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May 3, 2022 β€’ 19 tweets β€’ 7 min read β€’ Read on X
There is a beautiful way to derive the famous Euler’s formula!

In this🧡, I’ll explain a visual, intuitive derivation. It’ll fit into a single picture at the end.

#iteachmath #MTBoS #math #maths #Mathematics An equation with 𝑒^(𝑖𝑑) on ...
This is Thread 3 in a series, but don't worry! This thread is self-contained.
In Thread 1, we saw applications of a mass on a spring.

We solved for its future position, at time 𝑑. This led to 𝑒^(𝑖𝑑), but 𝑖 is "imaginary."

The position along the ground had to be β€œreal,” so we sought to interpret 𝑒^(𝑖𝑑) in terms of real numbers.
In Thread 2, we tackled the exponent 𝑖𝑑. Before simplifying, it had the form π‘₯+𝑖𝑦. We placed π‘₯+𝑖𝑦 at
(π‘₯, 𝑦) in the plane. Then we realized

𝑖 = 0+ 𝑖1
=> 𝑖 is the unit along the y-axis, located at (0, 1) A plane with a horizontal x...
We also found that multiplying by 𝑖 amounts to rotating by +90Β°. In particular, multiplying 𝑑 by 𝑖 yields 𝑖𝑑 on the y-axis. A plane with a horizontal x...
So, we know what 𝑖𝑑 looks like. Now, we’ll tackle 𝑒^(𝑖𝑑) as a whole: what does 𝑒^(𝑖𝑑) look like?

We expect 𝑒^(𝑖𝑑) to be a "complex" number π‘₯+𝑖𝑦. After all, complex numbers include all numbers we know, including 𝑒, 𝑖, and 𝑑. A plane with a horizontal x...
As 𝑑 is the only variable in 𝑒^(𝑖𝑑), π‘₯ and 𝑦 depend on 𝑑:
𝑒^(𝑖𝑑) = π‘₯(𝑑)+𝑖𝑦(𝑑).

So, 𝑒^(𝑖𝑑) is located at the point (π‘₯(𝑑), 𝑦(𝑑)). Since 𝑑 represents time, this is a moving point. What path will it take?
We could try to plot (π‘₯(𝑑), 𝑦(𝑑)) for many values of 𝑑, but we only know how to find 𝑒^(𝑖𝑑) when the exponent is real. That only happens at t=0. Then we get

=𝑒^(0)
=1
=1+0𝑖.

So, (1, 0) is the initial point. A plane with a horizontal x...
Fortunately, one point is enough!

Calculus lets us plot curves without a table of points: we calculate the direction, to anticipate where our starting point will move next.

In other words, we calculate the derivative.
Specifically,

(π‘₯(𝑑), 𝑦(𝑑)) is the position at time 𝑑
(π‘₯β€²(𝑑), 𝑦′(𝑑)) is the velocity, which gives the direction

How can we calculate the velocity? Our approach all along has been to calculate with 𝑖 as if it's real… A plane with a horizontal x...
This is fine.

As Herodotus apparently learned from his travels in Persia, it's often a good strategy to...

deliberate while drunk,
decide while sober.

We're in the drunken discovery phase right now. If we are so inclined, we can make definitions and apply modern rigor later.
For real r, the derivative of e^(rt) is re^(rt). That is, u=e^(rt) grows exponentially, at a rate proportional to the amount: u'=ru.

So, differentiating e^(rt) amounts to multiplying by r. If we treat 𝑖 the same way, differentiating e^(𝑖t) amounts to multiplying by 𝑖.
Remember, multiplying by 𝑖 causes a rotation by +90Β°. Hence, the velocity is obtained by rotating the position +90Β°.

Beautiful! It’s so much simpler than we would’ve initially guessed.

Can you guess the path that the moving point will follow? A plane with a horizontal x...
It’s a circle! The unit circle, to be precise.

Now, if only we knew formulas for a point moving around the unit circle… A plane with a horizontal x...
We know this from trigonometry:
(x(t), y(t)) = (cos(t), sin(t)).

(This assumes unit speed, but we’re safe. Each velocity vector is obtained by rotating a position vector of a point on the unit circle. Each position vector has length one, so the velocity vectors must as well.)
We’ve done it!

We’ve determined that 𝑒^(𝑖𝑑) is located at
(cos(𝑑), sin(𝑑)).

It follows that

𝑒^(𝑖𝑑) = cos(𝑑)+𝑖 sin(𝑑)

This is Euler’s formula. The following equation is t...
Now that we understand the pieces, we can understand the entire derivation with a picture! At the top of the picture i...
Notes:

I learned of Herodotus from @viktorblasjo:


I learned a version of this derivation years ago, on Reddit:
reddit.com/r/math/comment…

You can read all three threads in this series by starting here:
I share math explainers regularly, so if you enjoyed this, follow me for more!

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