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Those kind of « it’s defined like this » explanation has always confused me in math in general. there are two really different things : - the mathematical conc

by bsaul 7y ago

Those kind of « it’s defined like this » explanation has always confused me in math in general. there are two really different things : - the mathematical concept was initialy created as a way to solve this particular problem, and so this should come at no surprise And - after centuries trying to define this concept, we found that the best way to define this concept is like that. In your case, was « e » created to perform 2D rotations in the first place ?

Excel_Wizard 7y ago
If you explored the problem domain, starting with an understanding of trigonometry and calculus, it will naturally emerge as a result. It was not arbitrarily defined. Imaginary numbers were not arbitrarily defined either, they precisely describe an actual phenomenon. At a certain level, it is difficult to distinguish between "the reality we decided is true" for math vs. "the reality that must be true underneath". If you accept empiricism, such that you trust observations we make about the physical world, this problem becomes irrelevant to this discussion.
username90 7y ago
We can look at history: 1. e was discovered in 1618 2. i was introduced around 1637 4. calculus was formalized by Newton in 1687 3. derivatives of sine and cosine were discovered in 1722 4. de Moivre's formula (cos x + i sin x) ^ n = cos nx + i sin nx was discovered in 1730 5. Eulers fornula e ^ ix = cos x + i sin x was discovered in 1748 by using Taylor series (requires their derivatives). So we knew about e and i more than a century before Eulers fornula. However we discovered it pretty soon after we started applying calculus to trigonometrical functions. Also we can note that it is a strictly more powerful than de Moivre's formula since it shows that we can easily add angles just by multiplying their complex representations, so it is not just a simplification of old knowledge. So it seems like the discovery of Euler's fornula actually did help with working with rotations. However it was done during a time when we were still exploring the applications of Calculus so there were a lot of low hanging fruit like this to pick. So to directly answer your question, e was not created to perform 2D rotations, instead we discovered that putting i inside of e makes 2D rotations simple. But the actual important identity you care about is that you can add angles by multiplying which is easy to prove using eulers fornula: (cos x + i sin x) * (cos y + i sin y) = cos (x + y) + i sin (x + y)
- jacobolus 7y ago
  > Eulers fornula e^ix = cos x + i sin x was discovered in 1748 Discovered by Roger Cotes in 1714, in IMO its most natural form, ix = log(cos x + i sin x) > actual important identity you care about is that you can add angles by multiplying Personally I think the multiplicative concept of rotations is the natural one, with rotations associated to points on a circle embedded in the plane, rather than associated to arclengths. The amazing thing is that we can compose rotations (naturally multiplicative) by first taking the logarithm of the rotations (a.k.a. angle measure) adding them, and then taking the inverse logarithm. The logarithm is a tool which turns multiplication into addition. The protractor and the slide rule turn out to be more or less the same concept, which is why we can substitute the former for the latter in https://en.wikipedia.org/wiki/Prosthaphaeresis https://en.wikipedia.org/wiki/Prosthaphaeresis
creato 7y ago
There are many valid ways to arrive at e as an interesting constant. A particular way of arriving at e as an interesting comment might provide a connection to something else (such as 2D rotation), but that doesn't mean it was created for that purpose.
thaumasiotes 7y ago
No, as username90 says, e was not created to perform rotations. The aspect of e that comes into play for the relation to sine and cosine is that the function e^x satisfies the differential equation df/dx = f. This makes the Taylor series convenient by not introducing factors of (ln K)^n in each term. Thus, 2^ix = cis (x ln 2) = cos (x ln 2) + i sin (x ln 2). And since e is, by definition, the number that satisfies ln e = 1, e^ix can be stated more simply as cis x. And that identity, ln e = 1, is in fact the original motivating definition of e.

by bsaul 7y ago

Excel_Wizard 7y ago

If you explored the problem domain, starting with an understanding of trigonometry and calculus, it will naturally emerge as a result. It was not arbitrarily defined. Imaginary numbers were not arbitrarily defined either, they precisely describe an actual phenomenon. At a certain level, it is difficult to distinguish between "the reality we decided is true" for math vs. "the reality that must be true underneath". If you accept empiricism, such that you trust observations we make about the physical world, this problem becomes irrelevant to this discussion.

username90 7y ago

We can look at history: 1. e was discovered in 1618 2. i was introduced around 1637 4. calculus was formalized by Newton in 1687 3. derivatives of sine and cosine were discovered in 1722 4. de Moivre's formula (cos x + i sin x) ^ n = cos nx + i sin nx was discovered in 1730 5. Eulers fornula e ^ ix = cos x + i sin x was discovered in 1748 by using Taylor series (requires their derivatives). So we knew about e and i more than a century before Eulers fornula. However we discovered it pretty soon after we started applying calculus to trigonometrical functions. Also we can note that it is a strictly more powerful than de Moivre's formula since it shows that we can easily add angles just by multiplying their complex representations, so it is not just a simplification of old knowledge. So it seems like the discovery of Euler's fornula actually did help with working with rotations. However it was done during a time when we were still exploring the applications of Calculus so there were a lot of low hanging fruit like this to pick. So to directly answer your question, e was not created to perform 2D rotations, instead we discovered that putting i inside of e makes 2D rotations simple. But the actual important identity you care about is that you can add angles by multiplying which is easy to prove using eulers fornula: (cos x + i sin x) * (cos y + i sin y) = cos (x + y) + i sin (x + y)

jacobolus 7y ago
> Eulers fornula e^ix = cos x + i sin x was discovered in 1748 Discovered by Roger Cotes in 1714, in IMO its most natural form, ix = log(cos x + i sin x) > actual important identity you care about is that you can add angles by multiplying Personally I think the multiplicative concept of rotations is the natural one, with rotations associated to points on a circle embedded in the plane, rather than associated to arclengths. The amazing thing is that we can compose rotations (naturally multiplicative) by first taking the logarithm of the rotations (a.k.a. angle measure) adding them, and then taking the inverse logarithm. The logarithm is a tool which turns multiplication into addition. The protractor and the slide rule turn out to be more or less the same concept, which is why we can substitute the former for the latter in https://en.wikipedia.org/wiki/Prosthaphaeresis https://en.wikipedia.org/wiki/Prosthaphaeresis

creato 7y ago

There are many valid ways to arrive at e as an interesting constant. A particular way of arriving at e as an interesting comment might provide a connection to something else (such as 2D rotation), but that doesn't mean it was created for that purpose.

thaumasiotes 7y ago

No, as username90 says, e was not created to perform rotations. The aspect of e that comes into play for the relation to sine and cosine is that the function e^x satisfies the differential equation df/dx = f. This makes the Taylor series convenient by not introducing factors of (ln K)^n in each term. Thus, 2^ix = cis (x ln 2) = cos (x ln 2) + i sin (x ln 2). And since e is, by definition, the number that satisfies ln e = 1, e^ix can be stated more simply as cis x. And that identity, ln e = 1, is in fact the original motivating definition of e.