Is it by definition that i^2=-1

[pivoxa15]

Would you mind justifying that assertion, please?

On p120-121 in Morris Kline's 'Mathematics: The loss of certainty'
"Apparently Euler, too, was still not clear about complex numbers. ... He
also made mistakes with complex numbers. In his 'Algebra' of 1770 he
writes srt(-1)*srt(-4)=srt(4)=2 because srt(a)srt(b)=srt(ab)."

It is easy to see that he had to assume srt(-1)*srt(-1)=1 in order to get
2 as an answer.

More generally. if a=-1 and b=-1 than srt(-1)srt(-1)=srt(-1*-1)=srt(1)=1

since i=srt(-1)

i*i=i^2=1

We now know that the order of operation has been extended so that roots
must be evaluated or simplified before multiplication and division. Hence
srt(-1)*srt(-1) = srt(-1)^2 =-1

It is strange that Euler went by srt(a)srt(b)=srt(ab) because it would
have led to i^2=1 which would contradict his famous formula which he had
derived before 1770. In fact in 1751 he published "Investigations on the
Imaginary Roots of Equations." which related complex numbers to its polar
form.

 

[Doodle Bob]

I was looking into that text, but now seeing the contents, I'm doubting whether or not this book is really as good as the reviews say. I'm going to give it a shot regardless since there is no harm in doing so.

It just seems like there is an unecessary amount of material.

You're kidding, right?

This text does a wonderful job in explaining and illustrating the very many subtle points of complex analysis. And, at the same time, it shows how complex analysis is intrically involved with many other strands of mathematics: from non-euclidean geometry to Riemannian surfaces to number theory to Lie groups.

In fact, after reading the text, what first seems as unnecessary to the topic becomes absolutely essential to understanding it.

It's too bad that not many other math texts are written in this way.

 

[matt grime]

It is easy to see that he had to assume srt(-1)*srt(-1)=1 in order to get
2 as an answer.

and as I explained in reply to Orthodontist, that is morally different from asserting 'he thought that i^2=1' because he wasn't writing about what *we* now label as 'i' and treating it how we now treat it, and it is an important point.

We now know that the order of operation has been extended so that roots
must be evaluated or simplified before multiplication and division.

at least write it properly: we must chose a branch cut.

It is strange that Euler went by srt(a)srt(b)=srt(ab)

Not really. He believed it to be so probably for the same reasons you were wondering about in the first thread in this post, and as plenty of other people have wondered. The difference is that Euler was writing before these things had all been worked out thoroughly and to a point of logical consistency; the way of doing mathematics prior to the late 1800s and modern mathematical ideas are very dissimilar.

It is strange that Euler went by srt(a)srt(b)=srt(ab) because it would
have led to i^2=1 which would contradict his famous formula which he had
derived before 1770.

Again, I must be being very dense, how does one expression not about exponential forms say anything about another to do with exponential forms? You must remember not to use modern mathematical standards when thinking about pre 20th century mathematics.

 

[arildno]

This is, to me, reminiscent of the fierce debate that raged in the mathematical community in the 17th century as how to understand the magnitude of the reciprocal of a negative number.

Many dismissed this as essentially meaningless by the following argument:
Since it is true that for decreasing positive x's the reciprocals, 1/x, increases in value, and when x is 0 the "reciprocal" has reached infinity, it follows that, say, 1/(-1) must be bigger than infinity, since -1<0.

As long as clear definitions haven't been hammered out, such debates must be expected to occur in the "border zones" of what everyone "intuitively" understands.

 

[pivoxa15]

and as I explained in reply to Orthodontist, that is morally different from asserting 'he thought that i^2=1' because he wasn't writing about what *we* now label as 'i' and treating it how we now treat it, and it is an important point.

As far as I know, i=srt(-1) no more no less. So if Euler was using srt(-1) than we can we say he was using i? He obviously didn't have the background mathematics involving i as we do now. If you like I will repharase my claim that Euler thought srt(-1)*srt(-1)=1 instead of i^2=1 even though I see no difference between the two.

From the descriptions in the book, what do you think Euler thought srt(-1)*srt(-1) equal?To me it seems like he would have answered 1, given that he thought srt(-1)*srt(-4)=2 and also the general forumla given directly after it.

The reason why I think it is strange for him to think srt(-1)*srt(-1)=1 (if he ever did), after he derived his famous formula is not because of his logic but because his famous formula could only be derived if he allowed srt(-1)*srt(-1)=-1. He has simply contradicted himself by allowing 1 and -1. I assume he derived it in a similar way shown by AlphaNumeric which surely Euler was capable of doing given the existence of the Taylor series. Moreover, srt(-1)*srt(-1)=-1 is also a result of his formula when the correct angles are calculated.

 

[matt grime]

It appears you're writing as a platonist, whereas I am viewing it as a formalist.

And I still don't see you justifiying why:

his famous formula could only be derived if he allowed srt(-1)*srt(-1)=-1

You can let things behave differently at different times, you know, depending on circumstance. Now, of course, we don't, for sqrt(-1), but as I say that is a modern view on functions. Perhaps he manipulated things in whichever way he saw fit at the time, such as some people do with the axiom of choice or constructibility.

 

[arildno]

I would think that whatever complex maths Euler was doing, it was internally consistent.
That, however, does not mean he was doing complex maths as we choose to do it.

 

[pivoxa15]

It appears you're writing as a platonist, whereas I am viewing it as a formalist.

And I still don't see you justifiying why:

his famous formula could only be derived if he allowed srt(-1)*srt(-1)=-1

I am probably writing more as a naive mathematician but definitely not a platonist.

In this website http://en.wikipedia.org/wiki/Euler's_formula it seems that only if i^2=srt(-1)*srt(-1)=-1=-1 can Euler formula be derived. Are you suggesting there is a way to derive this formula by allowing i^2 to equal something else?

You can let things behave differently at different times, you know, depending on circumstance. Now, of course, we don't, for sqrt(-1), but as I say that is a modern view on functions. Perhaps he manipulated things in whichever way he saw fit at the time, such as some people do with the axiom of choice or constructibility.

That is interesting. It just shows how much more maths is a product of the human mind rather than some objective, indepedent reality.

 

[matt grime]

Are you suggesting there is a way to derive this formula by allowing i^2 to equal something else?

I am saying that there is quite possibly a method (for Euler) to demonstrate that

exp(ipi)+1=0

without ever stating what sqrt(-1)*sqrt(-1) is or isn't.