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Most mathematicians would agree that $$e^{i\pi}+1=0$$ is one of the most impressive formulas.

Imagine your students have just learned about the definition of $e$ or $i$ (just assume it's $e$, normally $i$ comes later in curricula); $\pi$ should be known for sure at that point.

Is it a good motivation for the new quantity (here $e$ or $i$) for the students if you tell them about the formula above (and maybe short explaining the unknown quantity) to emphasize the importance of the new variable ($e$ or $i$)? Or is such a thing distracting or even unhelpful since you overload the students with stuff they don't know at that point (and maybe don't care)?

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  • $\begingroup$ The questions was motivated by matheducators.stackexchange.com/questions/425/… -- where I was not sure if that is really a good answer or even a bad one. $\endgroup$ – Markus Klein Mar 18 '14 at 13:35
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    $\begingroup$ There is one serious problem: the complex exponentiation is usually introduced even later than the $i$ itself. $\endgroup$ – dtldarek Mar 18 '14 at 14:36
  • $\begingroup$ @dtldarek Yes, that's true. But at the motivation point of view I had in mind (and some of my teachers actually did!), the thing was more like: "Look! You just can't even know what the expression means. But isn't it beautiful and connecting?" $\endgroup$ – Markus Klein Mar 18 '14 at 14:38
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    $\begingroup$ Well, then perhaps you could try "Look! That airliner is flying! Right know you wouldn't grasp the math that allows it to fly, but isn't it beautiful and connecting? The $e$ constant is an essential in relevant formulas and, one day, that power might be yours!" $\endgroup$ – dtldarek Mar 18 '14 at 14:54
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    $\begingroup$ I thought the square root of -1 came far earlier than e. i, in 8th grade in the US, where e, not till 10th grade or so. $\endgroup$ – JoeTaxpayer Mar 18 '14 at 20:24
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I have never quite understood why it was impressive and/or beautiful, and it always frustrates me when people claim that it is. Therefore, I would say "no, it is not good motivation", because beauty is subjective.

On the other hand, if you explain to students that the formula is based on $e^{i\theta}=\cos\theta+i\sin\theta$ and this essentially allows you to combine circles with complex numbers and hence prove many wonderful things (e.g.), then I agree that this is good motivation.

My point is, introducing the formula on its own is pointless. Rather, it is what the formula tells you, and the subsequent applications, which are important and, perhaps, motivate the study of $e$ and $i$. I mention this point because it is often omitted when people mention the formula: they simply claim that it is beautiful "because it contains $e$, $\pi$ and $i$". However, (I believe that) the beauty lies in the application...

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    $\begingroup$ Ok. I just wanted to try and explain that this is (at least one of) the reason that many people think the equation is "beautiful". You are, of course, free to disagree. $\endgroup$ – Thomas Mar 18 '14 at 14:29
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    $\begingroup$ I agree, it is not so interesting or beautiful. More interesting and beautiful (and useful) is $e^{it} = \cos(t)+i\sin(t)$. $\endgroup$ – Gerald Edgar Mar 18 '14 at 16:15
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    $\begingroup$ Also agree that it isn't beautiful. Now johndcook.com/blog/2014/02/17/imaginary-gold shows a really beautiful equation. I also agree with the serious point of this answer. $\endgroup$ – Loop Space Mar 20 '14 at 10:11
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    $\begingroup$ The equation $e^{0 \pi i} = 1$ contains exactly the same 5 "fundamental numbers". $\endgroup$ – dan04 Apr 30 '14 at 2:32
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    $\begingroup$ I agree about the (lack of) beauty of it. If the formula is beautiful because it contains those numbers, then it is not beautiful in and of itself, but only borrows its beauty from the beauty of the numbers it contains. The numbers in their turn are not beautiful in and of themselves but because of their properties and connections to other ideas. 0 is beautiful, for example, because it's the additive identity and because of the long history of not being invented. Without these ideas, the numbers and the formula are not beautiful and so without these ideas the formula shouldn't be introduced. $\endgroup$ – DavidButlerUofA Aug 25 '14 at 21:59
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Unless you have introduced or are about to introduce exponentiation of complex numbers, this is no more than interesting trivia. It doesn't mean anything and doesn't help them do anything.

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I don't think the equation itself is a good motivation. To be seen as beautiful the students first need to have an understanding of $e$ and an understanding and its Taylor series (which is motivated by lots of applications). Then they need some appreciation of the algebra of the complex numbers (which again can be motivated in many ways). Then, once they know what $i$ and $e$ are, and they would generally agree the two constants are unrelated, you can do the magic of plugging in a complex number into the Taylor series for $e^x$ and get the lovely formula. Plug in $i$ and you get the (now beautiful) equation. If you just drop the equation on them, then they will see no beauty in it. It's beauty lies in the relationship it makes between two seemingly unrelated constants. The students need to first think the constants are unrelated.

An analogy would be: is it a good idea to present a Weierstrass nowhere differentiable continuous functions as motivations for differential calculus. Well, if the students don't know anything about continuity of differentiability, then they won't find it surprising that Weierstrass functions exist. To be surprised by their existence the students need to first develop the wrong feeling for these concepts, only so that their understanding can be honed by the surprises.

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I find it brilliant, wonderful, beautiful, but I should think it's a big turn-off for most students because it's so abstract; it's impossible to explain what it actually means.

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Let me take on a different meaning of "introducing" and argue why I think Euler's identity is a great way to introduce the concept:

I teach primarily remedial students in grade 8, 9, and 10, and I put the following classroom poster:

$ \sqrt-1 \space 2^3 \space \sum \pi...$ and it was delicious!

http://www.spreadshirt.com/1-2-i-8-sum-pi-i-ate-some-pie-shirt-C3376A10250297

None of my students need to know about $i$ or $\Sigma$ (yet) -- and a few of them are disappointingly clueless about $2^3$ and $\pi$ -- but nearly all of them were interested in the poster and wanted to know what it meant, and some students even did a bit of research outside of class.

Now, I have not introduced these concepts in a mathematically meaningful way. My students don't know what half of the symbols mean. But by casually mentioning to them that $i$ is not on but above the number line and $\Sigma$ is like + but not really, I hope I can pique their curiosity.

In the same way, I think that using Euler's identity to dive into a long lecture might undermine students' enjoyment, but perhaps leaving the identity for students to think about as they progress through learning about $e$, $\pi$, and $i$ would be motivating or at least interesting.

Finally, it must be noted that Euler's identity really is much better than the lame math pun I put up in my classroom because it's true! That fact might be lost on students, but it shouldn't!

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Usually not. The constants $e$ and $\pi$ have great applications, which should be their motivation.

Maybe in an algebra class that introduces $i$, if you're discussing the geometry of the complex plane, then Euler's formula could be relevant.

Definitely in a differential equations class, where solutions often involve exponentiating complex numbers, Euler's formula is important.

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the key is "..have just learned about the definition of e or i". I think it would be total overload to introduce this equation at this point. I would save it for the end of the school year, making clear that this is not going to be on an exam. I like user1729's point about the trigonometric identity, especially when there is a path parameter like t and you can make things go around in a circle.. Also, I think it is important to talk about the power series expansion in this context. I find that part most beautiful-seeing the connection between the e-function and sine and cosine, seeing how two infinite series are combined into one.

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