I would like to show you a number trick.
So regular scientific calculator
I'm not trying to conceal anything, but I would like you Brady to tell me what to write
I'm gonna put some fives into the calculator. Maybe you call this trick the surprising fives. I don't know
Maybe that's what I call it and someone else could think of a better name
But I'm gonna put the digit 5 into this calculator as many times as you want me to.
So tell me how many times you'd like to put the number five
Six
So five five five five five five
Whatever you have told me were to run with that and next two steps are the same whatever we do.
The first step is to take the reciprocal of this number. You can see why we're doing this on a calculator
I know that you could do it given time but the reciprocal by X to the minus 1 or 1 over the answer
Gives us this number handily known as 1 over 555 555
So far so good and now we take the sine of that answer.
So it's gonna be sine of the answer and
We get this result
Which is where I want to check what you notice...
It's pi!
Is it? It's pi-like
It's PI times 10 to the minus 8
It's PI like enough to cause a reaction in anybody who's seen PI before I think.
okay
So you gave me six fives, I want you to give me a different number of fives now
one
One five
Mathematician in the making here, Brady
you see something surprising, you test the extreme cases, really really sensible mathematical things
that reciprocal 1 over 5 is otherwise known as 1 over 5 or 0.2 and I take the sine of the answer
And it give me the answer 3.4906 times 10 to the minus 3
Not so PI-like
I kind of talked you into giving me more than one five
I deliberately said give me as many as you want
So I invite you to try this experiment again
How many fives you'd like this time?
13
13 fives, let's see if our calculator will cope
So that as a reciprocal is 1 point 8 times 10 to the negative 13
My calculator has given up trying to display it as a fraction.
This is a tiny number.
Take the sine of that
Yeah, pi-like
I mean if you know your pi expansion, this is pretty pi-like
Except for the whole thing at the end which you did spot pretty quickly
That times 10 to the minus 15 this time.
That's the trick
It's a classic example of what I like to call "unexpected pi"
Pi turns up literally all over the place in mathematics particularly when you don't expect it
Obviously when you're doing stuff to do with circles it turns up.
We haven't mentioned a circle so far, pi has turned up somewhere
There are several clues that I've kind of concealed from you but
In the rest of the video
I'd like to explain why this happens, but it's worth leaving as a surprise
If you want to go and
If anyone watching the video wants to figure out what's going on
Now's the time to do it before I give it all away
The only clue I'm going to give is your calculator should be in degrees mode
Otherwise you won't get as nice a result.
That's the clue, have a think.
The reason I like this trick
It's not so much a trick is that it's surprised and I think you need three bits of maths to understand it
There's a nice example of simple bits of mass coming together to give you a surprise
Chapter one reading. Oh
That's like an audio book
Brady do you know what a Radian is? It's like a unit for measuring angle, isn't it?
Yeah, we and we're used to using degrees
So if I drew a circle and chopped up into support a circle that right angle would be 90 degrees
Which means there's 360 degrees all the way around
360 is this arbitrary number that we sort of inherited we didn't have to use it
In fact, we still don't have to use degrees almost
Everybody learns about degrees of school and it's a useful way of talking about how much you turn 360 what they think is to do
With our year, we're 365 days is close to 360. So a sort of full cycle of a year is round about the 360
So there's a connection in history there. The Babylonians used counting's in base 60s, which is to do with
360 or
3600 and all of these things are sort of historical accidents and we still see this in our time measurements in hours and minutes
360 is convenient. You can chop it up a little bit, but there is a better way to measure angles
Let me show you the definition of Radian now
This circle has a certain radius
And if you start deciding I want to angles around the circle in a some non-arbitrary way
You could use 400 say and actually they do there they could call that gradients 400
gradients all the way around
If you go to your calculator, there's a mode button that goes between degrees and radians and gradients
There's a G in there as well. So gradient means 400 bits around a circle
It's kind of like metric degrees Li because then a right angle is a hundred
And we will kind of recognize right angles are important and why 90 then in some ways gradients are very sensible
I think the French tried it for a while
They liked metric izing lots of things, but it didn't really catch on and not a lot of people use it a lot
I'm sure people will comment if they found good examples of it being used but even that's arbitrary
I mean my eye pick 401
I'll go for a hundred when I go for a thousand we could do anything and actually our choice is free
We just have to accept the consequences. So what you might want to do is pick a measure that has fewer horrible consequences and
More nice consequences and everything to do with the circle is tied up with its radius
You can already see a clue in the name radians
That's why I'm going to talk about so this circle has a radius of our one complete turn seems important
But another amount of a turn would be if you start walking around this circle some distance. Why don't we pick a distance?
There's already part of the circle which is a radius
So if you know the radius what would happen if you walked around the circle a radius is 1 Radian
So the name is good. If you shown it to rad is it's one radius of a turn
So it's making length and angle measured the same way on a unit circle
It means if there's radius is 1 then you walk a distance R worn around the circle. You've gone one Radian
It does have a consequence which some people find
Nasty, but just deal with it. All right, is that all the way around the circle?
It's not a nice number anymore or rather is a nice number
but it's not a rational number because
you know the formula for distance around a circle is 2 pi off if the radius is 1 the distance around is 2 pi which
Must mean then to do a full turn is 2 pi radians, okay
So that's chapter 1 we're almost there
The little summary is that if you accept that all the way around a circle is 2 pi radians
It was also known in the olden days or in our previous less state as 360 degrees
I can now write this relationship down 2 pi radians is the same as
360 degrees which means that we could say that pi radians is 180 degrees again sort of this arbitrary
- and this gives us the relationship the kind of the exchange rate if you like
So one Radian is the same as 180 over PI degrees
This relationship is important that's chapter one of our little story
We've got to understand the radians are an alternative way to measure circles and they work like this chapter two then small
If I show you a picture of a graph his set of axes and here is what looks like a straight line graph
This is a relationship between two variables. This is a linear thing. They call it linear because of straight lines
But if I show you that same graph on a bigger scale, you could realize that I could have drawn a graph like this
So the thing I'm trying to point out. Is that a sine wave which is
Involved in our calculation is a graph this a function has an input and output but if you zoom in on this sine wave
Right about here
Can you see that it looks a bit like this?
So what we're saying is that although this is not a straight line function
part of it looks like it is so if I zoom in in fact if you work in a certain angle measurement
Which I'll label in a moment
Then this straight line function, which is only true at the beginning is actually the graph of y equals x
Whereas the original function is y equals sine X and I hope you recognize now that the title is kind of relevant when angles are
small
Sine x and x are basically the same thing
So here's my conclusion. You can go and look up some ways to prove it
but visually it's pretty convincing if you zoom in on the sine graph sine X is approximately
X2 caveats for small angles and only in radians
Which is important that we have chapter 1 available to us if you drew a graph of this in degrees
which then this straight line does not have a gradient of 1 which the claiming is if you if you think about this line being
Going up to 1 here and you drop a line down here. That's a it's about 60 degrees there
So the grading of that line in degrees, it's like 1 over 60 tiny number
So the question is what?
Angle measurement do you need to make it have a gradient of 1 which is why I'm claiming
well
you need that to be 1 and
Then it would be grating 1 only if you realize that those are 2 and then that's a 3
My scale is a bit rubbish. But that means this must be PI
And that's 2 pi and now we're back to our radians idea
This sign X is approximately X for small angles and if they're measured in radians end of chapter 2 chapter 3
Fraction you remember that in the trick? We did a reciprocal
In fact, we did one over five five five five five five
and actually what's important is to recognize that as a decimal that looks
Different and actually when you gave me six the calculator just spat that fraction back is like 1 over five five five five four five
Is one over five five five five five and slightly annoying in the calculator
telling you the question when you just asked it what the answer is so has a decimal we could do is do a bit of
Long division and work it out or I could force the calculator to tell me two six fives
I do the reciprocal of that one over that
You get Y over five five five five five five. Otherwise known has according to my calculator
0.000001 eight and very faintly you can see there's a dot above the second zero dot above the eight
Do you know what these dots mean?
In Europe, they wouldn't recognize them either in Europe
They might write this like this but that in Thai bar, but that also might make you look horrified it
Is that more familiar recurring? Yes?
For some reason Britain is inherited the dot notation
you've mark the
Place where it starts with carrying with a dot and the place where it stops were going with it and they need to carry on
Most people use in the rest of the world. I think use the line they call it a vinculum
It's the same line name as the line between a numerator and a denominator
They say there's a fine line between a numerator and the denominator and it's called a vinculum anyway
And this is one of them thing is it's a recurring decimal
So if I didn't put the dots on here if I wrote this carried out
It would be zero zero zero zero one eight zero zero zero one eight and so on so turning fractions into decimals
Now what we could do if you were so inclined maybe I encourage any watches of the video
You could go the other way around you could check this recurring decimal say if that's a recurring decimal
How do you turn that back into a fraction?
It's actually because of school level skill
But when it's this long a recurring thing takes a little bit of guts to go through the process, but for now
there's one thing in this decimal expansion which was not obvious in the fraction that gives us a clue about why this is working and
It's the only numbers that are kind of significant in here's these two
Numbers possibly followed by the zero after it and if I just refer back to a piece of paper from chapter
I'm trying to point out this number
Has turned up and suddenly we get a glimpse of why
These things are coming together the conclusion of all three chapters is when we put these facts together and we'll do the trick again
With all three facts now, it's all under our control and hopefully it becomes obvious why the unexpected Pyatt ends up?
So let's try it. First of all, you did one over five five five five five five
Do you accept that the answer to that calculation is small?
Yeah
Which means we're in this world of small angle approximation if I'm going to do a sine of something
I'm in this world of small-angle approximations
But I didn't say at the very start but I did admit it at the end
I needed your calculators be in degrees
Not these magic radians we've been talking about so if I do one over five five five five
Five five and then take the sine of it, which is what our whole calculation was
It is not gonna be equal to one over five by five five five five or even approximately because we weren't working in
Radians if we were working in radians, then the input would be the same as the output. That was my small angle business
So let's go back to our three chapters starting at the end chapter three the decimal version of your number
I'm gonna write it as one point eight times 10 to the negative something. Let's check
One two, three, four, five six. So ten to the negative six
But we're working in degrees. So if I wanted to turn it into radians
So that I can start using my small angle approximation up here. I need to check back chapter one
My conversion turn radians into degrees. In fact if we want to go the other way around we better just check
It's just the opposite exchange rate. I wanna turn my degrees which is what the count later was set up into radians
So I'm going to just turn it into radians by multiplying by the exchange rate
Which is PI over 180 that was in degrees, and this is now in radians
I've done the conversion, which means I can now use this
Chapter to business to say this is approximately equal because sine X and X approximate equal because this is small there's no question
This is a small number
It's approximately equal just to that number one point eight times 10 to the negative six
Times pi over 180 and you're beginning to see the light at the end of the tunnel. There's a PI in there
We can see what's coming from and there's some ones and eights possibly cancelling
So let's turn this into 100 eighty if I make that a hundred times bigger
I better make this a hundred times smaller to compensate times pi over 180
and now I think you can see that 180 divided by 180 cancels out and I'm left with
PI times 10 to the negative 8 which is the end of my trick. That's where the surprising pi comes from
you do a bunch of stuff to do with the number five and out pops pi to a very small power of 10 and the
Reason you are using fives is because of the relationship between fives and this 1/8
Yes it actually to do with the fact that there are five 18s in 90
So recurring decimal is quite often to do with the number nine and ones where?
Recurring is doesn't happen off that every single digit ends up to do with being 90s and nine hundreds
And so it's all to do with the fact that five times 18 is 90
Gives us a clue. It does mean it we should better predict what happens though?
I had to do a
Scaling of a hundred to get the cancellation to happen 10 to the negative to our hundredth to make it work
Otherwise what you'll see is that is to do with a number of fives over here
We had six and that six is not coincidence. So we're probably in a position to make a prediction
We've done the process once with your example. We should be able do a general one
You want to pick a number of fives you haven't tried yet?
I mean 10 10 5 1 2 3 4 5 6 7 8 9 10 there are 10 fives before we actually do the calculation
This is the power of mathematics. We should be at an a predict what's gonna happen? We're expecting a pie to turn up
But what power of 10 you reckon?
if you turn this into a reciprocal, I think you'll get a
power of 10 to the negative 10
but we're gonna have to adjust it in our calculations like we did before by another two, so I'm hoping
pie times 10 to the negative 12
Let's find out
Take the sine
3.14159 blah blah blah times 10 to the negative 12
Even if it's a pointless trick at least math the power of mathematics. Let us predict the outcome air. We are surprising Pi from surprising Phibes
you know when I graduated from high school, I felt like I was on top of trigonometry and now
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