Is it by definition that i^2=-1

[pivoxa15]

Is it by definition that i^2=-1 or is it worked out from more fundalmental laws?

i^2 could also have been 1. It depends how one evaluates the square.

1. Multiply the two numbers inside the two squareroots first than square root the product (i.e. sroot(a)sroot(a)=sroot(a*a)=a)
or
2. Cancel the squareroot and leave the number (that was before inside the square root) alone (i.e. sroot(a)sroot(a)=(sroot(a))^2=a)

From the definition of i^2, the second option was chosen.

For real numbers it did not matter which option one chose but complex numbers posed a problem. I think Euler first used option 1 for this operation. I got this information from 'Mathematics: The Loss of Certainty' by Morris Klein.

If it is an axiom than why option 2 instead of 1?

 

[rachmaninoff2]

1. Multiply the two numbers inside first than square root the product

This is a fallacy. You in general can NOT just move things in and out of square-roots, because they are multiple-valued (not functions!). You only get a legitimate function if you specify which root you're taking. With real numbers, this is defined to be the positive square root. In this instance the square root gets squared, so there's no ambiguity:

i^2=\left(\sqrt{-1}\right)^2=-1\neq\sqrt{(-1)^2}

 

[daveb]

This is a fallacy. You in general can NOT just move things in and out of square-roots, because they are multiple-valued (not functions!). You only get a legitimate function if you specify which root you're taking. With real numbers, this is defined to be the positive square root. In this instance the square root gets squared, so there's no ambiguity:

i^2=\left(\sqrt{-1}\right)^2=-1\neq\sqrt{(-1)^2}

The problem with this is that it is a multiple value function because of the way i is defined.

My understanding is that i is defined to be the solution to the equation x^2 +1 = 0, or x^2 = -1, then taking the positive square root of both sides so that i = sqrt (-1). Since at the time there was no such thing as the square root of a negative number, i was chosen to represent that number. They could have chosen to take the negative square root instead, which means the definition would have been i = -sqrt(-1). Those are the only possible definitions.

 

[VietDao29]

Is it by definition that i^2=-1 or is it worked out from more fundalmental laws?

i^2 could also have been 1. It depends how one evaluates the square.

Hmm, pivoxa15, we define i² = -1. We don't "evaluate" it.

The problem with this is that it is a multiple value function because of the way i is defined.

My understanding is that i is defined to be the solution to the equation x^2 +1 = 0, or x^2 = -1, then taking the positive square root of both sides so that i = sqrt (-1). Since at the time there was no such thing as the square root of a negative number, i was chosen to represent that number. They could have chosen to take the negative square root instead, which means the definition would have been i = -sqrt(-1). Those are the only possible definitions.

Or more precise, we define the imaginary unit i to be the pair of numbers (0, 1), since there are up to 2 square roots of -1.

 

[HallsofIvy]

Defining i by asserting that "i²= -1" doesn't work because there are two complex numbers that have that property (i and -i). Defining i to be \sqrt{-1}, or, equivalently, as the positive number such that i²= -1 both are invalid because the complex numbers cannot be made into an ordered field so we cannot distinguish between the two numbers in that way.

We can define the complex numbers to be pairs of real numbers (a,b) with addition and multiplication defined by (a,b)+ (c,d)= (a+c,b+d) and
(a,b)*(c,d)= (ac- bd,ad+bc). That can be shown to be a field. Also the sub-field of pairs of the form (a,0) can be shown to be isomorphic to the real numbers with natural isomorphism a->(a, 0). Finally, (0,1)*(0,1)= (-1, 0) so we call that "i", as VietDao29 said, and then write i²= -1. It is also true that (0,-1)*(0,-1)= (-1,0): (0,-1) is, of course, the additive inverse of (0, 1) and so we call it "-i".

 

[matt grime]

... then taking the positive square root of both sides so that i = sqrt (-1). Since at the time there was no such thing as the square root of a negative number, i was chosen to represent that number. They could have chosen to take the negative square root instead...

There is no such thing as an ordering on the complexes that makes i positive at all in any meaning full sense that behaves well with respect to arithmetic.

proof:
suppose i>0, since mutlipying an inequality by a positive number preserves the inequality i can multiply that by i on both sides to obtain -1>0. I leave it to you to see that assigning i<0 also makes no sense.

 

[daveb]

When I said positive square root I didn't mean to imply an ordering on the complex field. I just couldn't think of any other way to put it. My point was that the historical basis for i was when mathematicians tried to find solutions to the equation x^2+1=0. They couldn't find any solutions in the reals, so had to invent the imaginary numbers. At least that's what my calc teachers had told me way back when.

 

[pivoxa15]

We can define the complex numbers to be pairs of real numbers (a,b) with addition and multiplication defined by (a,b)+ (c,d)= (a+c,b+d) and
(a,b)*(c,d)= (ac- bd,ad+bc). That can be shown to be a field. Also the sub-field of pairs of the form (a,0) can be shown to be isomorphic to the real numbers with natural isomorphism a->(a, 0). Finally, (0,1)*(0,1)= (-1, 0) so we call that "i", as VietDao29 said, and then write i²= -1. It is also true that (0,-1)*(0,-1)= (-1,0): (0,-1) is, of course, the additive inverse of (0, 1) and so we call it "-i".

So this is the proper way to define complex numbers.

What about the two options I listed in post 1?

Is it just a coincidence or a consequence that i^2=-1 follows option 2?

 

[matt grime]

'The' proper way? No, not really though it is probably the best. The 'proper' definition is the algebraic closure of the reals, and the description given is one way to produce a model of this, as is the quotient ring R[x]/(x^2+1).

Often in maths there is some abstract 'proper' definition, and that doesn't help you ever work out what the thing looks like. For example, a 'proper' (by which I technically mean categorical) definition of something might be:

given two sets A, B we'll define the product to be a set C with maps to A and B such that for any for other set D with maps to A and B the maps factor uniquely through C.

But of course if you ever need to think about what the product of A and B is it is the set of ordered pairs (a,b), isn't it?

And you first post with the '2 options' doesn't actually make anysense. You don't, for instance, say what it is that you're squarerooting in your options.

In any case, the proper way to squareroot something is to pick an argument in some specified region (-pi/2 to pi/2 or 0 to pi depending on your preference). It is called taking the principal value.

 

[pivoxa15]

This is a fallacy. You in general can NOT just move things in and out of square-roots, because they are multiple-valued (not functions!). You only get a legitimate function if you specify which root you're taking. With real numbers, this is defined to be the positive square root. In this instance the square root gets squared, so there's no ambiguity:

i^2=\left(\sqrt{-1}\right)^2=-1\neq\sqrt{(-1)^2}

In light with the previous post I should say that option 2 is less wrong than option 1.

http://en.wikipedia.org/wiki/Order_of_operations
The order of operations is expressed in the following chart.
exponents and roots
multiplication and division
addition and subtraction

You have to cancel or combine the square-roots first which means you are left with the original term (inside the sqaure root priori to the squaring) alone. The result is not obtained from multipling the term inside the square roots first than squaring the whole thing. Because roots first than multiplication and not vice versa

 

[pivoxa15]

And you first post with the '2 options' doesn't actually make anysense. You don't, for instance, say what it is that you're squarerooting in your options.

In any case, the proper way to squareroot something is to pick an argument in some specified region (-pi/2 to pi/2 or 0 to pi depending on your preference). It is called taking the principal value.

I did say that I was operating on numbers (real or complex). They are the only numbers I had in mind. I was describing things are an intuitive or elementary level which I hope is consistent with the proper definitions.

 

[matt grime]

Can I suggest you reread your first post.

Take this for instance:

"1. Multiply the two numbers inside first than square root the product
or"

Which two numbers inside what?

 

[pivoxa15]

Can I suggest you reread your first post.

Take this for instance:

"1. Multiply the two numbers inside first than square root the product
or"

Which two numbers inside what?

Sorry I have made my original post clearer, I hope.

I realized that the two options yield the same result when considering real numbers only. Because sroot(a)*sroot(b)=sroot(ab) whether a=b or not.

But if operating on complex numbers than option 1 is completely different to option 2. i.e. sroot(a)*sroot(b), when the sroots are evaluated individually and than multiplied, does not equal sroot(ab) where a,b are complex numbers. Hence the oder of operation rules are important when operating roots of complex numbers.

 

[symplectic_manifold]

The question is a classic.
Look up here:
https://www.physicsforums.com/showthread.php?t=108693

 

[pivoxa15]

Using Euler's identity is a nice way of confirming that i^2=-1

I wonder how he came up with this way of representing complex numers. Why is the base e instead of something else. e doesn't come into play until you differentiate the complex number. So that might be the reason for e.

 

[Treadstone 71]

Why is it more "natural" to define complex numbers as ordered pairs? The definition of multiplication, for example, (a,b)*(c,d)= (ac- bd,ad+bc), is not intuitive at all from this POV. If you defined it as something of the form a+bi then it is obvious.

 

[AlphaNumeric]

I wonder how he came up with this way of representing complex numers. Why is the base e instead of something else. e doesn't come into play until you differentiate the complex number. So that might be the reason for e.

Once you have the series solution for e^{t} you could well ask yourself 'I wonder what happens when t = ix', probably what Euler did.

e^{t} = \sum_{n=0}^{\infty}\frac{t^{n}}{n!}

e^{ix} = \sum_{n=0}^{\infty}\frac{(ix)^{n}}{n!} = \sum_{n=0}^{\infty}\frac{i^{n}x^{n}}{n!} = \sum_{n=0}^{\infty}\frac{i^{2n}x^{2n}}{(2n)!}+\sum_{n=0}^{\infty}\frac{i^{2n+1}x^{2n+1}}{(2n+1)!}

= \sum_{n=0}^{\infty}\frac{(-1)^{n}x^{2n}}{(2n)!}+i\sum_{n=0}^{\infty}\frac{i^{2n}x^{2n+1}}{(2n+1)!} = \sum_{n=0}^{\infty}\frac{(-1)^{n}x^{2n}}{(2n)!}+i\sum_{n=0}^{\infty}\frac{(-1)^{n}x^{2n+1}}{(2n+1)!} = \cos x + i \sin x

Just a matter of playing around with the summations, though you get the series for e^{x} from it's property of differentiation as you mention.

Another way (which I didn't know, but mathworld just informed me) is to consider

z = \cos x + i\sin x
dz =( -\sin x + i\cos x )dx = i(\cos x + i\sin x )dx = izdx

Rearrange and integrate to get \ln z = ix so z = e^{ix} = \cos x + i\sin x

 

[dextercioby]

Using Euler's identity is a nice way of confirming that i^2=-1

Circular logic. We use i^2 =-1 to prove Euler's formula.

Daniel.

 

[HallsofIvy]

Why is it more "natural" to define complex numbers as ordered pairs? The definition of multiplication, for example, (a,b)*(c,d)= (ac- bd,ad+bc), is not intuitive at all from this POV. If you defined it as something of the form a+bi then it is obvious.

But how do you define i unambguously? That was the point.

 

[matt grime]

Why is it more "natural" to define complex numbers as ordered pairs? The definition of multiplication, for example, (a,b)*(c,d)= (ac- bd,ad+bc), is not intuitive at all from this POV. If you defined it as something of the form a+bi then it is obvious.

Because throoughout high school you are told "square roots of negative numbers do not exist". This is of course nonsense, I can define a square root of anything arithmetic as a symbol whose square is what we started with, but we don't teach maths like that to kids.

Thus, people come to imaginary numbers (even the name is silly) with a long history of thinking what you're about to do is absolutely wrong because their teacher told them. For some reason many kids think that their high school teacher is more of an authority on mathematics than anyone else.

With the ordered pair view point you are able to construct a field for them that is algebraically closed without any more ontological commitment: they know what real numbers are, they damn well ought to know about ordered pairs (or vectors, really), and this is a nice construction that completely sidesteps the need to introduce a new symbol that they are preconditioned to believe does not exist, nay, cannot exist.

You should not use this explanation to actually work with, but it is a very useful gadget to show that you're not doing something illegitimate after all.

You are just showing that R^2 can be endowed with the structure of a division algebra (that happens to be commutative); R^4 also has a division algebra structure, and this leads to many interesting questions about which other R^n are division algebras (possibly non-associative).

 

[pivoxa15]

Circular logic. We use i^2 =-1 to prove Euler's formula.

Daniel.

How would you derive Euler's formula from i^2=-1? By randomly guessing relationships between different areas in math? Finally come upon the a relationship between complex numbers and trig?

I imagine it would have been easier and more natural to derive Euler's formula from the two ways mentioned by AlphaNumeric and most likely how Euler orginally came up with his formula. i^2=-1 just became a consequence of this elegant relationship.

 

[D H]

How would you derive Euler's formula from i^2=-1? By randomly guessing relationships between different areas in math? Finally come upon the a relationship between complex numbers and trig?

Two things go against your reasoning: History and the playfulness of mathematicians. Bombelli developed the imaginary numbers in the 1570s. Leibniz revived the idea 100 years later. Taylor developed the mathematics behind power series in 1715. The power series representations of exp(x), sin(x), and cos(x) are obviously closely allied. All it took was a little imagination to see the relationship. Have you never just taken an equation apart and played with it?

I imagine it would have been easier and more natural to derive Euler's formula from the two ways mentioned by AlphaNumeric and most likely how Euler orginally came up with his formula. i^2=-1 just became a consequence of this elegant relationship.

AlphaNumeric derived Euler's formula given i^2 = -1.

 

[robphy]

This is a gem: "Visual Complex Analysis" by Needham. The website http://www.usfca.edu/vca/ has some information and excerpts.

 

[pivoxa15]

Two things go against your reasoning: History and the playfulness of mathematicians. Bombelli developed the imaginary numbers in the 1570s. Leibniz revived the idea 100 years later. Taylor developed the mathematics behind power series in 1715. The power series representations of exp(x), sin(x), and cos(x) are obviously closely allied. All it took was a little imagination to see the relationship. Have you never just taken an equation apart and played with it?



AlphaNumeric derived Euler's formula given i^2 = -1.

I did not look at the proof given by AlphaNumeric properly. i^2=-1 was an assumption built into the derivation. i^2=-1 is not a consequence but an axiom of this derivation.

Euler probably had access to the Talyor series when he developed his famous formula because in 1715 he was only 8 and surely he would have developed that formula at a later stage of his life.

So it is only by accepting i^2=-1 that Euler's formula can exist and the formula is certainly an elegant result. It could have been that he allowed i^2=-1 just so that this relationship could stand. Its a bit like 'the elegance of this formula proves that i^2=-1' since there was no logical basis at the time for i^2=-1 or 1. Remember that he once thought that i^2=1. Interestingly he did not consider representing the complex numbers on a plane which would have made his formula more plausible.

 

[JasonRox]

This is a gem: "Visual Complex Analysis" by Needham. The website http://www.usfca.edu/vca/ has some information and excerpts.

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.

 

[robphy]

It depends on what you'll use it for. As a textbook for a non-math major, it may not be appropriate. For me, it provides some interesting ways and insights into what is going on in the complex plane... (e.g., numerous ways to motivate the de Moivre theorem).

 

[matt grime]

Remember that he [Euler] once thought that i^2=1.

Would you mind justifying that assertion, please?

 

[0rthodontist]

Euler actually once thought that sqrt(-3)*sqrt(-2) = 6. It's at the bottom of page 1 of the PDF sample of Needham's book.

 

[matt grime]

For my money that is morally different from asserting 'he thought that i^2=1' as that comes with the implication that he thought that what we call i squares to 1, whereas what he was playing with wasn't what we now call i in some sense (ie he demonstrably had it behaving differently, and something is characterized by its behaviour), but this might just be my formalist sensibilities being offended.

 

[JasonRox]

There's nothing wrong with Euler making a mistake. He wasn't perfect. If anything, he probably thought he put a "-" in front, but because he had bad eye sight, he couldn't confirm it.

You can't be afraid to be wrong. You just have to accept that you were wrong and correct your logic.

Sometimes being wrong is better than being right. If you got bad logic running around and everyone (teachers) say you're right, that will hurt you a lot in the long run.

 

[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.