e to the pi i, a nontraditional take (old version)

What people said

Recommends Servaes · 2016-06 · Mathematics Stack Exchange · on “What is a geometric interpretation of multiplication/division in the complex plane?”

By rotation and stretching. Multiplication by $z=re^{i\theta}$ with $r$ and $\theta$ real corresponds to rotating the plane over $\theta$ radians, and stretching the plane in all directions by a factor $r$. I also find that this video [this video] has very nice animations illustrating the geometry of arithmetic on complex numbers.

Recommends Ben Grossmann · 2016-11 · Mathematics Stack Exchange · on “Euler's number in exponential form of complex numbers”

Take a look at this answer [LINK] and this one [LINK] . There are also some videos that might help, such as this (Kahn academy) [LINK] and this one (3Blue1Brown) [this video]

Recommends Ben Grossmann · 2017-02 · Mathematics Stack Exchange · on “Intuition for $e^{ix}$ lies on the unit circle.”

There are some videos that explain it nicely. See this one from 3Blue1Brown [this video] or this one from Mathologer [LINK] .

Recommends Ben Grossmann · 2017-06 · Mathematics Stack Exchange · on “False proof of $\pi=0$”

Here's a quick way to summarize the essence of your paradox: $$ e^{2 \pi i} = e^0 \overset{?}\implies 2 \pi i = 0 $$ The first equation is true, the second is not. The problem is that the log is no longer uniquely defined. If it were defined in that sense, then taking the "log of both sides" would take us from the first equation to the second. With real numbers, we can only have $e^x = e^y$ when $x = y$. This is not true if $x$ and $y$ are complex. Since you are learning about complex numbers for the first time, it is worth noting that for real numbers $x$, we have Eulers formula: $$ e^{xi} = \cos(x) + i \sin (x) $$ Notably, both cosine and sine are periodic, so this function will be periodi

Recommends Arthur · 2017-08 · Mathematics Stack Exchange · on “Is there a way to visualize, on the complex plane, complex numbers multiplication?”

I would recommend this video [this video] , by 3blue1brown. It begins with the complex plane around the 4-minute mark, although you should see from the beginning in order to understand what he's talking about. Although it's about something slightly different (the focus is understanding $e^{i\pi} = -1$), it shows a great way of thinking about multiplication and addition in the complex plane.

Recommends Arthur · 2017-08 · Mathematics Stack Exchange · on “What does multiplication mean in terms of $2\times \pi$?”

You could see this video [this video] (from about a minute in), as it explains one way to think about multiplication that doesn't have the problem you're describing. The video itself has the focus of explaining exponentials (and specifically $e^{i\pi}$), but the framework helps a lot here too.

Recommends M. Winter · 2018-02 · Mathematics Stack Exchange · on “How do I interpret Euler's formula?”

I can highly recommend this video [this video] for a rough geometric understanding of complex numbers and Euler's identity. It's only six minuts and can't hurt.

Recommends Mason · 2018-07 · Mathematics Stack Exchange · on “What is the physical interpretation of the real product of two complex numbers?”

I think this is a duplicate of this question: [LINK] in the comments there is this youtube video by 3blue1brown which I think will be what you want: [this video]

Recommends Arthur · 2018-09 · Mathematics Stack Exchange · on “How is the function $f(x)=2^x$ defined?”

(... or maybe this video [this video] would be more suitable to explain the idea; most of what he says about $e^x$ works for any other base.)

Recommends Mauro Giliberti · 2019-04 · Mathematics Stack Exchange · on “Equation of a line on a complex plane”

I don't fully understand your way if explaining the problem, but I think you're unfamiliar with the concept that multiplying on the complex plane means rotating (or stretching), and in particular multiplying by $i$ means rotating by $\frac{\pi} {2}$ , aka 90°, aka right angle, aka "perpendicular to". You can easily check it by taking a random complex number (and in particular a real number for your problem) and multiplying by $i$ . This is why, while on $\mathbb{R} ^2$ we have to do all the fancy stuff to get a perpendicular line to one given, on $\mathbb{C} $ we just multiply by $i$ . 3b1b made a very interesting video [this video] about this, you should check it out.