Could mathematics
ever be described
as beautiful?
If you are a religious person
then perhaps the answer might be yes
because mathematics is the language
that allows us to describe, with utter precision,
the intricate way in which God created his
universe.
If you are a scientist,
then perhaps you might think so too
because there are those formulae that link,
most beautifully, two areas of knowledge
that may have seemed hitherto unrelated.
The famous American physicist
Richard Feynman
seemed to think so about one formula in particular
which he called “our jewel”
and “the most remarkable formula in mathematics”.
He was referring
to Euler’s identity
first expressed in 1748
by Leonhard Euler.
Leonhard Euler
was born on the 15th of April 1707,
in Basel, Switzerland.
His father, Paul
was a Church minister
so religion was an important part
of his formative years.
His father was friendly
with the Bernoulli family
and it was Johann Bernoulli,
one of the foremost mathematicians of his
time
who convinced Paul
that his son may have a greater future
in the field of mathematics.
The Bernoulli family,
a dynasty of famous mathematicians,
has had a huge effect on the way we live our
lives today
as it was one of their number,
Daniel,
who would go on to describe what is now known
as the Bernoulli Effect,
the principle that enables aircraft to fly.
It was Daniel Bernoulli
who, in 1727,
secured Euler a post at the Imperial Russian
Academy of Sciences
in Saint Petersburg.
Euler would eventually replace Bernoulli
as the head of the mathematics department,
but concerned about the continuing turmoil
in Russia,
Euler left St. Petersburg in 1741
to take up a post at the Berlin Academy.
It was while Euler was in Berlin
that he published his
“Introduction to Analysis of the Infinite”
in which he expressed what is now known
as Euler’s formula,
a special case of which
led to Euler’s identity.
Strangely enough,
the special number “e”,
known as Euler’s number
and used in Euler’s formula
was not actually discovered by Euler himself.
It was discovered, instead
by another of the Bernoulli clan,
Jacob Bernoulli
who came across it whilst working on the principle
of compound interest.
Jacob Bernoulli found
that if he deposited $1
at the beginning of the year
and awarded himself a total of 100% interest
over that year,
no matter how many times he divided the interest
payments
throughout the year,
he couldn’t get past a grand total of
$2.72 by the end of the year.
The exact number is actually:
2.718281828459045235360287471352662497757…
It was Euler who named this number “e”
after his own name
and that has been that symbol that has stuck.
But how is “e” actually calculated?
The humble calculator knows only 4 mathematical
operations:
addition,
subtraction,
multiplication
and division.
The number “e”
can actually be calculated
by an infinite series
of additions,
multiplications
and divisions
like this:
We start with 1.
To this
we add
1 divided by 1 factorial.
This gives us 2.
We then add 1
divided by 1 x 2,
otherwise known as 2 factorial.
This gives us 2.5.
We then add 1
divided by 1 x 2 x 3,
or 3 factorial.
This gives us 2. 666
We then add 1
divided by 4 factorial.
This gives us 2.708
We then add 1
divided by 5 factorial.
This gives us 2.716
As we add more and more terms
into the equation,
we get closer and closer the precise value
of Euler’s number.
The beautiful thing about this equation is
that it is very easy to predict what the next
term will be
as each time, the denominator factorial
is increasing by one.
Now
e, as we have written it here,
is like writing e to the power of 1.
If we wanted to make this more general
and raise e to the power of x,
then the infinite series changes to:
1
plus x over 1 factorial
plus x-squared over 2 factorial
plus x-cubed over 3 factorial
plus x to the 4 over 4 factorial
and so on.
Now it just so happens
that e to the x
is not the only thing
that can be calculated by such an infinite
series.
The trigonometric functions
Sine and Cosine
can also be calculated by infinite series.
Sine of X is equal to:
X divided by 1 factorial
Minus x-cubed over 3 factorial
Plus x-to the 5 over 5 factorial
Minus x-to the 7 over 7 factorial
and so on.
Again it is easy to predict the continuation
of this series
as each time the denominator is increasing
by 2
as is the power to which we are raising x.
The sign of each term keeps alternating
between a plus and a minus.
Something similar happens with the Cosine
of x too.
1
Minus x-squared over 2 factorial
Plus x-to the 4 over 4 factorial
Minus x-to the 6 over 6 factorial
Plus x-to the 8 over 8 factorial
and so on.
Again,
like the Sine series,
the denominator and power terms
increase by 2 each time only this time,
only this time instead of starting from 1,
they begin from 2.
The infinite series for Cosine, Sine and e
look so similar to each other,
that you might be forgiven for thinking
that there is a relationship between them.
If we were to add the Cosine and Sine series
together,
would that give us Euler’s number?
Well, unfortunately not.
There's a little problem.
The problem is that the signs
between the terms in the Sine and Cosine series
keep changing
whereas in the series which calculates Euler’s
number,
they don’t.
If only there was some way we could sort out
this series,
some number we could multiply the x term by
to make the two equations equal each other.
In order to make the equations equal,
the something we would need to multiply x
by
when it was squared,
would have to equal -1.
The same would be true when
x was cubed,
again when x was raised to the power of 6
and 7
and so on.
The problem is, when we square a number,
the result is ALWAYS positive,
not negative.
What we need is a number,
that when squared,
is equal to -1.
The problem is,
such a number doesn’t exist.
The brilliant thing about mathematicians
is that
when they are on their way to some wonderful
mathematical discovery,
they don’t let a little thing like
“numbers not existing”
stop them.
After all, mathematics is the science of numbers,
so if a number,
rather inconveniently,
doesn’t exist…
well, they just jolly well go out and invent
one.
As it happens,
Euler’s identity
wasn’t the only place
where it would be useful
to have a squared number equaling -1.
The same problem was happening all over mathematics.
And so was born
the imaginary number “i”.
“i”, is the only number that,
when squared,
gives a negative result,
namely: -1.
By now,
you might have tried plugging the number -1
into your calculator
and hit the square root button.
Your calculator
unless it’s a very clever one
will have gallantly refused to give you an
answer.
Maybe the word “Error” is currently showing
on its readout.
That’s because your calculator
is a rational piece of electronics
and cannot deal with numbers which don’t
exist.
However, in the eyes of a mathematician,
just because a number doesn’t exist,
doesn’t mean that it cannot be useful.
What happens if we take Euler’s number
and raise it,
not to the power of x,
but to the power i
the square root of -1
times x?
This will give us:
1
plus ix
plus ix-squared over 2 factorial
plus ix-cubed over 3 factorial
plus ix-to the 4 over 4 factorial
and so on.
but we just said that i
is the square root of -1.
Therefore, if we square it
we'll just get -1.
Just look at that term over there:
there we have i x squared
so we could rewrite those brackets
as minus x-squared.
Now i-cubed
is the same as
i-squared times i,
so that term could be written as i squared
which is equal to minus 1
times i times x-cubed.
i-to the power 4
is the same as writing
i-squared times i-squared,
so -1 times -1 equals 1.
So that term there
is just x-to the power 4.
This is beginning
to look like the series we worked out before
for cos of x plus sin of x.
Now we have some terms that are real
i.e. are not multiplied by i
and some terms that are imaginary
the terms that are multiplied by i.
Let’s highlight the real terms in blue
and the imaginary terms in red.
Now
let’s move everything around
so that we group the real and imaginary terms
together.
All the imaginary terms are multiplied by
i
so we can factor i out.
Now the real terms are the terms we
use to calculate cosine of x
And the imaginary terms
are the terms we used to calculate sine of
x.
It’s just we’ve multiplied them all by
“i”.
So there is a relationship
between Euler’s number
and cosine and sine,
it’s just we have to raise e to the power
of i x
and multiply the sine term by i
in order to make the maths work.
This is Euler’s formula
which we mentioned at the beginning of the
video.
I said that a special case of Euler’s formula
led to Euler’s identity
which the physicist Richard Feynman found
so beautiful.
The special case occurs when x is equal to
pi.
Cosine of pi is equal to -1
and sine of pi is equal to zero,
so e to the i pi is equal to -1.
Rearranging this slightly
gives us Euler’s identity:
e to the i pi plus 1
equals zero.
What is so amazing about this little identity
is that it links together,
in one simple relationship,
four completely different concepts
in mathematics.
Euler’s number,
Pi,
The Cosine and Sine functions
and the imaginary number i.
And THAT,
in mathematical terms, is beautiful!
Filming is currently underway
on a special online video course
which explores the Fourier Transform
and how it works.
This video forms part of the module
on Complex Numbers,
the language used to express the Fourier Transform.
To reserve your free module from the course,
click on the information card
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