Talk:Euler's formula

e to the pi i

Somewhere in the introduction the familiar (and amazing!) expression e**pi*i = -1 needs to be shown. Casey (talk) 22:51, 25 September 2013 (UTC)

The equation that this article talks about is just a generalization of that expression. See this page.98.195.88.33 (talk) 01:20, 28 January 2017 (UTC)

Proposed merge with Cis (mathematics)

The cis function is rare and redundant to Euler's formula. A former AfD, Wikipedia:Articles for deletion/Cis (mathematics), was closed as no consensus. GeoffreyT2000 (talk, contribs) 05:34, 12 December 2016 (UTC)

• Oppose "Rare" is not "non-notable".

It is likely that any readers searching for cis() will find cis(), not Euler. There they are already given an adequate explanation of it in both trigonometry and history, with an appropriate pointer to this article. Readers finding Euler will find Euler, and that's what they need, im likely blissful ignorance of cis(). How does a merged article improve upon this? This is not paper. We are not short of pages. Andy Dingley (talk) 02:14, 13 December 2016 (UTC)

• Oppose. Obviously, Euler's formula and the cis() function are related but they have different histories to be told and different use cases, and we therefore would not do either of them a favour by discussing them in a single article. They kind of approach a problem from different angles. To someone, who hasn't learnt about exponential functions, the redundancy does not exist. To some, cis() is a sometimes very convenient abbreviation or shorthand notation, for others it is a vehicle in math education.

Regarding the previous AfD (which was still about an article in a much weaker state and with lacking sources), there were prior discussions in the past decade and they suggested to discuss the cis() info in a separate article, because readers felt that the info on cis() did not belong into the Euler article and it was inadequately covered there.

As has meanwhile been established by plenty of reliable sources, cis() has 150 years of history and is a notable topic by itself. Yes, it is not in main-stream use, but this doesn't make it non-notable. After all, we're an encyclopedia and it is our duty to document things from a neutral point of view and not suppress information.

--Matthiaspaul (talk) 12:43, 20 December 2016 (UTC)

• Oppose Many Wikipedia topics are rare. That is not a reason to make them go away because this is an encyclopedia and so should cover the full circle of knowledge. Andrew D. (talk) 14:24, 20 December 2016 (UTC)
• Oppose It's not even that rare. It comes in handy in computer documentation where superscript typography is not so easy. Here is a relatively high traffic example: http://en.cppreference.com/w/cpp/numeric/complex/exp I think the article's extremely negative intro needs to be toned down.--agr (talk) 15:19, 20 December 2016 (UTC)
• Oppose I just came here looking for cis and not for eulers formula. Someone might want to remove that sign above the article. 95.91.212.79 (talk) 01:15, 6 January 2017 (UTC)
• Oppose. One certainly could discuss cis within an article on Euler's formula, but cis (mathematics) has so much material on the usage of that specific notation that I think that it works better as a separate article. You wouldn't want to either get rid of that or try to stuff it all in here (which would make this article unbalanced). ―Toby Bartels (talk) 15:00, 10 January 2017 (UTC)
• I oppose. Cis(x) is an equation, and its uses are not limited to calculating e^ix. Also, Euler's formula can be written as e^xi=cos(x)+i×sin(x) without even acknowledging that there is a shorter way to write cos(x)+i×sin(x). I do not feel that the subjects of these two articles are closely enough related to merit merging.

98.195.88.33 (talk) 01:12, 28 January 2017 (UTC)

Just for the records and to help avoid unnecessary further discussion, User:GeoffreyT2000 closed the discussion as "Don't merge" on 2017-03-24T02:08:43‎. [1] Somewhere we have some nice wrapping template for closed discussions, but I can't seem to find it right now...

--Matthiaspaul (talk) 21:18, 27 March 2017 (UTC)

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Cotes equation

I'm confused and hoping for explanation. How is it rationally possible to posit or envision the Cotes equation cited in this article without x being in radians and fully understood (by Cotes) as a periodic function of pi, as is stated in the article? How did Cotes instead define x? Wikibearwithme (talk) 23:46, 13 January 2018 (UTC)

Thanks. The article statement is unsourced and seems thoroughly dubious, and I've added a Citation Needed requesting a supporting citation or a suitable re-wording. What Cotes actually used is not too clear to me (tho I think it is basically radians as found in an arc of a quadrant of a circle, seemingly with no need to consider periodicity for the specific problem that he was dealing with, irrespective of whether he was aware of such periodicity or not - although I suppose it is conceivable that ignoring periodicity may have introduced a flaw into his proof). What he actually wrote, plus a modern interpretation thereof, can be seen in the lengthy footnote (currently numbered [5]) in his biographical article Roger Cotes (although this footnote is basically just dealing with his conclusion, and is not dealing with his proof, which is available online, but possibly only in Latin - although his archaic English terminology would presumably also be a problem for any Wikipedian trying to read it).Tlhslobus (talk) 07:05, 16 January 2018 (UTC)

Hi - Thanks. I agree, and think without an explicit statement by Cotes of this supposed limitation of the argument, the supposed limitation of Cotes in this article does not seem to be a supportable inference. Wikibearwithme (talk) 08:57, 19 January 2018 (UTC)

Thanks. At the moment I'm waiting for a decent period of time (I'm never clear how long that should be) for somebody to come up with a supporting citation (which seems unlikely, but just about possible if there's an error in his proof as a result). However such waiting always carries the risk that it will never get fixed, so I won't object if anybody else decides to fix it straight away. I'm not 100% sure what the ideal rewording should be, but perhaps something along the lines of "The term Euler's formula is used because Euler's exponential formulation, although it came later, has been preferred to Cotes's logarithmic formulation because a complex number actually has an infinite number of natural logarithms due to the periodicity of trigonometric functions." Tlhslobus (talk) 06:11, 20 January 2018 (UTC)

A simple explanation/proof for Euler's formula

This is technically original research, but on the other hand anyone can show that it is true. Everything about e^ix = cos x + i sin x can be understood by thinking about i^x.

you can identify a point on a unit radius circle by saying by how much 1 + 0i would have to be rotated to get there.

you can rotate a point on the Complex plane by multiplying it by powers of i, or in other words, by i^x.

therefore you can identify a point on a unit radius circle by saying by how much 1 + 0i would have to be multiplied by i^x to get there.

therefore you can identify a point on a unit radius circle in terms of just i^x.

a multiplication by i rotates a point by quarter of a circle. Therefore, you can say that any point indicated by i^x can also be indicated by cos x + i sin x, where x, cosine and sine are working in a system that divides a circle up into 4 angles.

you can rephrase i^x so that x can be a value in degrees or radians or any other way of dividing up a circle: e.g. i^(x/90) works with x in degrees; i^(2x/pi) works with x in radians. It will still be the case that any of these new exponentials will still be equal to cos x + i sin x, where x, cosine and sine are working in that particular way of dividing up a circle.

you can rephrase those exponentials to be a base raised to an Imaginary power e.g. (The 180i root of -1)^ix works in degrees; (The i*pi root of -1)^ix works in radians. It's still the case that these are equal to cos x + i sin x, where x, cosine and sine are working in that particular way of dividing up a circle

you can swap these bases for Real numbers (by using knowledge of how to calculate e^i) to get a Real number raised to an Imaginary power. So, 1.01761^ix works for degrees; e^ix works for radians. And it is still the case that these will be equal to cos x + i sin x, where cosine and sine are working in the particular way of dividing up a circle. In other words: 1.01761^ix = cos x + i sin x, when x, cosine and sine are working in degrees; e^ix = cos x + i sin x, when x, cosine and sine are working in radians.

I explain it in depth here: http://www.wimtarriner.com/

Even if this understandably gets dismissed as original research, I think the article should point out that e^ix = cos x + i sin x only works in radians. You often see it used with x in degrees, which is wrong. Timtimw (talk) 12:15, 6 January 2019 (UTC)

This is not only original research, but also this contains many errors:

• This would require a definition of $i^{x}$ for non-integer x. The simplest definition passes by the formula $i^{x}=e^{x\log i},$ which, in turn requires the definition of log i. The common definition for this uses Euler's formula. So your whole reasoning is essentially circular.
• Euler's formula is an equality between complex valued functions of a real variable. No measure unit is involved. So your edit request, at the end is nonsensical, as well as all comments about how measuring angles.
• Your reasoning is sketchy on the most difficult part, the definition of exponentiation with a complex basis. It misses the fact that if x is not a rational number $i^{x}$ has infinitely many values, and these values are dense on the unit circle; that is, for every irrational x, and every complex number z of modulus 1 (that is lying on the unit circle), there are values of $i^{x}$ that are as close as one want from z.

These are only the most evident errors. D.Lazard (talk) 14:40, 6 January 2019 (UTC)

Thanks for your contribution!

i^x where x is not an integer is perfectly valid. The most obvious examples you will know are i^0.5 and i^pi.

i^x doesn't require knowledge of e^ix to be calculated -- You can calculate simple values of x using a compass, ruler, protractor and a bit of thought.

Multiplication of a point on the complex plane by e^ix rotates that point by x radians. You can test this is true by picking a complex number, plotting it on axes, drawing a circle centred on the axes with a circumference that goes through that point, then drawing a second point one radian around that is still on the circumference. Its position will be the original point multiplied by e^1i. You can try this with any number of radians too. If you multiply 1 + 0i by e^1i, you will get to a point that is at cos x + i sin x, when x, sine and cosine are in radians. If e^1i rotates by 1 radian, then it cannot be the case that e^1i will rotate by 1 degree. Therefore, e^ix = cos x + i sin x cannot make sense if x is anything but radians. If you want an exponential that rotates by degrees you need to use 1.017606491206^ix. [I'm leaving in the unnecessary 1s and 0s in the complex numbers to make this clearer]. Timtimw (talk) 09:45, 8 January 2019 (UTC)

Timtimw, you are missing several mathematical fundamentals that are required to discuss this in a rigorous sense. Your explanations consist purely of hand-waving. Using a visual example (or even any finite number of them) cannot by its nature constitute a mathematical proof of Euler's formula; there is no such thing as a "proof by example".

1. The complex logarithm is a multivalued function and therefore one cannot talk about "i^x" without ambiguity, unless one first specifies a particular branch cut. You fail to provide a rigorous definition of that function.
2. The very geometric interpretation (rotating) that you talk about is only possible in the first place with Euler's formula already established.
3. To answer your original post, sine and cosine are understood to use radians in an analytical context like this one. It may indeed be useful to clarify that, but if so, only one single sentence, because the vast majority of readers will have been exposed to radians.

I appreciate your curiosity in this subject, to be clear, but it should also be noted that this is WP:NOTAFORUM for discussing the mathematics of your suggestion beyond why it is unsuitable for the article. Further questions should be directed to Wikipedia:Reference desk/Mathematics.--Jasper Deng (talk) 10:07, 8 January 2019 (UTC)