Welcome to a presentation
of Euler's Identity, also known as Euler's Equation.
Euler's Identity is one of the most surprising
and beautiful formulas you'll find in mathematics.
Euler's Identity is the equation
e raised to the power of i times pi plus one equals zero.
Notice how the equation contains three basic operations.
We have addition, multiplication of the exponent
and also because of the exponent, we have exponentiation.
The equation also contains
five of the most famous constants.
First we have the number zero,
which is the additive identity, meaning if we add
zero to any number we get the same number.
The equation contains the number one,
which is the multiplicative identity, which means if we take
any number and multiply by one, we get the same number.
The equation also contains the number pi,
where we can obtain pi by taking the circumference
of any circle and dividing by the diameter.
So pi is a constant, but it's also an irrational number
which is approximately 3.14159.
The equation also contains e which is the base
of natural log, e is an irrational constant
just like pi, except e is approximately 2.71828.
And then finally our equation contains the number i,
which is the imaginary unit of the complex numbers,
where i is equal to the square root of negative one.
So the question is why does e raised to the power
of i pi plus one equal zero?
This is actually the result of a special case
of Euler's formula, where Euler's formula is
e raised to the power of i x equals
cosine x plus i sine x, where x is measured in radians.
So let's take a moment and review why this equation
is true using power series.
So we'll begin with three known power series,
shown here below.
We have the power series for e to the x
cosine x as well as sine x.
And power series are usually a topic of Calculus Two,
but even if you haven't taken Calculus Two yet,
I think you'll still be able to appreciate
this discussion.
So we'll begin with the power series
for e to the x, and we'll determine the power series
for e raised to the power of i x.
So to do this, we'll substitute i x for x,
which is shown here.
Notice how we have one plus, instead of x we have i x
and instead of plus x square divided by two factorial,
we have plus i x squared divided by two factorial,
and so on.
And now we'll simplify the terms in our power series.
And for quick review, remember i to the first
is equal to i, i squared is equal to negative one,
i to the third is equal to negative i,
i to the fourth equals one, and then the values
start to repeat.
So i x doesn't simplify, but for this term here,
since i squared is equal to negative one,
this third term becomes minus x squared divided
by two factorial.
In the next term we have i cubed, which equals negative i.
So this term here simplifies to the minus i
times x cubed divided by three factorial.
The next term contains i to the fourth which equals one,
so this simplifies to plus x to the fourth
divided by four factorial, i to the fifth is equal to i.
So we have plus i times x to the fifth divided
by five factorial and so on.
Notice how some of the terms contain a factor of i,
and others don't.
And for the terms that contain i here,
I changed their color to red.
So now we'll group the terms that contain i,
and we'll group the terms that don't contain i.
So the terms that don't contain a factor of i
are listed here, and the terms that do contain
a factor of i are listed here.
But notice how we also factor out the common factor
of i.
And we should recognize that this is the power series
for cosine x, and this is the power series for sine x.
Which gives us Euler's formula.
We've just shown that e raised to the power of i x equals
cosine x plus i sine x, sometimes abbreviated as cis x.
So going back to our original question
of why e raised to the power of i pi plus one equals zero,
we can now show why this equals zero using Euler's formula
when x is equal to pi.
So if we substitute pi for x, we would have e
raised to the power of i pi equals cosine pi
plus i times sine pi, where cosine pi is equal
to negative one.
And you can determine this value on the unit circle
or looking at the graph of the cosine function
shown here.
Notice that pi, the function value, is negative one,
and sine pi is equal to zero.
Looking at the graph of the sine function,
here's the point where x equals pi.
And notice how the function value is zero.
And therefore e raised to the power of i pi equals
negative one plus zero.
So if we add one to both sides of the equation,
we have Euler's Identity or Euler's Equation,
e raised to the power of i pi plus one equals zero.
And I think you'll admit this has to be one
of the most surprising and beautiful equations
you'll see in mathematics.
Thank you for watching.