Why do we need the set of complex numbers to solve?

[lavinia]

What is intuitive seems to refer to what can be visually seen or imagined and circular motion, (rotation through an angle), is intuitive as is changing the length of a measuring rod that is pointed in some direction. These two operations, change of length and rotation through an angle, are the two components of complex multiplication. If one thinks of i as rotation through 90 degrees, i is just as intuitive as 1 which is rotation though zero degrees or -1 which is rotation through 180 degrees.

So if real numbers are intuitive or "real" so are the complex numbers.

What is less intuitive - at least to me - are the real numbers themselves. For instance,one can think of rational lengths as the total of all possible divisions of integer lengths (some number of measuring sticks laid side by side)using ruler and compass constructions. After a bit of thought one can see how one might go about making one that you are given, that is you could define a mathematical procedure for making it. This involves a high level of mathematical abstraction. It is not intuitive but rather deductive or at least requires deduction before a visual picture can be achieved if even possible.

If one extends the idea of ruler and compass constructions to the plane, one gets some irrational numbers and again with some thought one can imagine how one might go about constructing such a number e.g. the square root of 2. But again we are in a world of abstraction and mathematical procedures. What does it even mean to say that one can intuit the number sqrt( 2 + sqrt( 2 + sqrt(2))/27?

Gauss showed that all linear lengths that can be constructed with a ruler and a compass are iterated square roots but this does not come close to all of the real numbers. There are even lengths of regular polygons that can not be constructed in this way e.g. the length of the side of a regular septagon. Is this length intuitive?

Mathematicians conceived of the real numbers in various ways but not using algorithms or intuitions. Rather they came up with the idea of the metric completion of the rational numbers. But one can not intuit these numbers because there are uncountably infinitely many of them. There is no procedure even to generate any except a paltry countable subset.

There is a subfield of the real numbers for which there do exist algorithms, that is if a computer were allowed to run forever it would be able to express the number in terms of limits of rational numbers. These are called computable numbers. For instance the square root of 2 and π are both computable. One might say that in principle a number is "real" or worse "intuitive" if it is computable. There are only countably many computable numbers so this excludes almost all real numbers.

 

[sophiecentaur]

What is intuitive seems to refer to what can be visually seen or imagined and circular motion, (rotation through an angle), is intuitive as is changing the length of a measuring rod that is pointed in some direction.

It all depends upon who you are dealing with here. I think you should look at this.

 

[mathwonk]

lavinia of course is right. The difficult concept is the real numbers. the complex numbers are an almost trivial extension. but to go from rationals to reals, that is a tour de force. With reference to another thread asking for the greatest mathematicians, I think this raises the name of the seldom mentioned genius Eudoxus very high.

 

[xAxis]

lavinia of course is right. The difficult concept is the real numbers. the complex numbers are an almost trivial extension. but to go from rationals to reals, that is a tour de force. ...

I just can't see this. I would think real numbers were "intuitive" at least from the time of Zeno and his motion paradoxes. For Greeks mainly geometry was the mathematics. I would think once a Greek mathematician draw the number line with his ruler, some idea of continuum would (or should have been) there. Pythagoras knew that sqrt of 2 was not rational, and yet he could place that number on the line. It was easy for them to "see" that 4/3 is bigger than 5/4, which is bigger than 6/5 an so on. On the other hand they measured weights, volumes and other things and expressed them with the same numbers. So I think that first intuition about numbers was something like "Bigness."
Sure it's easy to visualise complex numbers by varying length and angle, but had not complex numbers been already invented, could you think of them by just looking at varying length and angle?
And let me be the first one in this insightful thread to say that I like the name 'imaginary' for imaginary numbers. I couldn't think of better name today, let alone in time they were invented.
Sophie great insight into the cause of great variety of terms for a group of objects. This morning in my backyard I saw a murder. (And later a glaring) :)

 

[mathwonk]

well look at it this way. a complex number is just two real numbers (a,b) with multiplication defined by (a,b).(c,d) = (ac-bd, ad+bc). That's it. You will notice that (0,1).(0,1) = (-1,0). So if you regard the subset of real numbers as the pairs of form (a,0), then (0,1) is the square root of (-1,0).

But to describe a real number it takes an infinite sequence of rational numbers. And it has to be a "Cauchy sequence", so a sequence {xn} of rartional numbers defines a real number if and only if for every e>0 there is a positive integer K such that for every n,m > K, we have |xn-xm| < e.

Then two Cauchy sequences {xn}, {yn} define the same real number if and only if for every e>0, there exists a positive integer K such that whenever n,m > K we have |xn-ym| < e.

So a real number is an equivalence class of Cauchy sequences of rationals. Then you have to define the ordering on them and prove the basic "least upper bound" property for them, that for every non empty, bounded above, set of reals, there exists a unique least upper bound. All this is very tedious and lengthy.

Isn't that more complicated? I.e. going from rationals to reals is very sophisticated, even if it was understood by the ancients to some extent, but going from reals to complexes is pretty simple, once you get over the psychological barrier of not wanting to believe in them. And exhibiting a complex square root of -1 is easy, we just wrote it down. Admittedly the fundamental property of complex numbers is not so easy, that every equation of positive degree has a complex root. But defining them is much easier than defining reals.

 

[FactChecker]

I'd like to elaborate a bit on the point that complex numbers do have a nice graphical interpretation. Set $a + ib = (a,b)$, a point in the Argand plane. Multiplying by $i$ rotates the position vector for this point counterclockwise by 90$^{\circ}$. More generally, using the polar forms, multiplying two complex numbers gives

$z_1z_2 = r_1e^{i\theta_1}r_2e^{i\theta_2}= r_1r_2e^{i(\theta_1+\theta_2)}$,

So $z_1$ stretches $z_2$ by amount $r_1$ and rotates it by an amount $\theta_1$. There's a delightful book that shows the unfolding of complex analysis from this geometrical perspective, Needham's Visual Complex Analysis.

I second the recommendation of Needham's book. The definition of multiplication that gives you rotation in the complex plane is very important. The usual XY plane does not have a basic rotation operation unless matrices are introduced.

 

[Allin]

The set of complex numbers should be taught at the primary /elementary school. The radical of integer -1, is an operator. Taken 4 times as a factor: the product is integer +1.

 

[chad kimsey]

If you just touch the fundamentals of mathematics, as well as any subject really, you just get to see the ice above the water. When in reality, the majority of the iceberg is under the water and you can't see it yet.

I don't know how far you have studied, but this same question stumped me all the way through high school and up to a point in college. Even at differential equations, I know complex numbers are necessary but why are they "imaginary"? And, what good is it really? So you solve a cubic equation, who cares right? Why would you ever need this crap??

The answer to that question is deeper at the fundamental understanding, the root of how things work and why. Going as far as engineering, creating things from scratch, modeling any world situation with blocks of differential equations impossible to solve by hand...understanding that level is awesome. However, it still never truly hit me until physical chemistry my senior year as a chemical engineer. This was by far the most outrageous, difficult class I ever had (and I'm really good with math/engineering way of thinking). Guess what popped up? Complex numbers. They don't make sense unless you go into quantum mechanics, where they are essential to understanding anything in the atomic world. Touching on the Schrödinger Equation, movement and behavior of electrons, and describing MO theory all use complex sets. It still blows my mind how "imaginary" numbers were created to describe things that can't normally happen fit perfectly in the world of the atom.

And one other thing bothers me, I try not to think about it but I can't help myself...is why is there two separate sets of rules, equations, physics, whatever you want to call it, that describe the nano and macro worlds? If everything is made of atoms, then why does it behave according to completely different laws? Why do properties and models fall apart, and exactly where does that happen? Gather enough mass, and you have gravity warping the space-time around the body. All that mass is a collection of endless atoms...but in the atomic world there is no gravitational force. Whats that all about? I love science, the more I learn the more I want to know

 

[Allin]

In any equation of one unknown value, of the first degree, the implicite coefficient 1 is identifiable to: (square root of -1) exponent 0. The first fudamental theorem of algebra should be superficially shown to the pupils at the last quadrimester of primary studies. In our world, rondness and circular movements are omnipresent and in diverse ways, thus the operator "square root of -1" is usefull and most practical for measuring and evaluating distance, center of mass, instant accelation, energy-productive impact, collapse critic point, etc. Generally speaking, wherever the infinitesimal (afterwards called "differential & integral") calculus is concerned, and its offspring mathematical subjects as ODEs (ordinary differential equations), PDEs, integral équations, etc, the complex set of numbers _imaginary cousins of the set of real numbers_, this operator is ubiquous i.e. most necessary in the calculations /computations. In a 3D cartesian coordinates system, the course of the extremity of an arrow of wrist watch, moreover the course of the lone electron tourning around the core of an atom of hydrogene, moreover the random-like traveling of an abandonned debris of satellite moving marygoround our planet Earth, could be computed and drawn with formulas containing the operator "square root of -1". ODEs are so difficult to compute /solve that even the most powerfull computers cannot succeed, just can make successive approximations (thence the reniewed methods of making approximations, called "numerical analysis", compulsory in the baccalaureate in engineering' (almost nothing to do with infinitesimal calculus analysis that is in the curriculum of major-level fundamentals mathemathics). The studies of engineering in physics, includes 3 to 6 credits in numerical analysis /methods.