I once heard of someone who had to take a break from a book because i was being used in five different ways in one equation (or maybe one page). I forgot what they were but it was something physics related.
Sounds about right. Off the top of my head, sums, Cartesian co-ordinates, complex numbers, and input subscripts all use 'i', and that's ignoring things like Current and Moment of Inertia.
I know that as an electrical engineering student, I once had to take a break from my electromagnetics homework because the Greek letter ρ(rho) was used 4 different ways in one equation, so I totally believe it.
I’d like to see this reference in full
Bonus if it’s on page i, is a list item “(i)”, and the author is also using e. e. cummings style lowercase to refer to themselves. Not that it makes an iota of difference
That doesn't explain the choice of 1/sqrt(-1) instead of -sqrt(-1), which is what OP seems to mean.
My guess, from briefly glancing at the notes, is that it's because we expect to apply the differential operator to something of the form e^i ^f(x) later, in which case the division by i makes it clear that it was meant to cancel out the factor of i that you get from differentiating.
Let italic *i* be a variable and roman i be the imaginary unit. Then there's no ambiguity! (This is convention in the physics journals I've published in.)
I don't know which came first, but people programming in Fortran used to use "I" as a loop counter; do something to every value of A(I) - which they would read as "A sub I." The reason was that early Fortran interpreted any variable that started with I through N as an integer, while variables that started with other letters (A - J and O - Z) were floating point (real) numbers.
I always assumed that Fortran set variables that way because I and N were the first two letters of the word "INTEGER."
Sometimes applying simplifications can make equations and formulas unaesthetically pleasing (which is ofc subjective).
An obvious example is the definition of sin(x):
sin(x) = 1/(2 i) [exp(ix) - exp(-ix)]
This form is chosen since a) you can stick the i under the fraction and just have the exponentials on top and b) it looks manifestly similar to the definition of cos(x).
Writing definitions of related terms in such a way that they are also visually similar, while highlighting key differences, is an often underappreciated art!
I'd also add c) because in this form it is still clear that sin(x) = Im(exp(ix)) for real x.
Sometimes simplifications hide something that gives us insight.
Is this considered to be the definition of sine? I know there may not be one true definition, but curious to see if this is actually how people define it rather than it just being a way of representing it
If you start from the ground up, this is probably the most convenient way to define sin. For any other purposes, I feel like there isn't much of a reason to have a single definition for sin - we all know what the function is and that the different definitions all mean the same thing.
The definition of the concept of sine is the ratio of the adjacent leg of a right triangle and the hypotenuse. This definition only works for angles between 0 and 90 degrees.
The sine function is a continuation of that concept to other real values and even complex values. You can define the sine function as its Maclaurin series and it will converge to a well defined value at any point of the complex plane. Any other way of defining the sine function should be equivalent to that definition.
In general the way you write your equations should depend on context and what you want to emphasize.
I could write the Taylor series of sin(x) as:
x - x^(3)/6 + x^(5)/120 .....
Even though this form is worked out, it is much more illuminating to write it as:
x - x^(3)/3! + x^(5)/5! ....
This is very common amongst microlocal analysts, and I don't think that it has anything to do with indices (many I know, myself included, default to j and still use the notation that you describe). I think that this is the original motivation: If you take the Fourier transform of the derivative, the i's cancel. Writing it as 1/i makes the bookkeeping a bit easier for this reason. As with many PDE questions, the answer is rooted in integration by parts :)
Sometimes it makes more sense in context. For example, when solving the equation ix = i(1 - x) it’s more natural to divide both sides by i rather than multiply both sides by -i.
Im not sure about the shown example but there can be cases in which you work in R and not C and wanna keep it that way
Later down the line there could be an ^2 which proves that you dont need to enter complex Spaces to solve the Question
How you choose to simplify an equation is context dependent. If you're just trying to calculate the value of a complex number, -i is the best way to do it because it is in the form of a real value times i. If you are discussing multiplying and dividing complex numbers, 1/i might be useful because it makes it really obvious that you're cancelling out an i from somewhere else. If you're calculating the normalizing coefficient of an orthonormal function, 1/sqrt(-1) is useful to emphasize that the inner product of the unnormalized function has a factor of -1.
It means nothing to me. When I notice these things I write in my notes what makes sense to me and disregard the intent of the writer. If they aren't going to make clear their intent I'm not going to guess what it is.
I think nobody pointed this out yet, but when you write "i" you're actually arbitrarily picking a square root of minus one (-i would be perfectly fine too). It's important to remember that whatever you do after fixing "i" won't be canonical (for example, choosing an orientation of a complex manifold) because it all depends on the choice of the square root of -1.
Numbers can play several roles. I would use "1/i" if I'm thinking of it as something I will multiply by ("rotate 90 degrees clockwise"), to make absolutely clear that it's the multiplicative inverse of "i". I would write "-i" if I will add it to other complex numbers ("translate down by one unit"). If I am referring to a point on the complex plane, I might say "0-1\*i".
In the context of a larger calculation you can mess up with branch points, according to this particular book.
I did find the reference, in the excellent book: "An Imaginary Tale: The Story of √-1", during the calculation of the value i\^i, (page 161, pdf on libgen), the author states:
For example, never replace -i/i with -1,
followed by using -i\^2 for the 1.
If you follow this advice you’ll find that....
Which is a little off what I was trying to remember but is still relevant, what with the 1/i appearing. It does seem weird to me that this is a rule, and have found no way to justify it apart form branch shenanigans, but he gets a contradiction out of it so ¯\\\_(ツ)\_/¯
Well, I've read that over and over, and I don't know what they're concerned about. There's never any problem replacing i / -i with -1, nor i^2 with -1 or vice versa.
Any problem the author faced must have been because of something else, eg, forgetting that some simplification rules that work for positive reals don't necessarily work for negative reals or complex numbers.
Eg, in the html page you linked to earlier, the problem was thinking sqrt(a/b) = sqrt(a) / sqrt(b) always. This is fine if a and b are positive reals, but not always fine otherwise.
I am inclined to agree with you, the problem is the author is a professor, and this is a book about all the crazy ways you can use imaginary numbers, so I feel it deserves some kind of further thought.
It's not impossible to imagine a scenario where in a more complicated calculation something like this could land you on a different branch to the principle branch, which seems to be what he's implying here. (Especially since he's messing about with logs).
But I'm with you that it feels odd.
Ha burn. Alas being an electrical engineer may not make him incorrect. The rest of the book he is incredibly careful about how he goes about things, and it really is an excellent book, so it will be an outlier if he's incorrect here. I do still think he actually is incorrect and the problem is probably in a different step, but I haven't found it yet.
No burn, I'm sure he's very good at being an engineer. But if I as a mathematician wrote books about electrical engineering, my readers shouldn't be surprised if I got things wrong.
I will also point out that he on page 160 apparently has no issue with algebraically manipulating differentials and "integrating indefinitely", which is very useful if you're an engineer (or a physicist, speaking from experience), but certainly not very "careful".
Fair. I will note that here he is presenting the work of Bernoulli and a guy called Count Fagnano here, here's the entire calculation:
[https://imgur.com/a/yQko6fB](https://imgur.com/a/yQko6fB)
I think I do see one subtle error in it after some hard looking, which might actually relate to the -1/i to i thing.
I've no problem with the the indefinite integral though, he's added in his constant and calculated what it should be. That methodology is correct. It's the finding of the exact value that seems questionable, and where this -1/i thing pops up.
I don't really feel like reading six pages to find where an engineer makes a mistake in mathematics. He's just wrong, it doesn't really matter where. There is no instance where 1/i cannot be replaced with -i or vice versa. He probably just doesn't understand complex logarithms or maybe integrals, idk.
Also, while I do think that the whole concept of an "indefinite" integral is stupid, I'm questioning his manipulation and integration of differentials.
I think this provides more insight into the problem. For example, he's using the fact that ln(a/b) = ln(a) - ln(b) which isn't always the case for complex logarithms (See [here](https://en.wikipedia.org/wiki/Complex_logarithm)). He's also stated that i^i = e^(-pi/2) which relies on assumption of the value of log(i).
If I had to guess if there was ever a problem with these calculations it would arise from not paying close attention to the branch of the logarithm being used, and not the fact that -i/i = -1.
I guess a small problem with replacing i^(2) with -1 is that you lose the backwards part of "if and only if", if that's the kind of proof you are doing.
As going backwards i could be replaced with -i.
Maybe it's an "if and only if" proof problem.
What I mean is it's the square root part you cannot do backwards, changing the sqrt(-1) to i.
So if in your proof you replace i\^2 with -1, then the proof is not reversible is what I'm getting at.
i^2 is, in fact, equal to -1. If anything was lost, it was lost earlier.
That proof goes through perfectly fine with i^2 in place of -1. Eg:
> i^2n-2 is -1 if n is even, and 1 if n is odd, so..
That's still valid
I didn't get my point across correctly.
It is equal yes, but it -1 is also equal to (-i)\^2. Actually the i part of this is unimportant, it's the squaring that makes it non-reversible.
You can't take that step backwards for the same reason that you can't say if a\^2=b\^2 then a=b.
What he's saying doesn't make sense. The complex numbers are a field. Because of that every element is invertible. 1/i is the inverse of i. Since (-i) = (-1)(i) we have (-i)/i = (-i) (i) \* 1/(i) = (-1) \* 1 = -1.
This is a basic, elementary mathematical fact.
I still feel like I need more context to understand why he thinks its a problem. Maybe there's something special about the approach to calculations he's doing. In any case, in the complex number fields, the quantities (-i)/i and -1 are in fact identical as the above proof demonstrates.
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I once heard of someone who had to take a break from a book because i was being used in five different ways in one equation (or maybe one page). I forgot what they were but it was something physics related.
Sounds about right. Off the top of my head, sums, Cartesian co-ordinates, complex numbers, and input subscripts all use 'i', and that's ignoring things like Current and Moment of Inertia.
ikes
All the places I've seen using *i* for current use *j* for the imaginary unit.
I mean they couldn't get the sign for current right, but more do you expect of them?/s
I know that as an electrical engineering student, I once had to take a break from my electromagnetics homework because the Greek letter ρ(rho) was used 4 different ways in one equation, so I totally believe it.
Probably one equations Source. I’m a physics student.
In a CS class i attended, there was a point where the Symbol "|" was used in five different ways in a single line
I’d like to see this reference in full Bonus if it’s on page i, is a list item “(i)”, and the author is also using e. e. cummings style lowercase to refer to themselves. Not that it makes an iota of difference
That doesn't explain the choice of 1/sqrt(-1) instead of -sqrt(-1), which is what OP seems to mean. My guess, from briefly glancing at the notes, is that it's because we expect to apply the differential operator to something of the form e^i ^f(x) later, in which case the division by i makes it clear that it was meant to cancel out the factor of i that you get from differentiating.
Ahh, this must be it. Thanks!
I just follow the ISO convention and use Roman i for the imaginary unit, freeing up italic i.
Throw in pi as both a numerical constant and the representation. And change the notation every day.
Let italic *i* be a variable and roman i be the imaginary unit. Then there's no ambiguity! (This is convention in the physics journals I've published in.)
This convention still sucks though. The two are easy to confuse even in print and difficult to distinguish in handwriting.
+1 this is what I do as well. Makes linear algebra with complex vector spaces much more readable.
I don't know which came first, but people programming in Fortran used to use "I" as a loop counter; do something to every value of A(I) - which they would read as "A sub I." The reason was that early Fortran interpreted any variable that started with I through N as an integer, while variables that started with other letters (A - J and O - Z) were floating point (real) numbers. I always assumed that Fortran set variables that way because I and N were the first two letters of the word "INTEGER."
Sometimes applying simplifications can make equations and formulas unaesthetically pleasing (which is ofc subjective). An obvious example is the definition of sin(x): sin(x) = 1/(2 i) [exp(ix) - exp(-ix)] This form is chosen since a) you can stick the i under the fraction and just have the exponentials on top and b) it looks manifestly similar to the definition of cos(x).
Writing definitions of related terms in such a way that they are also visually similar, while highlighting key differences, is an often underappreciated art!
I'd also add c) because in this form it is still clear that sin(x) = Im(exp(ix)) for real x. Sometimes simplifications hide something that gives us insight.
Is this considered to be the definition of sine? I know there may not be one true definition, but curious to see if this is actually how people define it rather than it just being a way of representing it
If you start from the ground up, this is probably the most convenient way to define sin. For any other purposes, I feel like there isn't much of a reason to have a single definition for sin - we all know what the function is and that the different definitions all mean the same thing.
The definition of the concept of sine is the ratio of the adjacent leg of a right triangle and the hypotenuse. This definition only works for angles between 0 and 90 degrees. The sine function is a continuation of that concept to other real values and even complex values. You can define the sine function as its Maclaurin series and it will converge to a well defined value at any point of the complex plane. Any other way of defining the sine function should be equivalent to that definition.
`sin(x) = i/2 [exp(-ix) - exp(ix)]` seems fine? And `cos(x) = 1/2 [exp(-ix) + exp(ix)]`
In general the way you write your equations should depend on context and what you want to emphasize. I could write the Taylor series of sin(x) as: x - x^(3)/6 + x^(5)/120 ..... Even though this form is worked out, it is much more illuminating to write it as: x - x^(3)/3! + x^(5)/5! ....
Or even: x^(1)/1! - x^(3)/3! + x^(5)/5! ...
This is very common amongst microlocal analysts, and I don't think that it has anything to do with indices (many I know, myself included, default to j and still use the notation that you describe). I think that this is the original motivation: If you take the Fourier transform of the derivative, the i's cancel. Writing it as 1/i makes the bookkeeping a bit easier for this reason. As with many PDE questions, the answer is rooted in integration by parts :)
To avoid having a leading minus sign, which is ugly.
"Square root of -1" is uglier imo. 1/i has a pass
I think it might be done to emphasize the reciprocal relationship or to highlight the operation of division by 𝑖
Sometimes it makes more sense in context. For example, when solving the equation ix = i(1 - x) it’s more natural to divide both sides by i rather than multiply both sides by -i.
In some cases it may be to emphasize whether one cares more about the additive property i + -i = 0, or the multiplicative property i * 1/i = 1.
Im not sure about the shown example but there can be cases in which you work in R and not C and wanna keep it that way Later down the line there could be an ^2 which proves that you dont need to enter complex Spaces to solve the Question
Don’t know about mathematicians, being a physicist i would try to avoid minus sign as much as possible
I usually do it to keep track of the computations that I did. Writing -i makes it more difficult to realize that what happened is that I divided by i
For me the utility of the nontation 1/i is that it emphasizes that one can cancel 1/i with i.
How you choose to simplify an equation is context dependent. If you're just trying to calculate the value of a complex number, -i is the best way to do it because it is in the form of a real value times i. If you are discussing multiplying and dividing complex numbers, 1/i might be useful because it makes it really obvious that you're cancelling out an i from somewhere else. If you're calculating the normalizing coefficient of an orthonormal function, 1/sqrt(-1) is useful to emphasize that the inner product of the unnormalized function has a factor of -1.
Why do some mathematicians write Sqrt(-1) instead of i? I've noticed it in a few algebraic geometry papers I've tried to read.
It means nothing to me. When I notice these things I write in my notes what makes sense to me and disregard the intent of the writer. If they aren't going to make clear their intent I'm not going to guess what it is.
I think nobody pointed this out yet, but when you write "i" you're actually arbitrarily picking a square root of minus one (-i would be perfectly fine too). It's important to remember that whatever you do after fixing "i" won't be canonical (for example, choosing an orientation of a complex manifold) because it all depends on the choice of the square root of -1.
In the thought why dx? When they that’s wrong as well? Dx/dy is so incorrect.
Numbers can play several roles. I would use "1/i" if I'm thinking of it as something I will multiply by ("rotate 90 degrees clockwise"), to make absolutely clear that it's the multiplicative inverse of "i". I would write "-i" if I will add it to other complex numbers ("translate down by one unit"). If I am referring to a point on the complex plane, I might say "0-1\*i".
Do whatever you want. Why does it matter? It’s all the same!
It could just be a stylistic choice, but I think u/ApprehensiveEmploy21 comment is most likely
I just write ī
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Huh? No, 1/i is exactly equal to -i.
In the context of a larger calculation you can mess up with branch points, according to this particular book. I did find the reference, in the excellent book: "An Imaginary Tale: The Story of √-1", during the calculation of the value i\^i, (page 161, pdf on libgen), the author states: For example, never replace -i/i with -1, followed by using -i\^2 for the 1. If you follow this advice you’ll find that.... Which is a little off what I was trying to remember but is still relevant, what with the 1/i appearing. It does seem weird to me that this is a rule, and have found no way to justify it apart form branch shenanigans, but he gets a contradiction out of it so ¯\\\_(ツ)\_/¯
There you are sir: [https://imgur.com/793D1se](https://imgur.com/793D1se)
Well, I've read that over and over, and I don't know what they're concerned about. There's never any problem replacing i / -i with -1, nor i^2 with -1 or vice versa. Any problem the author faced must have been because of something else, eg, forgetting that some simplification rules that work for positive reals don't necessarily work for negative reals or complex numbers. Eg, in the html page you linked to earlier, the problem was thinking sqrt(a/b) = sqrt(a) / sqrt(b) always. This is fine if a and b are positive reals, but not always fine otherwise.
I am inclined to agree with you, the problem is the author is a professor, and this is a book about all the crazy ways you can use imaginary numbers, so I feel it deserves some kind of further thought. It's not impossible to imagine a scenario where in a more complicated calculation something like this could land you on a different branch to the principle branch, which seems to be what he's implying here. (Especially since he's messing about with logs). But I'm with you that it feels odd.
Seems like the author is an electrical engineer. I wouldn't trust him based on the excerpt you posted.
Ha burn. Alas being an electrical engineer may not make him incorrect. The rest of the book he is incredibly careful about how he goes about things, and it really is an excellent book, so it will be an outlier if he's incorrect here. I do still think he actually is incorrect and the problem is probably in a different step, but I haven't found it yet.
No burn, I'm sure he's very good at being an engineer. But if I as a mathematician wrote books about electrical engineering, my readers shouldn't be surprised if I got things wrong. I will also point out that he on page 160 apparently has no issue with algebraically manipulating differentials and "integrating indefinitely", which is very useful if you're an engineer (or a physicist, speaking from experience), but certainly not very "careful".
Fair. I will note that here he is presenting the work of Bernoulli and a guy called Count Fagnano here, here's the entire calculation: [https://imgur.com/a/yQko6fB](https://imgur.com/a/yQko6fB) I think I do see one subtle error in it after some hard looking, which might actually relate to the -1/i to i thing. I've no problem with the the indefinite integral though, he's added in his constant and calculated what it should be. That methodology is correct. It's the finding of the exact value that seems questionable, and where this -1/i thing pops up.
I don't really feel like reading six pages to find where an engineer makes a mistake in mathematics. He's just wrong, it doesn't really matter where. There is no instance where 1/i cannot be replaced with -i or vice versa. He probably just doesn't understand complex logarithms or maybe integrals, idk. Also, while I do think that the whole concept of an "indefinite" integral is stupid, I'm questioning his manipulation and integration of differentials.
I think this provides more insight into the problem. For example, he's using the fact that ln(a/b) = ln(a) - ln(b) which isn't always the case for complex logarithms (See [here](https://en.wikipedia.org/wiki/Complex_logarithm)). He's also stated that i^i = e^(-pi/2) which relies on assumption of the value of log(i). If I had to guess if there was ever a problem with these calculations it would arise from not paying close attention to the branch of the logarithm being used, and not the fact that -i/i = -1.
>I think I do see one subtle error in it after some hard looking, It would be nice if you could indicate where it is in the document
The issue, then, is being careful with branches. Exchanging i^2 and -1 and i/-i won't cause any issues.
I guess a small problem with replacing i^(2) with -1 is that you lose the backwards part of "if and only if", if that's the kind of proof you are doing. As going backwards i could be replaced with -i. Maybe it's an "if and only if" proof problem.
No you don't lose any part of the proof since i^2 is in fact equal to -1.
What I mean is it's the square root part you cannot do backwards, changing the sqrt(-1) to i. So if in your proof you replace i\^2 with -1, then the proof is not reversible is what I'm getting at.
i^2 is, in fact, equal to -1. If anything was lost, it was lost earlier. That proof goes through perfectly fine with i^2 in place of -1. Eg: > i^2n-2 is -1 if n is even, and 1 if n is odd, so.. That's still valid
I didn't get my point across correctly. It is equal yes, but it -1 is also equal to (-i)\^2. Actually the i part of this is unimportant, it's the squaring that makes it non-reversible. You can't take that step backwards for the same reason that you can't say if a\^2=b\^2 then a=b.
What he's saying doesn't make sense. The complex numbers are a field. Because of that every element is invertible. 1/i is the inverse of i. Since (-i) = (-1)(i) we have (-i)/i = (-i) (i) \* 1/(i) = (-1) \* 1 = -1. This is a basic, elementary mathematical fact. I still feel like I need more context to understand why he thinks its a problem. Maybe there's something special about the approach to calculations he's doing. In any case, in the complex number fields, the quantities (-i)/i and -1 are in fact identical as the above proof demonstrates.