Solving the problem of similarity between exponents and trigonometric functions by introducing the imaginary number i resulted in the most beautiful equation - Euler's identity.

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IN CAME EULER
PM [B010]

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Leonhard Euler
The French mathematician
Leonhard Euler (1707-1783) was
fascinated by the similarity of these
last three formulas.
He ingeniously solved the problem
by introducing the imaginary
number 𝑖𝑖 in the equations.
He had worked it out in a way
different from what we are doing
now. It was more analytic and
harder to work out.
Neither did he leave an equation
exactly the same as we have now.
But the credit of finding the solution
was accredited to him.

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Expansion of 𝑒𝑒𝑖𝑖𝑥𝑥
By the Maclaurin method, the vector 𝑒𝑒𝑖𝑖𝑥𝑥
can also be expanded into a similar infinite
series:
𝑒𝑒 𝑥𝑥
= 1 + 𝑥𝑥 +
𝑥𝑥2
2!
+
𝑥𝑥3
3!
+
𝑥𝑥4
4!
+
𝑥𝑥5
5!
+
𝑥𝑥6
6!
+
𝑥𝑥7
7!
+ · · ·
𝑒𝑒𝑖𝑖𝑥𝑥
= 1 + 𝑖𝑖𝑖𝑖 −
𝑥𝑥2
2!
−
𝑖𝑖𝑥𝑥3
3!
−
𝑥𝑥4
4!
+
𝑖𝑖𝑥𝑥5
5!
−
𝑥𝑥6
6!
− 𝑖𝑖
𝑥𝑥7
7!
+ · · ·
𝑒𝑒 𝑥𝑥
→ 𝑒𝑒𝑖𝑖𝑥𝑥

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Shuffling the Terms
Rearranging the terms into two groups:
𝑒𝑒𝑖𝑖𝑖𝑖
= 1 + 𝑖𝑖𝑖𝑖 −
𝑥𝑥2
2!
−
𝑖𝑖𝑥𝑥3
3!
−
𝑥𝑥4
4!
+
𝑖𝑖𝑥𝑥5
5!
−
𝑥𝑥6
6!
− 𝑖𝑖
𝑥𝑥7
7!
+ · · ·
𝑒𝑒𝑖𝑖𝑖𝑖
= 1 −
𝑥𝑥2
2!
−
𝑥𝑥4
4!
−
𝑥𝑥6
6!
+ · · · + 𝑖𝑖 𝑥𝑥 −
𝑥𝑥3
3!
+
𝑥𝑥5
5!
−
𝑥𝑥7
7!
+ · · ·

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The 𝑖𝑖 does it
Compare this:
𝑒𝑒𝑖𝑖 𝑖𝑖
= 1 −
𝑥𝑥2
2!
−
𝑥𝑥4
4!
−
𝑥𝑥6
6!
+ · · · + 𝑖𝑖 𝑥𝑥 −
𝑥𝑥3
3!
+
𝑥𝑥5
5!
−
𝑥𝑥7
7!
+ · · ·
With the cosine and sine equations again:
𝑐𝑐𝑐𝑐𝑐𝑐 𝑥𝑥 = 1 −
𝑥𝑥2
2!
+
𝑥𝑥4
4!
−
𝑥𝑥6
6!
+ · · ·
𝑖𝑖 𝑠𝑠𝑠𝑠𝑠𝑠 𝑥𝑥 = 𝑖𝑖 𝑥𝑥 −
𝑥𝑥3
3!
+
𝑥𝑥5
5!
−
𝑥𝑥7
7!
+ · · ·
It is not hard to see that:
𝑒𝑒𝑖𝑖𝑖𝑖 = 𝑐𝑐𝑐𝑐𝑐𝑐 𝑥𝑥 + 𝑖𝑖 𝑠𝑠𝑠𝑠𝑠𝑠 𝑥𝑥

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𝒙𝒙 → 𝜽𝜽
Now, it’s time for the medusa to
come in. She connects the linear
world with the circular universe
but in a slightly different way
from the centaur. She links up
the arc with the radius:
𝜃𝜃 =
𝑠𝑠
𝑟𝑟
She is a hybrid and easily slip
into the equation which
becomes:
𝑒𝑒𝑖𝑖 𝜃𝜃 = 𝑐𝑐𝑐𝑐𝑐𝑐 𝜃𝜃 + 𝑖𝑖 𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃
𝜃𝜃

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Where is the Centaur?
But this is not the famous Euler
equation - it is only the origin of the
Euler equation.
When the vector rotates to the
other side of the real axis, that is, to
where −1 is, the angle becomes 𝜋𝜋
(in radian measure).
𝑒𝑒𝑖𝑖 𝜋𝜋
= 𝑐𝑐𝑐𝑐𝑐𝑐 𝜋𝜋 + 𝑖𝑖 sin 𝜋𝜋
= −1 + 0
= −1
𝑒𝑒𝑖𝑖 𝜋𝜋 + 1 = 0
Reputed to be the most ‘beautiful’
equation in science and
mathematics.
𝑂𝑂
𝐴𝐴
𝜃𝜃 = 𝜋𝜋
𝜋𝜋

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The Beauty of Equation
Now, maybe you’ve never
thought of math equations
as “beautiful,” but look at
that result:
It combines the five most
fundamental numbers in
math − 0, 1, e, 𝑖𝑖, and 𝜋𝜋 −
𝑖𝑖𝑖𝑖 a relation of irreducible
simplicity.
Lee Simmons@actual_self
𝜋𝜋
𝑖𝑖
𝑒𝑒
𝜃𝜃

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The beautiful Equation
Even more astonishing if you slog
through the proof, which involves
infinite sums, factorials, and
fractions nested within fractions
within fractions like matryoshka
dolls. And remember, e and 𝜋𝜋 are
infinitely long decimals with
seemingly nothing in common;
they’re the ultimate jigsaw puzzle
pieces. Yet they fit together
perfectly—not to a few places, or a
hundred, or a million, but all the way
to forever.
Lee Simmons@actual_self
𝑒𝑒𝑖𝑖 𝜋𝜋

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The Most Beautiful Equation
But the weirdest thing about Euler’s
formula—given that it relies on
imaginary numbers—is that it’s so
immensely useful in the real world.
By translating one type of motion
into another, it lets engineers convert
messy trigonometric problems into
more tractable algebra—like a
wormhole between separate
branches of math.
It’s the secret sauce in Fourier
transforms used to digitize music,
and it tames all manner of wavy
things in quantum mechanics,
electronics, and signal processing;
without it, computers might not
exist. Lee Simmons@actual_self
𝑒𝑒𝑖𝑖 𝜃𝜃

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