The Wizard of Oz: From Fractions to Formulas
The Wizard of Oz: From Fractions to Formulas
David
Hello, I’m David Brannan. Films have often been used as creative tools with which to teach
mathematics. One of the best known examples is the 1939 classic “The Wizard of Oz”,
particularly the famous scene where the Scarecrow gets his brains and recites Pythagoras’s
Theorem – albeit incorrectly!
V/O
“The sum of the square roots of any two sides of an isosceles triangle is equal to the square
root of the remaining side”.
Of course the Scarecrow should have referred to a right-angled triangle not an isosceles
triangle, and to squares not square roots, so that:
“The square of the hypotenuse of a right-angle triangle is equal to the sum of the squares of
the other two sides”.
In this podcast we’re going to take you on a mathematical journey called the Wizard of OHZED, or OH-ZEE if you are American.
It’s a parody of the Wizard of Oz, and is based on an Open University radio programme; but
instead of a Scarecrow, talking Tin-Man and cowardly Lion, our Dorothy encounters some
rather challenging mathematical concepts!
V/O
“Integers, fractions, equations – Oh My; Integers, fractions, equations – Oh My; Integers,
fractions, equations – Oh My…”
Phil
With the help of Dorothy, we’ll meet fractions and complex numbers – including a rather
colourful character “I”– before a climactic finale involving one of the most fascinating
equations of all time “Euler’s Magical Formula”.
Audio Clip
Dorothy
Toto, I have a feeling we are not in Kansas anymore. Oh dear! I’ve said that already. Who are
these tiny creatures? Hello? Hello? Who are you?
Fraction
We are the fractions.
Dorothy
Fractions? My, aren’t you small?
Fraction
Well, that’s because we are proper fractions. We are less than a whole.
Dorothy
Isn’t something less than a hole still a hole? You can’t have half a hole you know?
Fraction
Not the empty sort of hole, the full sort, unity, completeness, oneness, that sort of whole. ‘wh’
whole, ‘w’ whole.
Dorothy
Oh, are there improper fractions?
Fraction
Of course! Over there! Look!
Dorothy
What, the fraction standing on its head?
Fraction
That’s right. Any of us become improper, if we turn ourselves upside down.
Dorothy
Of course, and there are so many of you.
David
So there we just heard how Dorothy met the fractions. But what are fractions?
And what’s the difference between improper and proper fractions?
Phil
Well, David, we can explain fractions by describing a number line.
First, we mark some point on the line that we call zero. Then in one direction, to the right, we
mark successive equally-spaced points as 1, 2, 3, and so on, just like a ruler. And in the
opposite direction we mark successive equally-spaced points as minus 1, minus 2, minus 3,
and so on. That gives us all the whole numbers, or the integers. Then we start again.
But this time, we mark equally-spaced points in the positive direction at distances of one-half
from each other. From zero the first is called one over two, the second, two over two, the
third, three over two, and so on. In the opposite direction we mark successive points as minus
1 over 2, minus 2 over 2, and so on. This gives us all the points on the line that represent
fractions with denominator, that’s the number on the bottom, 2. Some of them are repeats of
the integers of course, for example 2 over 2 is the same as 1. Then we repeat this process for
each whole number, q, say, where the distance between successive points is 1 over q instead
of a half. The points that we get then represent all the fractions, p over q. If a fraction is p over
q and q is bigger than p, then we say that the fraction is a proper fraction; and p over q is less
than 1, otherwise we call it an improper fraction.
David
And is that all the numbers there are?
Phil
No: there are lots more numbers that somehow lie between these fractions. The fractions are
often called rational numbers, because they are ratios of whole numbers. The other numbers
are then called irrational numbers - not because they’re illogical, but because they are not
ratios of whole numbers. You can approximate these irrational numbers by a process of
taking limits of rational numbers. Taken together all these numbers are called real numbers.
All real numbers can be written as decimal numbers, which are whole numbers plus so many
tenths, plus so many hundredths, plus so many thousandths, and so on, possibly for ever and
ever - that’s an example of a limit.
Let’s just look at the tenths, hundredths and so on, the proper fraction bit. If we write the
multiples of the tenths, hundredths and so on, we get a whole sequence of numbers between
0 and 9 in succession. If the original number is a fraction, then these numbers eventually
become a sequence of batches repeated, we say that the decimal expansion is recurring.
If the original number isn’t a fraction but one of the other real numbers, then the sequence
isn’t a set of repeated batches, we say the decimal expansion is non-recurring.
David
Now, it looks like Dorothy’s about to meet a 300 year-old witch with a famous curve – called
the Witch of Agnesi – so let’s see what happens.
Audio Clip
Dorothy
I don’t want to disturb anybody.
Fraction
Well, you will certainly disturb her.
Dorothy
You mean that angry old lady over there with the flowing bell shaped cloak?
Fraction
She’s the Witch of Angesi, a famous curve. Almost 300 years old.
Versiera
You’ll regret this.
Dorothy
What is she so angry about?
Fraction
She’s misunderstood, or rather mistranslated. She is Italian, and her name is Versiera, that
means turning.
Dorothy
That’s not a bad name for a curve.
Fraction
No, but she’s been translated into English as ‘a versiera’, which is ‘witch’.
Dorothy
Which is which? Oh, I see what you mean, which is ‘witch’!
Fraction
Right, and she’s been pretty unpleasant about it ever since.
Dorothy
I’m sure if I’m nice to her she’ll help me. Ah, Versiera?
Versiera
Versiera, not a versiera!
Dorothy
Can you help me please?
Versiera
Probably, but that doesn’t mean that I’m going to. What do you want?
Dorothy
I want to get back to where I came from please.
Versiera
Back to your origin you mean?
Dorothy
Please.
Versiera
Well I’m no use to you. I don’t pass through the origin.
Dorothy
Oh, do you go anywhere that’s useful?
Versiera
Well, I approach the x-axis, the origin is on the x-axis you know.
Dorothy
Is it? Well yes, I suppose it is. So if I follow you to the x-axis…
Versiera
I said I approached it, not that I got there. I am asymptotic, I don’t get there until infinity.
Dorothy
So do I have to go to infinity then?
Versiera
Yes!
Dorothy
Why?
Versiera
It spoils the story if you don’t.
Dorothy
What’s in infinity?
Versiera
Well, the ends of things, infinite things that is. And the Wizard!
Dorothy
The Wizard?
Fractions
The Wizard?
Versiera
The Wizard of OH-ZED!
Dorothy
Shouldn’t that be the Wizard of …oh, no I suppose it shouldn’t, it would spoil the story.
Versiera
You are learning. You should have seen the Wizard, and perhaps he will help you get back to
the origin. (7’46)
Dorothy
Won’t you come with me then?
Versiera
How can I? We don’t start from the same place.
Dorothy
So where do you start from then?
Versiera
From minus infinity of course.
Dorothy
Minus infinity? Oh dear, this is all terribly confusing.
Versiera
You’ll probably find me along the way.
Dorothy
Or joining me you mean?
Versiera
No, of course not! Getting closer and closer the further you get along the x-axis. What do you
think asymptotic means?
Dorothy
I really don’t know.
Versiera
I don’t know what education’s coming to these days! Get on your way!
Dorothy
Which way?
Fraction
That way! Ho ho!
David
Can you explain infinity and minus infinity?
Phil
Well, if you start at any point on the number line and walk in the positive direction - that is, to
the right- keeping walking so that you pass every point on the line, then we say that you have
gone to infinity. And if you walk in the opposite, the negative, direction, keeping walking so
that you pass every point, then we say that you have gone to minus infinity. Actually this links
with the idea of series, sometimes called infinite series. If you have a list of infinitely many
numbers, then you can try to add them all up: so, you write the first one plus the second, plus
the third, then plus the fourth, and so on for ever. This expression is called an infinite series,
or just a series for short.
Sometimes the values of the successive sums get closer and closer to a particular number, in
fact indefinitely close as you go on. Then we say that the series converges and the sum of the
series is that number. For example, if you start with the list a ½, a ¼, a 1/8 and so on, and
then add them up this way, then what you get is a ½, ¾, 7/8, and so on, These numbers
converge to 1, so the sum of the infinite series ½+1/4+1/8 and so on…is 1. Otherwise we say
that the series diverges.
David
Actually, the Witch of Agnesi curve has got nothing to do with witches, as you heard, but it’s a
very good name for a curve. Another thing, what did the witch mean by Asymptotic?
Phil
Ah, this is where things become two dimensional! You look at a plane, not just a line.
So, we take a horizontal number line in the plane that we call the x-axis, and a vertical
number line that we call the y-axis; with the two axes meeting at zero on each number line –
we’ll call that point of intersection the origin. Suppose next that you have a curve in the plane
that meets each vertical line just once and that goes infinitely far to the right. Now suppose
that the vertical distance between the curve and the x-axis gets closer to zero, in fact as close
to zero as you please, the further and further you go to the right, then we say that the curve is
asymptotic to the x-axis.
David
OK, shall we get back to Dorothy’s search for infinity?
Audio Clip
Dorothy
Oh my, what a long way this is. And oh no, there’s a fork in the road, which way do I go? You
are absolutely at right angles.
i
Howdy!
Dorothy
Who’s that? Why there’s a strange little man standing a little way up this branch? Ah, hello?
Who are you?
i
I’m i.
Dorothy
I beg your pardon, I don’t mean to be rude but shouldn’t that be ‘I’m me’?
i
Not at all, I’m ‘i’, imaginary, you know? This is the imaginary axis.
Dorothy
Do I want the imaginary axis? I’m looking for infinity. I need to find the Wizard. I was following
the x-axis.
i
All axis’s lead to infinity eventually. But it’s dramatically more convenient if you stay on the
other road - the real axis. Wait!
Dorothy
Where have you gone?
i
I’m over here! Back where you came from.
Dorothy
Oh what did you do then?
i
I square myself. You know. i square equals -1, I’m real now. Weren’t you real before then?
Not at all. I was imaginary.
Dorothy
But I can’t imagine what an imaginary number could be.
i
It’s quite easy. Think of minus one like I am now.
Dorothy
I don’t have any problems with that.
i
Now I’ve taken my square root…I’m back over here! i is the square root of -1!
Dorothy
Oh, shouldn’t that be ‘I am the square root of…’ no, I suppose it shouldn’t. So I should take
the real axis then that way?
i
That will certainly take you to infinity, but I warn you, it’s a long way!
Dorothy
Oh dear.
i
Hey, never mind, I come along with you. Besides, I can ask the Wizard to help me too!
Dorothy
What with?
i
I’m too simple, I get ignored. People don’t believe in me, I want to be more complex.
Dorothy
A complex number? But aren’t you already…? Oh, never mind. I expect it would spoil the
story. Don’t you need to square yourself again then?
i
Oh yeah, yeah, sure. There! Let’s go!
David
You’re listening to ‘Wizard of OH-ZED: ‘From Fractions to Formulas’ from The Open
University. We just heard Dorothy meeting an imaginary axis on her route to infinity.
Can you explain ‘I’ and the ‘square root of -1’ next?
Phil
Well, that brings us back to the complex numbers that Dorothy met earlier. We look again at
the plane, with the x- and y-axes in it. Any point in the plane can be described by saying how
far along the direction of the x-axis it is from the origin, a distance x say; and how far along
the direction of the y-axis from the origin it is, a distance y say. Then the numbers x, y are just
the coordinates of the point. Now here’s the cunning bit! The points on the x-axis correspond
to real numbers, so let’s called the points on the y-axis ‘imaginary numbers’. Then the point 1
on the y-axis we’ll denote by the symbol ‘I’, where i stands for imaginary. So the point 2 on the
y-axis is 2i, and so on. Then we’ll denote the point in the plane with coordinates x and y by
the single expression x plus iy. Then we’ll call x plus iy a complex number. And so we don’t
lose all connection with reality, let’s agree that complex numbers have to satisfy all the usual
rules of arithmetic – that is, the rules for adding, subtracting, multiplying and dividing
numbers. And this works fine, just so long as we add the extra assumption that multiplying the
imaginary number i by itself gives minus 1; that is, I squared equals minus 1 as you heard.
David
We should find out how Dorothy’s getting on in her quest to get to infinity.
Audio Clip
Dorothy
Hoh, we’ve been walking for ages now, it never seems to get any near!
i
That’s the problem with infinity you know.
(Versiera’s laughter)
Versiera
Not as easy as you thought it would be aye? But there is a way!
Dorothy
Well, how?
i
I’m not sure a witch’s advice can be trusted.
Versiera
You speed up. Get faster and faster regularly. Look, in the last hour, you did two miles. In the
next one, do three, and in the one after that, do four and so on.
Dorothy
Well I still think it will take forever.
Versiera
Maybe, but it’s something to think about. I’ll be seeing you again.
Dorothy
I think I’m just going to go as fast as I can and see what happens. I’ll race you to that pile of
junk over there…
e
Oh don’t, that’s not the way to refer to a distinguished infinite series.
Dorothy
Who are you?
i
What me?
Dorothy
He!
e
Not he, e!
Dorothy
Is every body in this country named after a letter?
i
Oh, it’s variable, but most of us are.
e
Literally, I’m the exponential series or function. Tsk tsk tsk, everyone’s so casual nowadays.
But you can call me e, most people do.
Dorothy
Well, can you elp, er, help us? We are trying to get to infinity. But it seems to be taking rather
a long time.
e
Well, I’m rather good at getting to infinity. You’ve heard the exponential growth I persume?
Dorothy
Ah huh.
e
Yes, well, I can get to infinity faster than anything else you care to name.
Dorothy
So if we went along with you, will we get there quicker?
e
Oh, quicker, certainly. But it would still take forever.
Dorothy
Oh dear, what can we do?
i
Let me think about it, I’m sure there’s a way. I suppose we change how we get faster.
Dorothy
How?
i
Well, now, we are just speeding up one mile at a time, perhaps we can do better than that.
Couldn’t we go two miles faster in the next hour, then four, then eight, sixteen so on, I’m sure
there’s something there.
e
I can assure you that no good will come out of it. Why is i travelling with you?
Dorothy
He wants the Wizard to make him more complex.
e
But isn’t he already? No, never mind. I expect it would spoil the story. Does the Wizard give
people things they want?
Dorothy
So I’m told.
e
Then would you mind if I joined you.
Dorothy
What do you want then?
e
To have some value! It’s all very well being infinite, but I never seem to have any value.
Dorothy
But surely…oh, never mind.
Dorothy/e
It would probably spoil the story.
David
Now, there we met the exponential function! What’s that? And ‘e’ – can you explain the value
of e?
Phil
First e. We have to use the shorthand term n factorial to represent the numbers 1, 2, 3 and so
on up to n all multiplied together. So 2 factorial Is 1x2, 3 factorial is 1x2x3, that’s 6, and so on.
So, take the infinite series 1, plus 1 over 1factorial, plus 1 over 2 factorial, plus 1 over 3
factorial, and so on indefinitely. It turns out that this is a convergent infinite series whose sum
is approximately 2.718281828 and so on; and that’s a non-recurring decimal. And we call this
sum the number e, also known as the Euler’s number. Euler was a famous Swiss
mathematician who became the court mathematician of Catherine the Great in St Petersburg.
David
And the exponential function?
Phil
Right: take the infinite series 1, plus x over 1 factorial, plus x squared over 2 factorial, plus x
cubed over 3 factorial, and so on indefinitely. This infinite series also converges, whatever
value x happens to be; and its sum is e to the power x. That’s an amazing fact about the
number e.
David
Of course we need the e and i to come together with π for Euler’s Magical Formula: eiπ = -1
Audio Clip
Dorothy
Oh, we’ve been using the Witch’s method for ages, I’m not sure that she was trying to help us
at all.
(Vasiera’s laughter)
i
Just leave me alone for a bit longer, I’m sure I’m getting somewhere.
e
Well we are not. Infinity is no nearer than it was before. And I’m getting very puffed up with all
this speeding up.
Dorothy
It’s no better than if we were just going around in circles.
e
Speaking of going round in circles…
Dorothy
Why is that lion chasing its own tail round and round? He will wear himself out? Hello? Who
are you? Are you an imaginary lion?
π
No, I’m a real lion. There are lots of us lions around here. Parallel lions, intersecting lions,
straight lions, curve lions, the lion at infinity…
Dorothy
The lion atinfinity? Can he help us? We are trying to get to infinity just to see the Wizard you
know.
π
I’m sure he’ll be useful. But he’s the lion at infinity, you see, so he’s not here just now, but this
road is going the right way.
Dorothy
I see, you are a real lion, do you have a name?
π
I’m π, you see, I’m not a whole lion, just a lion segment, not a very long one either, but a very
repetitive one, I just go round and round in circles, round and round this unique diameter. I
know it’s irrational, but I’m very tired of recurring this.
Dorothy
But surely…oh, never mind, this is obviously the way the story goes.
π
Do you think if I came with you to the Wizard, he could help me?
Dorothy
To stop recurring?
π
Hmm.
Dorothy
Oh yeah, do please join us. It’s so nice to meet someone who is just not a single letter.
π
Well, I’m afraid that that isn’t true. It’s my foreign ancestry. I’m not p, i, e you know, it’s π,
Greek.
David
What’s the value of π then? And didn’t we meet non-recurring before?
Phil
Well, the parameter of a circle of diameter d is πd, so the number π is just the parameter of a
circle of diameter 1. And the value of π is approximately 3.1415926535 and so on, a decimal
expansion that never repeats itself – another one, that’s non-recurring. And astonishingly if
you put iπ in place of x in the series for exponential function, that’s e to the power iπ, it turns
out after a bit of work that the sum of that infinite series is exactly minus 1. And that’s Euler’s
Magical Formula eiπ = -1. You can rearrange that formula as eiπ +1= 0, a formula which
involves the 5 most important numbers in mathematics 0, 1, π, e and i. A remarkable formula!
David
Now, they’re coming to the end of their journey which means we must be close to meeting the
Wizard and Euler’s Formula too?
Audio Clip
Wizard
And just who do we have here?
Dorothy
Please sir, I’m Dorothy, and this is e, i and π.
Wizard
And just what do you all want with me?
i
Please sir, if it’s not too much trouble, I’m tired of being imaginary, and I’d like to be a little
more complex.
Wizard
Ah uh.
e
And please Mr Wizard, I want to have some value, it’s not a good life as an infinite series.
π
And I’m tired of recurring sir.
Dorothy
And please sir, I want to go home. I want to get back my origin. Please sir. Can you help us?
Wizard
Oh dearing me, what a collection of misdirected non-problematic people we do have here. Oh
yes, what a collection. But a rather special collection mind you.
Dorothy
Then you can help us?
Wizard
Oh yes oh yes. Come here, i.
i
I want to be complex.
Wizard
But you are complex.
i
No I’m not. I’m wholly imaginary.
Wizard
It’s all a matter of how you look at things. All you need is a different label.
i
What label would that be?
Wizard
I think that this one…should do nicely.
i
0+1? Ah, I’m complex! I’m complex! I’m a complex number! Oh thank you, thank you, thank
you ever so much!
Wizard
Nothing to it, literally. And now, let’s see about the rest of you. You, e.
e
Yes, sir.
Wizard
What is this about not having value?
e
Well, sir, I’m just an infinite series.
Wizard
Just?
e
1 plus 1 plus 1 over 2 factorial, plus 1 over 3 factorial, plus 1 over 4 factorial and so on…
Wizard
Definitely!
e
Oh, phew, rather indefinite actually. I just go on and on forever, on and on to infinity.
Wizard
Well, now you’ve got to infinity. You’ve added all your terms. And I think you might find that
you’ve got a value.
e
2.718281824! Ha ha!
Wizard
Ha ha, better now?
e
Oh yes, yes.
Wizard
But don’t tell everyone about it. It’s not as easy as that for all infinite series. There are quite a
few divergent ones around you know?
e
2.718281824! I’ve got a value, I’ve got a value!
Wizard
Now π, what was your problem again?
π
Oh, I’m fed up with recurring over and over, just going round and round the same circles.
Wizard
And you think that’s what you are doing?
π
Yes, every 3.14159, I get back to where I started.
Wizard
Every how many?
π
3.141592653589793… (don’t stop)… 23846…
Wizard
I haven’t heard you recur yet!
π
26433…
Wizard
Enough! π you can go on all day forever and you will never recur. It’s just a matter of how you
look at things.
Dorothy
Please sir.
Wizard
Yes, Dorothy.
Dorothy
What about me sir, I want to get back to the origin.
Wizard
Aye, yes, I hadn’t forgotten you. But this is something that I think your friends are better place
to help me with.
Dorothy
How is that sir?
Wizard
π, stop murmuring digits to yourself and go over there and stand next to i.
π
Like this?
i
iπ?
Wizard
iπ, we are getting there. Now where is e?
i
e is behind you
All
Oh no, he isn’t.
Wizard
Stop this nonsense. e, get over there and exponentiate them!
Dorothy
eiπ?
Wizard
eiπ. We are almost done now. It’s just one more thing I need. Let me see, I need 1 of
something. Ah, yes, Toto, you had nothing to do in this story so far. I need to add 1 eiπ but not
yet, as soon as we do that, you’ll be back to the origin Dorothy.
Dorothy
How come sir?
Wizard
Well, as Euler, one of the greatest mathematicians observed that eiπ =-1, so…
Dorothy
So that eiπ +1=0 and I can go home now!
Wizard
Indeed my dear, and frankly I think you should.
David
Is it possible to find out more about the complex formulas that Dorothy encountered?
Phil
At the Open University we have a wide range of maths courses that explore the numbers and
ideas that Dorothy encountered. Our introductory course Y162, Starting with Maths looks at
fractions and simple formulas; while the course MU123, Discovering Mathematics looks at the
use of functions as models of real-world phenomena, including exponential function and other
topics like applications of basic algebra.
Then the second level course M208, Pure Mathematics deals with real and complex numbers,
as well as looking at functions, graphs and exponential functions in greater depth, and the
second level course MST209 Mathematical Models uses complex numbers to help solve
differential equations.
Finally, the third level course M337, Complex Analysis explains complex numbers in much
more detail, going all the way from Euler’s Magical Formula to topics like complex functions
applied to fluid flow and even fractals like Julia sets and the Mandelbrot set.
David
One last thing – what happened to the witch?
Phil
I think the Wizard has the last word.
Audio Clip
Wizard
Well, that’s just about that except that I think I can hear someone in the background.
Versiera? Come on out!
Versiera
Oh, I’ve got here too late, they’ve gone!
Wizard
Oh really my dear, you should try a more sedulous study of convergent.
Versiera
Well, now I’m here, at infinity at last, what happens next?
Wizard
Well, I think this could be the beginning of a beautiful friendship, or was that a different
movie?