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I've seen a number of very interesting ways to propose on YouTube.
One of the most fascinating to me, recently, was the young fellow who proposed to his girlfriend
using binary notation -- that is, just reciting a series of zeros and ones.
Now, I'll create a link to that particular video in the description below mine.
What I propose to do in this video is to create yet another
'nerdy' way of proposing and that is to graph, with one equation,
'WILL YOU MARRY ME?'
And I'd like you to come on a journey with me and see how to construct the whole thing.
Welcome to Crystal Clear Mathematics where it is easier than you think.
I'm your host, Graeme Henderson.
This particular equation that we're going to build
is composed of 55 separate elements.
Now, each element (or part) looks like a little straight line
but in fact it's an ellipse.
Every component is based originally on the equation for a circle.
So let's start there and, in three steps, see how to build the equation.
The basic equation for a circle looks like that.
It looks like Pythagoras' Theorem where x and y are the x and y values
at any position on the circle and r is the radius of the circle.
And you can see that,
if that's the x distance to the point, and this is the y distance to the point,
that x-squared plus y-squared equals r-squared (because we have a right-angled triangle).
That means, for example, that x-squared plus y-squared equals one-squared or one
is a circle with a radius of one. And it would go through
one and minus one on the y-axis and one and minus one on the x-axis.
Now, we can see that fairly clearly if we talk about the intercepts.
And to do that we replace x with zero or y with zero.
If we replace x with zero --
we know that zero squared is zero (it happens to be my favourite times table)
but if we replace that with zero we get y-squared equals one.
And the solution to that is that y can be plus one (because one times one is one)
or it could be minus one (because minus one times minus one
is also plus one ... two minuses multiplied ... two negative numbers multiplied make a positive).
In the same way, if y is zero, we get x-squared equals one,
and the solution to that is that x can be plus one or minus one when the y value is zero.
So, there we have a very simple circle.
Now, if we wish to distort the circle so that we can squash it in,
or squash it vertically, to make it elongated, what do we do?
It turns out to be a very, very simple process.
We simply divide the x-squared or y-squared by some other number, preferably a square.
Let's see what effect that has.
If y is worth zero then y-squared is zero and it doesn't matter what's on the bottom,
zero over any denominator is zero, so x-squared is one
and the solution is that x is plus or minus one.
However, if x is worth zero we now have this equation.
If we multiply both sides by three-squared (which happens to be nine)
and solve this equation we find that y can be +3 or -3 ... no longer one!
That means it can go down as far as -3 on the y-axis or up as high as +3
(and I obviously didn't plan this well because it's going to go off the top of my board here) --
but we end up with an ellipse.
It still passes through +1 and -1 on the x-axis, as we discussed,
but it now goes through 3 and -3 on the y-axis.
This is how we can compress and enlarge the dimensions of a circle.
I wonder how we might make an ellipse very, very narrow and a precise length.
Let's have a look at that next.
Let's imagine that I want an ellipse so narrow that when I graph it
it looks like a straight line, or an interval.
We'll draw it on the graph first.
Let's say we wanted to go through 2 on the y-axis, and -2.
But, to make it incredibly thin, we would like it to go through
1/1000 and -1/1000 on the x-axis.
So, if we drew this ellipse (obviously I'm not going to be able to do this)
it would be long and thin and so incredibly narrow that,
to the naked eye, it looks like an interval.
What would that look like?
Well, we start with our circle -- and, remember, to get it to go through + and -2 on the y-axis
we divide by 2-squared so that, when we solve that equation, y could be +2 or -2.
This means that, when I solve this equation to find the x positions,
I want x to be worth 1/1000 or -1/1000 -- and that's what we need.
Now that's not the simplest way to write this.
I suggest that it's actually a little bit easier to use scientific notation
(or at least index notation).
One thousand is 10x10x10 which is 10-cubed. We can simplify that even further.
10-cubed squared is 10-to-the-six, or a million.
And I will spell it out -- instead of dividing by one millionth
you know that we can multiply by the reciprocal, one million over one.
This is a pattern that we use in fractions.
Instead of dividing by this fraction, we multiply by its reciprocal,
and therefore, (whoops) therefore we can write one million x-squared
-- and that does the same job as this.
So, this equation produces that line.
Now, if I wanted a horizontal line -- let's do that in a different colour --
if I wanted a horizontal line to go from -1 to +1 on the x-axis,
then I would start my equation x-squared plus y-squared equals one.
This time my x values go from +1 to -1 and I want my y values to go
from plus 1/1000 to -1/1000 and we can achieve that by using the same technique
we used with the 'x's before. So this equation gives that line.
So, now we have the basic tools to construct little line intervals on a graph,
ready to construct our letters.
Now for step number two!
It's all very well being able to draw (on our axes) a vertical line or even a horizontal line
centred on the origin but, if that's all we could ever manage
we'd only end up with a cross and we wouldn't be able to construct letters at all.
We must be able to move these lines around the coordinate plane,
and how do we manage that?
Well, let's write our basic equation. There it is.
That's our ellipse that goes from +2 to -2 on the y-axis
and from +1/1000 to -1/1000 on the x-axis. So, that's our interval.
What if I wanted to move it to -5 on the x-axis?
All I need to do is to make this alteration.
And you can see that, when x is worth -5, this becomes worth zero (or, its value is zero)
and this portion behaves exactly as it did back over here
where y would have values of +2 and -2.
And it turns out that, when y is zero, at this position, x would move
from +1/1000 to -1/1000.
So, we have in fact moved the interval five spaces to the left
just by making that slight alteration.
Now, in a similar way (let's change this number to -2)
and let's have a bit of fun and change that to +3.
You can see the same basic structure of the equation.
It's an x-squared plus a y-squared equation but, with these modifications,
this now has a centre at x = +2 and y = -3.
On the y-axis it moves from +2 to -2 and, on the x-axis,
it moves from +1/1000 to -1/1000 so we have a vertical line here.
How would we get a horizontal line?
Well, we would simply swap these two coefficients and we would get something like
10-to-the-six here ... and let's use the same number here.
Let's change the position -- I'll use a +2 and a -3 this time.
The centre of this graph is at (-2,+3).
It's not well positioned there (-2, and +3) so that's the centre
of this particular line interval (or very thin ellipse).
Notice, this time, on the y-axis it goes from +1/1000 to -1/1000,
so it's extraordinarily thin vertically.
But, along the x-axis it goes from +2 to -2 away from the origin
(from the centre of this particular ellipse).
So, 1,2 ... 1,2 ... and we've now constructed that line.
So, we now have the skill to create these very thin ellipses
that look like intervals and to move them anywhere on the graph that we wish.
Now for step number three!
Again, it's all very well being able to move lines around the graph
but, if we could only ever graph one of them, we can't construct letters.
So, how can we, with one equation, have separate little portions of line
scattered all over the graph?
That's a very interesting question.
Let me discuss it with you this way.
There's our very basic ellipse at the origin.
If I rearrange it (rearrange this equation) by simply subtracting one from both sides
I now have an expression that is equal to zero.
This is the same equation as that,
with the same information, drawing exactly the same graph.
Now, mathematicians love zero when it comes to constructing equations.
That's why, for some (I think, for the last four or five videos),
I've been talking about zeros.
I'm going to write it out again. That's that equation.
Let's construct another one.
You can see that it's almost identical -- certainly identical in structure.
All I've done is I've changed the origin, so I've moved ...
I've created another line in a different part of the graph.
Let's set this equal to zero.
What's actually happened here?
I have two expressions multiplied together to make zero.
Now, if I challenge you to think up any two numbers that multiply to make zero
it won't take you long to realise that one of them must be zero.
It happens to be my favourite times table.
Zero times anything is zero.
So, if I have an expression times an expression equals zero then either that is zero or that is zero.
Now, if this is zero it creates this line; if this is zero it creates that line.
In fact, any point on this line will make this equation true
because it will make that zero;
any point on this (equation) ellipse will make this part of the equation zero
which will make the whole expression zero.
We can actually stack these multiple times in an equation
-- which is exactly what I've done -- I have 55 of them!
Each individual bit makes one little ellipse.
If I graph this very, very quickly
(I haven't left a lot of room ... but let's see what it looks like)
-- this one is at the origin, it goes from +2 to -2 and it's 1/1000 wide --
there it is.
This one is at x equals +1 and y equals -3 ...
there's the centre.
It goes 2 units up and down from the centre, up the y-axis and down the y-axis,
and you can see that it's 1/1000 wide.
So, those two lines are described by that equation.
That one equation graphs those two separate parts on the graph.
Let's put that all together to see how we can construct our equation
stating, "Will you marry me?"
I think you'll enjoy it.
So, here we have the first of the expressions in my equation.
If I graph this
-- and I'm going to draw a very long horizontal line with the x and y axes and the origin --
this has a centre at -25 (x equals -25 ... I'll just estimate down here)
and at y equals zero, so there's the centre of this particular ellipse.
Notice that four equals 2-squared so it goes from -2 to +2 on the y-axis
(or up 2 and down 2), so it creates a little line there
and it's 1/1000 either side of the centre ... so, again, that's our very thin ellipse.
That's the first term. Let's look at the second one.
As you can see, it looks almost identical with the first one.
But this one now goes through x equals -22, not quite as far as -25.
And you can see that it goes through y equals zero and, again,
we have 2-squared on the bottom so it goes up 2 units and down 2 units.
And we have the same coefficient of x-squared which is 10-to-the-power-six which means
it's going to go 1/1000 either side of the centre of that particular ellipse.
In a similar way, all the following terms start drawing vertical lines
and then horizontal lines building up the letters --
with the one exception of the last expression which is, in fact, a small circle --
and we'll look at that now and then put it all together.
Now, I think you can see that it has the structure of a circle.
It's an x-squared plus a y-squared equals (if we took this over here)
... equals something squared.
In fact, if I rewrite it -- 1/100 is, in fact, 1/10 squared!
Where's the centre of this circle?
It's at x equals +25 so, unlike these ones it's up this end, at y equals -2,
and it has a radius of 1/10 (a very tiny radius) so as to make a little circle.
The reason this is last is that's the final dot on our question mark
which I think, from memory, I've drawn something like that.
I've restricted myself to straight lines because creating curves is a little problematic
and we might discuss that in another video but, for the moment,
that's the kind of structure that we have and this is the last element in the entire equation.
So, let's put it all together and I just want you to see it unfold on the screen
and to see the complete equation.
Now, as I've mentioned before
this is going to be provided in a PDF file that's downloadable.
The link will be in the description below this video
or you can go across to my website and find it there.
Now that you've seen the equation it's probably good that I show you
how the graph is constructed so you should see
that unfolding on your screen now.
Here's my challenge -- or, at least, I have three of them.
The first is that, if you like what you've seen,
then please experiment with the ideas or the concepts that I've shared.
You've seen three very good skills that can be used to build up
quite complicated relationships on graphs.
The first is how to take the equation for a circle
and modify it to make ellipses ... even very, very thin ones.
The second is you've learned how to move graphs around the coordinate plane.
And the third is you've learned how to put equations for graphs together
into one large expression to build up quite a complicated picture.
I encourage you to explore that.
The second challenge is this.
If you feel you understand the concepts well enough,
then why not make up your own statements?
Build up an equation that, when you graph it,
says something else on the graph.
And thirdly, if you are just a little bit 'nerdy'
and wishing to propose to your boyfriend or girlfriend in a particularly unusual way
then, by all means, feel free to use this idea.
If you do so, I would love to hear from you -- I'd love to see the video.
And if you could create a link back to this to show people where the idea originated,
I would really appreciate that.
This is Graeme Henderson, for Crystal Clear Mathematics,
where it is easier than you think -- and I hope you thought it was!
Please like this video, please leave a comment,
and subscribe to my channel ...
and I look forward to seeing you in the next video. Thank you.