SpeedOfSound wrote:Is that last part about isomorphism or is that something else?

and so ughaibu's quaternions were not really on the same level as i?

Yes, an infinite field that isn't strictly bigger than C is isomorphic to C or one of its subfields.

I don't know much about quaternions, but I do know that in the quaternions multiplication isn't commutative i.e. A x B isn't the same as B x A. That means that, by definition, the quaternions aren't a field so the isomorphism result doesn't apply.

So am I safe in saying that real numbers are a sort of a 'collapse' of imaginary numbers. They add some new tricks to the mix. They are isomorphic under addition and multiplication but doesn't it stop there?

Intuitively they seem to extend the real number line into a plane.

I'm not sure what you mean. The reals are indeed isomorphic to the purely imaginary numbers and they are also a subfield of the complex numbers.

It's quite correct to say that another dimension gets added when we go from the reals to the complex numbers. The complex numbers are a 2 dimensional vector space over the reals so it's exactly like moving from a line to a plane.

susu.exp wrote:It means that exponents are well defined for all numbers. For a,b in R ab isn´t generally in R, but for a,b in C ab is in C as well.

I must be missing something. Isn't a real number raised to a real power always real?

logical bob wrote:

susu.exp wrote:It means that exponents are well defined for all numbers. For a,b in R ab isn´t generally in R, but for a,b in C ab is in C as well.

I must be missing something. Isn't a real number raised to a real power always real?

No, try e.g. (-1)^(1/2) or (-3)^(3/14) or whatever of that form.

If you restrict the base to positive numbers, you always can get a real solution, but if negative bases are permitted, there's no guarantee there'll be any real solution.

Here is how I, a lay person, think of imaginary numbers. Imaginary numbers are multiples of i where i = (-1)½
This means that i ½ = -1
Such multiples would be 7i or ¾ i

But no real number can be squared to produce a result of -1
But we can imagine that such a thing exists
What use is this?
It is a tool - a mathematical tool
Usually I think of it as an intermediate step in a calculation
We can allow i to crop up in a calculation, so long as in getting the final answer, we get rid of it again

Having said that, i can be cited in a specified quantity in isolation, (ie. not as part of any particular calculation), so long as I accept that the value is to be used as a tool in some calculation at some later stage.

i often comes with a coefficient, as in 7i
Also i may come with other numbers, as in i + 5 or 7i + 5
Such a term is called a complex number, because it has a real part, (eg. + 5) and an imaginary part, (ie. i)
In a complex number, the imaginary part has the qualities I mentioned above.

That's how I see it any way.

In direct answer to the OP:

At bottom, all extensions to the number system beyond the 'natural numbers' are human inventions, and there's an aphorism by the mathematician Leopold Kronecker to this effect.

In order to solve the equation x+a=0, where a is a natural number, we had to invent the negative numbers, and thus generate the integers.

In order to solve the equation mx+c=0, where m and c are integers, we had to invent the rational numbers.

In order to solve equations such as x2-2=0, we had to invent the irrational numbers. Or, more correctly, discover that between any two rational numbers, there exists at least one number that cannot be represented as a rational number. I'll spare you the distinction between algebraic and transcendental irrationals, because that's not needed here.

In order to solve equations such as x2+1=0, we had to invent the complex numbers.

However, once we had proceeded through those steps, and arrived at the complex numbers, something wonderful happened. The need to extend the number system beyond complex numbers ceased. Because it was found that the complex number system was immensely rich, unifying, and delivered results that were impossible wet dreams in the world of the reals. To begin with, the complex numbers allow the Fundamental Theorem of Algebra to be a genuine, rigorous theorem, without having to deal with "pathological cases" - the complex numbers make that theorem unified and simple, which it isn't in the reals.

Next, in the world of calculus, determining whether a function of the reals, other than the elementary functions, is differentiable (i.e., the derivative exists and is defined), is a seriously non-trivial task. A naive view would consider that a continuous function would be differentiable, but this is not necessarily the case. The Weierstrass function is an example of a function that is continuous everywhere, but which is nowhere differentiable. Now it so happens that the converse is relatively trivial - if a function is differentiable, then it is continuous. But simply determining differentiability of real functions, other than the elementary functions, is a taxing problem.

Move over to the world of complex functions, on the other hand, and a number of scintillatingly beautiful results arise quite naturally, once certain basic concepts are in place. For example, if one extends the concept of a continuous function in the reals, to the corresponding concept in the world of complex functions, then one arrives at the concept of an analytic function, and one of the marvellous results arising from the world of complex numbers is that any function that is analytic on the complex plane not only has a derivative, but, wait for it, is infinitely differentiable, a result that has no counterpart within the world of real-valued functions. More concisely, holomorphic functions (functions which are differentiable within a neighbourhood of any point for which the function is defined) are also analytic functions (i.e., they are locally representable by a convergent power series such as a Taylor series), and this result has powerful implications for the behaviour of complex functions, including the fact that what one might term 'well-behavedness' for complex functions implies stricter conditions than for real functions, and departures from 'well-behavedness' as it were, involve a range of well-defined and classifiable singularities.

Another powerful feature arising from the analysis of complex functions, is that if you only have a function defined for a subset of the complex plane, you can, in effect, 'stitch together' a tapestry of functions to extend the definition of the function of interest, provided that the functions in your 'tapestry' give identical values wherever individual subsets of the complex plane in your 'tapestry' overlap. This process is known as analytic continuation, and allows you to do wonderful things, such as extend the definition of the logarithm function to negative numbers, which is impossible in the reals.

Actually, you don't even need analytic continuation to do that for the logarithm, but with analytic continuation in place, the process is an absolute breeze, once the groundwork has been done and the supporting concepts mastered. In short, you can achieve results within the world of complex functions, that would lead to hair-tearing frustration if you tried to achieve the same results in the world of the reals, because the complex number system is immensely more rich in terms of structure. To learn why this is, you need to delve into some of the more rarefied aspects of modern algebra - in particular, some of the reasons why the complex plane, and functions thereupon, exhibit the lovely features that they do, requires you to go all the way into the wild, untamed landscape that is category theory, and I would NOT recommend that to someone with little more than high school mathematics behind them! But I digress. The fact remains, that the world of complex numbers allows a whole range of powerful results to be delivered, that would be difficult or even impossible to achieve in the world of real numbers.

Now, as an example of an application, of complex numbers, I have a lovely one for you. Erect a complex plane (represented by z = x+iy), and erect another complex plane (represented by w=u+iv). Now, on the z-plane, draw a unit circle (i.e, |z|=1). Now define a transformation from the z-plane to the w-plane, and see what happens to that unit circle when it is mapped from the z-plane onto the w-plane. This is a technique called conformal mapping, and one of the beautiful results arising from the world of complex numbers is that conformal maps are, in general, well-behaved - a powerful theorem by Riemann guarantees this, and guarantees furthermore that conformal maps sensu stricto, in the sense of transformations that preserve angles, will always exist. What use is this? Well, it transpires that one such transformation, w = z + 1/z, maps the unit circle onto a shape that looks very much like an aerofoil, and indeed, the resulting shape is, in effect, a mathematically perfect aerofoil known as a Joukowski aerofoil. Other transformations can be devised to produce other aerofoil shapes with various desirable characteristics, which means that when you board a modern airliner, chances are someone has been running some conformal maps through a computer in order to design the aerofoil cross section of the wings. What makes this even more beautiful, is that the behaviour of fluid flow around a body with a circular cross-section is well-known and well-understood, and the details of that fluid flow can be transformed using the same transformation, in order to produce an analysis of the behaviour of fluid flowing over the aerofoil. Which is one reason why aircraft designers are pretty sure that their planes will fly, because they know how their aerofoils behave, courtesy of work such as this.

Zwaarddijk wrote:No, try e.g. (-1)^(1/2) or (-3)^(3/14) or whatever of that form.

Indeed.

logical bob wrote:Don't get hung up on asking what complex numbers are. Do you feel confident about what negative numbers are?

To put that comparison into a historical context, I like to point out that negative numbers and imaginary numbers entered modern European mathematics in the 15th century and for exactly the same reason: they are notational devices which make it much easier to solve quadratic, cubic and quartic equations.

I stress "notational devices", because at this time, neither negative numbers nor imaginary numbers made any sense. Only positive numbers did. For thousands of years, mathematics boiled down to geometry, and numbers boiled down to the lengths of line segments. There's no such thing as a line segment with negative length, so there is no such thing as a negative number. And thus, those who used them gave them derogratory labels such as "imaginary" and "false." Even Descartes was calling negative numbers "false" when he used them to describe coordinate systems.

In the 18th century, geometry lost its grip on the foundations of mathematics, and the use of convenient notational devices became more accepted. Then, in the 19th century, everything became notational device as mathematics took an ever abstract turn. Nowadays, we can often just think about the different kinds of numbers in terms of the structures they form under operational rules, so the question of what imaginary numbers are is well out of scope.

Thanks Vaz. Excellent stuff as always.

The 'magic' of complex numbers is that, just by defining the imaginary number i, a set of numbers is created with an astoundingly rich structure. As mentioned, complex numbers are algebraically closed, and they reveal surprising connections like Euler's famous formula.

The power of complex numbers becomes very clear in calculus, with the concept of holomorphic functions, Cauchy's integral formula and the very important Residue theorem. The latter is a central tool to calculate integrals that would be much more difficult to do with real numbers alone. A classic example is



This can be shown by extending the real integral to the complex plane and using the residue theorem.
Another application is the solution of integral equations, which pop up frequently in physics. To solve these, one usually applies a transform to the complex plane, like a Laplace transform or a Mellin transform, then solves the simplified equation, and applies the inverse transform, which involves complex integrals.
The inverse Mellin transform is actually very interesting, because almost every special function (gamma, elliptic, bessel, hypergeometric,...) can be written as an inverse Mellin transform, in the form of a Meijer G-function.

Finally, complex numbers are essential in quantum physics.

Another example: in high school, I wondered where identities like



came from. Later, when I learned de Moivre's formula,



it all made sense.

As for hypercomplex numbers, like the quaternions and octonions, those are much less useful: they don't add much extra 'richness'; in fact, quaternions are no longer commutative, and octonions aren't even associative. Still, there are applications, like the Kustaanheimo-Stiefel transform.

logical bob wrote:Don't get hung up on asking what complex numbers are. Do you feel confident about what negative numbers are? The point isn't so much that complex and negative numbers correspond to something out there in the real world as that they give us a language that can be used to talk about the real world.

I'm not really clued up on the specific real world applications. Pure maths was always my thing. In any situations where you're dealing with polynomial equations, equations like

x3 + 2x2 - 3x + 1 = 0

you're dealing with complex numbers. If you only allow x to be a real number then some polynomials don't have any solutions. At the simplest level

x2 + 1 = 0

will fail. All such failures come down to the problem of negatives having square roots, so if x can be complex there is always at least one solution. The up yourself way of putting this is that the complex numbers are the algebraic closure of the real numbers.

Pure mathematicians don't like special cases and being unable to do stuff so they're naturally drawn to working in the complex numbers for the sake of generality. Also, calculus using complex numbers is actually really nice, in the sense that it works out far more neatly and aesthetically satisfying than calculus with real numbers. There are general results which hold that simply don't work out in the reals.

Thank you!

If complex numbers are complete then
consider

x^2 = -i

whats x?

CarlPierce wrote:If complex numbers are complete then
consider

x^2 = -i

whats x?

x = √2/2 - i√2/2 or -√2/2 + i√2/2

better written as x= +/- (1-i)√2/2
To see why it works:

((1-i) * √2/2)2

= (1-i)2 * (√2/2)2

= (1-i)(1-i) * 2/4

= (1 - i - i +i2) * 1/2

= -2i * 1/2

= -i

logical bob wrote:

CarlPierce wrote:If complex numbers are complete then
consider

x^2 = -i

whats x?

x = √2/2 - i√2/2 or -√2/2 + i√2/2

better written as x= +/- (1-i)√2/2
To see why it works:

((1-i) * √2/2)2

= (1-i)2 * (√2/2)2

= (1-i)(1-i) * 2/4

= (1 - i - i +i2) * 1/2

= -2i * 1/2

= -i

Thx I got this after I'd posted.....dam.

So is it true that a complex number X = a + bi exists for all real A,B
where X^2 = A + Bi

I suppose it must be as you can represent any complex number as a rotation + a distance from the origin on the complex plane. So every complex number must have a complex square root.

CarlPierce wrote:So is it true that a complex number X = a + bi exists for all real A,B
where X^2 = A + Bi

Yes.

X2 = (a + bi)2 = a2 - b2 + 2abi

This gives us a pair of simultaneous equations:

a2 - b2 = A and 2ab = B

which can always be solved to find two values for the square root you're looking for.