they can do that in any order.
Let's say they pick 8 and 5.
So they can write down 8 and 5 in a column like that.
Now you ask them to add those together, and to write the result underneath.
So they're going to write down 8 plus 5 is 13--underneath, like that.
To make the fourth number, well, you add the third to the one above it.
So in this case, 5 plus 13 is 18, and they're going to carry on like that,
and you want them to do ten. Ask them to write a column of ten numbers, just like this.
So if I write 13 plus 18, we get 31.
And now we're going to get 49.
80
The next number is 129.
209
and 338.
and not only will you look amazing when you finish this,
they get to practice some simple arithmetic too.
Now you get to turn back, look at this,
and instantaneously, you're going to be able to write down the sum
You draw a line, and you can tell them that the sum of those numbers is 880.
And you will look so cool.
So the trick here is to look at the fourth number from the bottom.
The fourth number from the bottom, in this case, was 80.
And you take that number, and you multiply by 11,
and that's your final answer.
So it's the fourth number from the bottom,
80 in this case, and multiplied by 11.
Now, why does this work? Well, you tell me.
But now you might be thinking, "Gosh, now I have to start multiplying by 11 instantaneously. How am I going to do that?"
Let me show you an easy way to multiply by 11.
So, it's actually quite easy to multiply by 11.
Let me show you how. I've got three examples here.
In each case, we're looking at the fourth number from the bottom.
We're going to be multiplying this by 11.
In this first example, we've got 71.
To multiply that by 11, all we do is we add the two digits together
and put it in the middle.
So for 71 here, it will start with a 7, it will end with a 1,
and then we add the two digits together--so 7 plus 1 is an 8
and we put that in the middle. So 781 is the correct answer.
That's 71 multiplied by 11.
Now, sometimes you're going to have to carry over,
like in this second example. 57 is the number I want to multiply by 11.
But if I add those two digits together, the 5 and the 7, I get 12,
and I'm not going to put 12 in the middle.
No, instead, I do end with a 7, like before,
I add the two digits together, I get a 12. I'm going to put 2 here,
and that 1 will carry over--it will add to 5, and become a 6.
So 627 is the correct answer in that case.
For a three digit number--same idea, really.
In this example, we've got 181. This will begin and end with the 1 and the 1.
In the middle, we add the digits together.
We add the last two digits--the 8 and the 1, to give me a 9.
And we add the first two digits, which is 8 and 1 again in this example,
so I've got another 9--carry over, if I have to.
And that's all there is to it!
And just to finish off, I want to show you another way to work out this sum.
What if I wanted to add up n of these numbers?
Now notice that this is a Fibonacci-style sequence,
which means that a number is the sum of the previous two.
With the Fibonacci sequence, that starts with a 1 and a 1, and then you carry on.
But it doesn't have to start that way. You could pick any two numbers and do the same thing,
which is what I've done here.
Let's say you wanted to add up the first n of that sequence.
So let's say you start with the first number--we'll call F1, and you want then to add together with F2,
and you want to do this all the way up to the last one--let's call it Fn.
So what does that equal?
Well, there's a formula for this,
and the formula is--this is equal to--
it is the n plus 2 Fibonacci-style number, minus the second number [F2].
So there we have a formula.
In other words, if you wanted to work out the sum of the first n of these numbers,
you just do a couple more steps--do a couple more steps, and subtract the second step,
and that is your answer.
Now, you can prove this. Induction fans, you can prove this formula
using induction--not too hard to do.
You check your base case, and then if it's true, or you assume it's true for an arbitrary step,
make sure it's true for the n plus 1 step.
And you can prove that formula too.
But in this case, let's see if we can do it.
So we had the first ten numbers here.
Let's just continue the sequence for a couple of more steps.
So the eleventh step is 547,
and the twelfth step--so if I add, now, ten and eleven together, I will get 885.
And look, it is true. The twelfth step minus the second step is the sum of the first ten--880, like I said.
So there you have it! A little trick, and a bit of maths
and now start practicing multiplying by 11, and show off to your friends.
And if you have been, thanks for watching.