Are there imaginary numbers other than i?

[filter54321]

Are there "imaginary" numbers other than i?

I'm taking a class in complex analysis and the professor wrote the textbook so I'm getting most of it. There is one elephant in the room though, and I haven't been able to make office hours to clear it up.

Are there "imaginary" numbers other than I. We declared that i existed because it was useful (as we constructed negative numbers in real analysis because they were useful). We did some proofs to show that algebra would still work, then we constructed C using the real line and complex numbers of the form a+bi.

While jumping right to C will let you play with functions of complex numbers, the professor never really addressed if there are other "fake" numbers. Is there for example, a j=1/0 or other nonsense that adds dimensions to C? Could you then compute with something like a+bi+cj? If utility and computability is the standard for creating a number, it would reasonable that there should be other "complex families" of them out there.

 

[Office_Shredder]

The quaternions and octonions are extensions of the complex numbers that add extra square roots of -1. You can find articles describing these on wikipedia: They aren't nearly as nice as the complex numbers though (for example, multiplication is no longer commutative).

There's also something called the hyperreals, which adds infinitesimal and infinitely large elements to the standard real numbers. It doesn't defined 1/0 or anything, but just adds a new element which is larger than every natural number, and then its reciprocal is smaller than every 1/n. These are actually used to do calculus in a wholly different manner (dx is an actual infinitesimal, and then when you're done doing calculations you just round to the nearest real number)

This list probably isn't comprehensive of course

 

[Hurkyl]

The projective reals numbers are quite useful.


One of the topics of study in abstract algebra is the construction of commutative rings -- number systems with + and * (defined on any pair of numbers) that behave "similarly" to familiar arithmetic.

Two basic operations are:

• Construct a polynomial ring
• Impose an identity

For example, the usual construction of the complex numbers from the real numbers proceeds by first constructing a polynomial ring -- the number system consisting of all polynomials in one variable whose coefficients are real numbers. Let's say the indeterminate variable is 'i'. Once we have the polynomial ring, we impose the identity i²+1=0, and we get the complex numbers.


Some other examples are:
Starting with the integers, we might impose the identity that 3=0. The result is modulo-3 arithmetic.

Starting with the rational numbers, if we impose the identity 3=0, then we get what is called the zero ring -- a number system in which all numbers are equal.

 

[statdad]

You've received good references (very good) references for your questions, and I won't try to add to them. Instead...

you need to do away with the notion that these are "fake" numbers. . They are no more fake than the numbers with which you were familiar prior to your complex analysis class. Complex numbers (and the others mentioned) serve a need; they are a tool, but the notion of number, any number, is a creation of the human mind. calling one type fake is making a distinction that isn't valid. (the term "imaginary" makes this easier, unfortunately)

 

[PlasticOh-No]

You've received good references (very good) references for your questions, and I won't try to add to them. Instead...

you need to do away with the notion that these are "fake" numbers. . They are no more fake than the numbers with which you were familiar prior to your complex analysis class. Complex numbers (and the others mentioned) serve a need; they are a tool, but the notion of number, any number, is a creation of the human mind. calling one type fake is making a distinction that isn't valid. (the term "imaginary" makes this easier, unfortunately)

Your points are wrong, and if taken as is, they are self-contradictory. You say:
- Complex numbers are not "fake"
- They are a simply a tool to solve problems

- They most certainly are not real. They are fake. There is nothing naughty about admitting it. Kroenecker said that God made the integers, everything else is man-made. That having been said, the Mona Lisa is also man-made, but is of course an object of profound beauty, and a deeply meaningful representation.

- They are either profound, or they are base. I say they are profound. You say that they are a "tool," and that they "serve a need." This means they are simply used to manufacture things, then I would say you've utterly missed out. They are not screwdrivers or drills (or stone axes.) They have deep meaning.

 

[Office_Shredder]

Your points are wrong, and if taken as is, they are self-contradictory. You say:
- Complex numbers are not "fake"
- They are a simply a tool to solve problems

- They most certainly are not real. They are fake. There is nothing naughty about admitting it. Kroenecker said that God made the integers, everything else is man-made. That having been said, the Mona Lisa is also man-made, but is of course an object of profound beauty, and a deeply meaningful representation.

- They are either profound, or they are base. I say they are profound. You say that they are a "tool," and that they "serve a need." This means they are simply used to manufacture things, then I would say you've utterly missed out. They are not screwdrivers or drills (or stone axes.) They have deep meaning.

So you think that the rational numbers are fake numbers also? This post just obfuscating the point at hand: that the complex numbers are no different than the real numbers are when you get right down to it

 

[Hurkyl]

Sigh, it would be nice to have a number system that wasn't named by people with an agenda.

At least Gauss managed to get people to use "complex number" instead of "imaginary number", which does some good in preventing people from being tricked by language.


Anyways, people, let's keep to the original topic.

 

[BruceG]

I believe the reply on quaternions and octonions is worth expanding on here as this is illuminating.

There is a most beautiful theorem called Hurwitz Theorem (1898).

This asks the quesions how many algebras can we create which obey the composition law.

So we start with any n-dimensional vector space (x1,x2,...xn) with vector addition.
Then we add a norm so we can measure vector length (typically x1^2 + x2^2 + ... xn^2)
Then we try to define a multiplication (composition) of vectors which respects the norm.

That is N(z1.z2) = N(z1).N(z2)

In two dimensions this can be done in one and only one way thanks to the following square identity for real number:

(x1.y1 - x2.y2)^2 + (x1.y2 + x2.y1)^2 = (x1^2 + x2^2).(y1^2 + y2^2)

You can check the left hand side is what you get if you multiple z1=x1+i.x2 and z2=y1+i.y2
and take its norm.

Hurwitz theorem then goes on to argue that just as we went from R to C by adding a vector i normal to R, we can continue this 'doubling' process and still satisfy the composition law.

So we have:
Complex Number (2-dimensions) -> Quaternions (4-dimensions)
Quaternions (4-dimensions) -> Octonions (8-dimensions)

However at each doubling we lose something from our algebra:
C->Q : commutativity is lost x.y = y.x
Q->O : associativity is lost xy.z = x.yz
But Hurwitz showed that once associativty is lost we can't go on.
So we could create a 16-dimensional algebra called the sedenions (thats a word you won't see much), but this won't obey the composition law.

 

[Mentallic]

The quaternions and octonions are extensions of the complex numbers that add extra square roots of -1

Why were they invented? I can see that complex numbers were invented to solve polynomials such as x²=-1 but why quaternions and octonions?

 

[Hurkyl]

I can see that complex numbers were invented to solve polynomials such as x²=-1

Actually, I believe they were invented to explain how the cubic formula worked -- square roots of negative numbers were involved in the calculation of real numbers.

but why quaternions and octonions?

And I believe these were an intellectual exercise.

Of course, we know that both turned out to be quite useful since their invention.

 

[BruceG]

Why were they invented? I can see that complex numbers were invented to solve polynomials such as x²=-1 but why quaternions and octonions?

I hope my post shows that they were 'discovered' not 'invented'.
Pure mathematics is largely the search for structures with 'nice' properties.
As such Q and O are the natural extensions of R and C.

It is too early to really say to what extent physics makes use of Q and O, but there are some tantalising ideas here. Heres a couple of examples.

1. Superstring theory works in 10-dimensions, so when a superstring moves in 10 dimensional space-time the degrees of freedom of the string normal to the 2-dimension world sheet is 8-dimensional.

2. M-theory works in 11-dimensions, so if you want to directly get a realistic 4-dimensional space-time you need to compactify 7-dimensions. A nice way to do this is use a G2-manifold. The existence of G2-manifolds is intermately related to octonians.

 

[Tac-Tics]

Quaternions are the standard extension.

Also, check out dual numbers (http://en.wikipedia.org/wiki/Dual_number), which introduces a new dimension using the unit ε, which is a non-zero number which obeys the identity εε= 0. It is effectively an extension of the reals which introduces infinitesimal and infinite numbers.

But whether or not ε could be considered "imaginary" depends on how you define imaginary.

 

[statdad]

"But whether or not ε could be considered "imaginary" depends on how you define imaginary."

The Bugatti Veyron in my garage is imaginary.

 

[D H]

Your points are wrong, and if taken as is, they are self-contradictory. You say:
- Complex numbers are not "fake"
- They are a simply a tool to solve problems

How is that self-contradictory? All numbers are simply tools to solve problems. Our innate abilities are limited to cardinality. Crows know the difference between one and five, but not twenty and thirty. The same goes for humans, where the magical number seven (plus or minus two) is about as far as our primitive brain can go without symbols. Humans invented the counting numbers to overcome these innate limits.

Some algebraic problems had negative solutions. Mathematicians and accountants rejected those negative solutions as nonsense for a long, long time (up 'til the 17th century!). The acceptance of zero as a concept is also quite interesting. Some algebraic problems had non-integer solutions. Those were rejected for a long time, too (but at least they were accepted as being "rational" solutions). On the other hand, solutions to a problem that could not be expressed as a ratio of two integers were literally treated as absurd. We still call numbers like √2 "irrational", and the name this symbol, √, is "surd".

Rhetorical question: Are things like zero, negative numbers, rational numbers, and irrational numbers "fake"? Of course not. Neither are the imaginary numbers and complex numbers. "Imaginary" is yet another derogatory label, as are "negative" and "irrational." BTW, "real" is a back-formation coined after "imaginary" numbers started gaining acceptance.

Bottom line: Please drop the "fake" label. Numbering has more than enough derogatory labels already.

Why were they invented? I can see that complex numbers were invented to solve polynomials such as x²=-1 but why quaternions and octonions?

Hamilton envisioned the quaternions while on an evening walk. Lacking any writing materials, he scrawled some graffiti on a bridge, i²=j²=k²=ijk, just in case he forgot his insight on the way home. Acquaintances of Hamilton saw a way to extend things even further and quickly came up with the octonions.Aside: Have you ever wondered why physicists often use i-hat, j-hat, and k-hat as the canonical unit vectors in three space? The answer lies in the quaternion origins of vector analysis as used by physicists.

 

[PlasticOh-No]

by the way just wanted to note that my earlier post was not indented to seem argumentative, actually meant to be in the fashion of spirited discourse. In other words just trying to contribute.
Thanks