[math] "Obtaining" complex numbers through abstract algebra

[fonz]

If you are familiar with complex numbers, they were most likely first introduced to you as an extension of the set of real numbers, something along these lines:

• With natural numbers one can solve the equation x+1=2, but not x+2=1.
• By extending the natural numbers with negativity, resulting in the set of integer numbers, one can solve x+2=1. One can also solve 2x=4, but not 2x=1.
• By extending the integer numbers with fractions, resulting in the set of rational numbers, one can solve 2x=1. One can also solve x^2=4, but not x^2=2.
• By extending the rational numbers to the set of real numbers (by adding algebraic numbers, transcendental numbers and perhaps more?) one can solve x^2=2, but not x^2=-1.
• By extending the real numbers with the imaginary unit i, one can solve x^2=-1 (as well as do some other neat new tricks).

When introduced in such a way, the concept of the number i appears somewhat artificial or even magical: "i is some funky abstract number such that i^2=-1. Don't ask how they came up with that idea, it just is - and it works." However, I remember having once seen a mathematician "construct" the complex numbers in a very elegant and natural way, using group theory. Unfortunately, I can't for the life of me remember exactly how he did that and I can't find it on the Web or in any book I own either. I suspect it had something to do with polynomial rings or field theory or something like that, but I'm not quite sure.

Is there anyone here who is sufficiently well-versed in complex numbers and group theory and who would care to enlighten the dumbass that is me?

 

[cpm@]

It was Caspar Wessel, but according to the book "An Imaginary Tale: The Story of [the Square Root of Minus One]", page 53, by Paul J. Nahin reads:

Historians generally credit Wessel with being the first to associate an axis perpendicular to the real axis with the axis of imaginaries. However, there are indications that this idea, just before Wessel, was one whose time was ripe.

 

[Crivens]

I clicked on this thread and expected some fishing expedition for some homework, but I found something really interesting. Well done!

 

[SirDice]

I think I first got to work with complex numbers when I had to calculate resonance, phase shift and other values with electronics. Mostly capacitors and coils.

The fun thing about them is that if you didn't have i there would be no sane way to calculate some of these things. Just because the intermediate answers are imaginary, it doesn't mean the final answer has to be

 

[Crivens]

Just because the intermediate answers are imaginary, it doesn't mean the final answer has to be

In contrast to global finance and banking. (!S)CNR

 

[RichardM]

If I'm remembering correctly... you can construct the complex numbers from the real numbers by defining the complex numbers as ordered pairs (x, y) of real numbers, and then defining addition, multiplication etc, proving all the group and field properties, and probably ending up with continuity and differentiability. But I don't have a reference - long time ago!

Then i just happens to be defined as (0, 1) and you can show (0, 1) * (0, 1) = (-1, 0), i.e., i^2 = -1.

 

[kpa]

The pairs or real numbers (x,y) that represent the complex numbers are visualized as points in the complex plane, that's the crux of the biscuit

 

[SirDice]

The pairs or real numbers (x,y) that represent the complex numbers are visualized as points in the complex plane, that's the crux of the biscuit

Aren't they called vectors? Two dimensional in this case.

 

[fonz]

Historians generally credit Wessel with being the first to associate an axis perpendicular to the real axis with the axis of imaginaries.

That's the geometrical interpretation of complex numbers. I know that already and it's not what I mean. The construction I witnessed was entirely algebraic, using group(/ring/field) theory. But thanks for the history lesson, though.

expected some fishing expedition for some homework

Homework? Thanks, but I'm not that young anymore

I think I first got to work with complex numbers when I had to calculate resonance, phase shift and other values with electronics.

I remember impedance being complex as well.

Just because the intermediate answers are imaginary, it doesn't mean the final answer has to be

Exactly. Take solving cubic equations for example. A cubic (with real coefficients) may have up to three real (and distinct) roots, but as far as I know there's no way to find them without at least temporarily using complex numbers. And another prime example is of course differential equations.

If I'm remembering correctly... you can construct the complex numbers from the real numbers by defining the complex numbers as ordered pairs (x, y) of real numbers, and then defining addition, multiplication etc, proving all the group and field properties, and probably ending up with continuity and differentiability.

Thanks, that's more along the lines of what I was looking for. I'm not sure it's exactly what I witnessed, but I'm going to try it out. Thanks again.

 

[mix_room]

I remember impedance being complex as well.

The great thing about complex values of impedance is that you end up with complex values for the power. So now you have real power and imaginary power flowing through the cables.

 

[Crivens]

I have seen many uses for complex numbers. But all that pales against two 'simple' things - Mandelbrot and Julia Sets. That gets you hooked, and then you end up in sitting in lectures about theory of conductors, or signal processing, pattern detection and suddenly you have a degree. And then you wonder what got you started

Oh, and of course: e^(i*PI)=0

 

[Daisuke_Aramaki]

Oh, and of course: e^(i*PI)=0

It should be
e^(i*PI)+1=0

 

[fonz]

I have seen many uses for complex numbers. But all that pales against two 'simple' things - Mandelbrot and Julia Sets.

I think I know what you mean Math has a reputation for being boring, but I can't think of anything demonstrating the beauty of mathematics better than fractals.

It should be
e^(i*PI)+1=0

I once heard someone say that the above is the coolest formula in mathematics because it expresses a relationship between the five most important numbers: 0, 1, pi, e and i.

 

[Crivens]

Argh, got me. I shall derive e^x untill it is zero.

And I heard of one teacher who claims to teach arts because he lacked the imagination for mathematics - too true.

 

[cpm@]

The moniker "imaginary" is unfortunate, since in one sense, all numbers are imaginary in that they exist only in our minds. But, in mathematics, there is a distinction made between real and imaginary numbers. The real numbers are those that show up on the
number line. Imaginary numbers arise because mathematicians could not find a solution to the equation x^2+1=0 in the set of real numbers. So, they decided to designate the square root of negative one by the small letter case i.

William Rowan Hamilton found another way to express complex numbers where he never had to introduce the word "imaginary". Hamilton's solution was to expand the definition of number, he decided that our ordinary "real" numbers are a subset of a larger set of numbers that are referred to as "ordered number pairs", and written (a,b), in which a and b are positive or negative numbers, including zero (in other words, in which a and b are real numbers).

The rules of arithmetic must be altered for ordered number pairs. Letting letters represent real numbers:

• Sum operation: (a,b)+(c,d)=(a+c,b+d)
• Subtraction operation: (a,b)-(c,d)=(a-c,b-d)
• Multiplication operation: (a,b)*(c,d)=(ac-bd,ad+bc)
• Division operation: (a,b)/(c,d)=((ac+bd)/(c^2+d^2),(bc-ad)/(c^2+d^2))

The ordered number pair (a,b) is equivalent to the complex number a+ib. That is, if b is zero, then (a,0) and a+i0 behave algebraically as the same "real" numbers. If a is zero, then (0,b) and 0+ib behave algebraically as the same "imaginary" numbers. Finally, if neither a nor b is zero, (a,b) and a+ib behave algebraically as the same "complex" numbers.