Solving the Mystery of i & j: The Square Root of -1

[Cheman]

Square root of -1...

We say that the square root of -1 is equal to i ( or j ), and that this is therefore not a real number - but what is this fact actually useful for? Why over complecate things with something not on the number line - why is it so useful to treat the square root of -1 as a number?
Thanks in advance.

 

[whozum]

Because it is one, mathematics in the complex plane is a branch of its own and has proven to be important even in the physical sense. The two branches of numbers are real and imaginary, but both are equally numbers.

 

[HallsofIvy]

For one thing it simplifies enormously sin(x) and cos(x) so that they can be written as exponentials. That allows engineers to use the much simpler exponential function to represent waves rather than sine and cosine.

 

[dextercioby]

Even though the introduction of complex (imaginary) numbers in mathematics happened when 3 italian fellows were studying the 3-rd order algebraic equation with integer coefficients,i'm sure they were really happy to say that the equation

x^{2}+1 =0 has solutions.Two of them.

And then claim that every possible polynomial of 2-nd,3-rd & 4-th degree has roots.

Daniel.

 

[uart]

Cheman, one simple answer to your question is to point out that for many mathematical problems that involve only real inputs and only real outputs that the simplist solution method can involve complex numbers at intermediate steps.

 

[Hurkyl]

Another interesting fact for those who prefer to live in the reals...

There exist real numbers that can be expressed through the application of arithmetic and n-th roots to integers, but there absolutely, positively must be complex intermediate values to do so.

 

[εllipse]

Another interesting fact for those who prefer to live in the reals...

There exist real numbers that can be expressed through the application of arithmetic and n-th roots to integers, but there absolutely, positively must be complex intermediate values to do so.

Could you elaborate or provide a link? I'm curious.

 

[Gokul43201]

but what is this fact actually useful for?

Complex numbers are useful for a whole lot of stuff and essential for most of the following - here's a few randomly selected choices :

1. AC electrical theory
2. Vibration analysis
3. Number Theory (to study something even as "real" as the distribution of the prime numbers)
4. Quantum Mechanics, Field Theories and Relativity
5. Digital Signal Processing

 

[Hurkyl]

Unfortunately, I can't. I think I first heard the fact when we were going over Galois theory in my algebra class, but I can't remember exactly how it worked.

 

[Nomy-the wanderer]

It worked well for me in ac circuits...

When iw as at school and i asked my teacher about it, he told me it was practically usefull for engineers in everything, mostly for prediction..

 

[robert Ihnot]

Nobody saw fit to mention, that one of the very first reasons "i" was useful is because of this kind of mistake:

\sqrt-1*\sqrt-1=\sqrt{-1*-1}=\sqrt1

 

[Hurkyl]

The mistake there is that √(-1) is not defined in the reals. So, if you're living in real number land, using √(-1) in an arithmetic expression is exactly as meaningless as doing division by zero.

 

[robert Ihnot]

HurkyL: Unfortunately, I can't. I think I first heard the fact when we were going over Galois theory in my algebra class, but I can't remember exactly how it worked.

Well, what about this? Define \omega=\frac{-1+\sqrt-3}{2}, a cube root of 1, and let \omega' represent it conjugate. Then define Y as:

Y = (\omega)^\frac{1}{3} +(\omega')^\frac{1}{3}

If we cube this value we get the equation Y^3-3Y+1=0. An equation with three real roots.

 

[Hurkyl]

The problem is that I don't remember how to prove that there isn't some other method of constructing Y that does not involve complexes.

 

[robert Ihnot]

Hurkyl: The problem is that I don't remember how to prove that there isn't some other method of constructing Y that does not involve complexes.


Evidently not, from what I am reading: "Well, there: you've got an answer using the tools of Galois theory and solvability. Note that there isn't a single reference to the reals or complexes. They are not really part of Galois theory, although they can be incorporated into the discussion if one wishes. Most of us usually don't wish to do so; I take it you do.

Very well then, your question seems to be, "Is there always a sequence of fields F_i _all of which are contained in the real numbers_ which have the properties (1)-(3) stated above? (That's not what "solvability" means in Galois theory, though!) To this question, the answer is no -- in fact, never, for irreducible cubics. Obviously condition (2) would require all three roots to be real, but that is the so-called "casus irreducibilis" which seems to prompt your query. A cubic with three real roots which factors in F_{i+1} would also factor in F_i. This is an exercise in Garling's Galois Theory, p. 138.

I'm not sure what the point of this is. It is easily proved that the roots of (say) x^3-3x-1 are all real, and can be expressed in terms of radicals like (1/2 + i sqrt(3)/2)^(1/3) + (1/2 - i sqrt(3)/2)^(1/3). This is an exact expression involving complex numbers whose imaginary part is clearly exactly zero. It may be manipulated like any other complex number. The result I quoted you above shows that its real part cannot be expressed using a finite number of radical operations applied only to real numbers; so? Sounds to me like a good reason to use the complex form -- this way we have something to write down. If you don't like this form, you may use the trigonometric forms; I hardly think these are "better" but of course may be better suited to some particular purposes. Chacun a son gout.

Dr. Rusin calls the above solution (for one of the three roots) a solution in "complex radicals" and declares that it qualifies as "a solution in radicals" according to the Galois theory. Such a solution would not be accepted here or in any other shop that seeks practical solutions to practical problems. In this case the solution given cannot be expressed in terms of real radicals because the given equation is an irreducible cubic according to the definition given on the previous page. The root can be expressed in trigonometric terms, namely 2cos(π/9). This is an exact expression because it can be evaluated to any desired degree of accuracy." http://www25.brinkster.com/ranmath/misund/poly02.htm

 

[Hurkyl]

As I recall, this fact was one of the major motivations for accepting the complexes in standard mathematics. Mathematicians of the time were happy enough to say that quadratics that had no real solutions simply didn't have a solution -- they saw no point in accepting the complexes simply so they could say all quadratics had a root.

But the cubic and quartic formulas changed that -- even if you only cared about real roots, you still needed to take a tour through the complexes to get there.

Historically speaking, this result was significant.

 

[Manchot]

On a more general level, what exactly makes the set of complex numbers more useful than the set of reals? Is it the fact that they're algebraically closed?

 

[Icebreaker]

What exactly does the "hypercomplex" numbers not satisfy that the reals and complex do?

 

[lurflurf]

What exactly does the "hypercomplex" numbers not satisfy that the reals and complex do?
The main thing they lack is the hypercomplex nombers are noncommutative.

On a more general level, what exactly makes the set of complex numbers more useful than the set of reals? Is it the fact that they're algebraically closed?

Being Algebraically closed is quite nice.
x^2+1=0 has solutions.
Outside of that poles at complex points often help explain "mysterious" behaviour on the real line. Like why the Taylor series of 1/(1+x^2) about 0 diverges if |x|>0. Many integrals, products, and sums are easy to compute using residue theory. Inverting integral transforms such as those of Laplace is sometimes easier using the complex inversion formula than the real one. It is often nice (as has been mentioned) to allow intermidiate steps or small errors to involve complex numbers. Many formulas look nice using complex numbers. Like (cos(x)+i sin(x))^n=cos(n x)+i sin(n x) for example. Some problems in R^2 are nice when expressed in C^1. Many time a method is general with complex numbers, but several cases must be dealt with over reals. For example consider the differential equation y''+1=0 if complex variables are used the solution is in the form C1 exp(r1 x)+C2 exp(r2 x) where r1,r2 are roots of r^2+1=0. This is similar to what happens with y''-1=0, but if complex numbers are not used we have to treat y''+1=0 as a special case and use sine and cosine. Also many other equations can be solved like sin(x)=17 and exp(x)+1=0.

 

[lurflurf]

Hurkyl: The problem is that I don't remember how to prove that there isn't some other method of constructing Y that does not involve complexes.


Evidently not, from what I am reading: "Well, there: you've got an answer using the tools of Galois theory and solvability. Note that there isn't a single reference to the reals or complexes. They are not really part of Galois theory, although they can be incorporated into the discussion if one wishes. Most of us usually don't wish to do so; I take it you do.

Very well then, your question seems to be, "Is there always a sequence of fields F_i _all of which are contained in the real numbers_ which have the properties (1)-(3) stated above? (That's not what "solvability" means in Galois theory, though!) To this question, the answer is no -- in fact, never, for irreducible cubics. Obviously condition (2) would require all three roots to be real, but that is the so-called "casus irreducibilis" which seems to prompt your query. A cubic with three real roots which factors in F_{i+1} would also factor in F_i. This is an exercise in Garling's Galois Theory, p. 138.

I'm not sure what the point of this is. It is easily proved that the roots of (say) x^3-3x-1 are all real, and can be expressed in terms of radicals like (1/2 + i sqrt(3)/2)^(1/3) + (1/2 - i sqrt(3)/2)^(1/3). This is an exact expression involving complex numbers whose imaginary part is clearly exactly zero. It may be manipulated like any other complex number. The result I quoted you above shows that its real part cannot be expressed using a finite number of radical operations applied only to real numbers; so? Sounds to me like a good reason to use the complex form -- this way we have something to write down. If you don't like this form, you may use the trigonometric forms; I hardly think these are "better" but of course may be better suited to some particular purposes. Chacun a son gout.

Dr. Rusin calls the above solution (for one of the three roots) a solution in "complex radicals" and declares that it qualifies as "a solution in radicals" according to the Galois theory. Such a solution would not be accepted here or in any other shop that seeks practical solutions to practical problems. In this case the solution given cannot be expressed in terms of real radicals because the given equation is an irreducible cubic according to the definition given on the previous page. The root can be expressed in trigonometric terms, namely 2cos(π/9). This is an exact expression because it can be evaluated to any desired degree of accuracy." http://www25.brinkster.com/ranmath/misund/poly02.htm

That quoted page has a small error it interchanges x^3-3x+1=0 and x^3-3x-1=0.
Clearly as they are odd functions the roots of one are the additive inverses of roots of the other.
i.e.
2cos(pi/9) is a root of x^3-3x-1
-2cos(pi/9) is a root of x^3-3x+1
Also care must be taken as the roots obtained by using particular cube roots do not correspond.

 

[Icebreaker]

What exactly does the "hypercomplex" numbers not satisfy that the reals and complex do?
The main thing they lack is the hypercomplex nombers are noncommutative.

Can you give an example?

 

[matt grime]

yes., since the first thing one learns about the hypercomplexes is that they are thigns of the form a+bi+cj+dk where... ij=-ji... (other rules omitted)

 

[robert Ihnot]

lurflurf: That quoted page has a small error it interchanges x^3-3x+1=0 and x^3-3x-1=0.

I noticed that, and I did not make that error myself on the previous page where I said:
If we cube this value we get the equation Y^3-3Y+1=0. An equation with three real roots.

 

[robert Ihnot]

Hurkyl: But the cubic and quartic formulas changed that -- even if you only cared about real roots, you still needed to take a tour through the complexes to get there.

Rafael Binbelli wrote a book in 1572: He then showed that, using his calculus of complex numbers, correct real solutions could be obtained from the Cardan-Tartaglia formula for the solution to a cubic even when the formula gave an expression involving the square roots of negative numbers.

He is credited as the first person to give the rules for the arithmetic of complex numbers, and was considered ahead of his time. http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Bombelli.html

 

[robert Ihnot]

Hurkyl: But the cubic and quartic formulas changed that -- even if you only cared about real roots, you still needed to take a tour through the complexes to get there.

Rafael Binbelli wrote a book in 1572: He then showed that, using his calculus of complex numbers, correct real solutions could be obtained from the Cardan-Tartaglia formula for the solution to a cubic even when the formula gave an expression involving the square roots of negative numbers.

He is credited as the first person to give the rules for the arithmetic of complex numbers, and was considered ahead of his time. http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Bombelli.html

Footnote: I would like to add that Cardan published his work on the cubic in 1543, so there was about 30 years between this and Binbelli's work.

Cardan at times showed an interest in imaginary numbers, posing the problem: If X+Y=10 and XY=40, find the numbers. However, he dismissed the result calling it: As subtle as it is useless.
http://www.und.nodak.edu/instruct/lgeller/complex.html

 

[mathwonk]

read the thread above: importance of complex analysis.

 

[pamichel]

I think the basic reason for inclusion of 'i' is that such a root of -1 is not possible. Let's see how: 25 = 5*5; So square root of 25 is 5. Now take an example of -25. How will you find out the root of -25? we know, -25 = -5*5. So what is the square root of -25? It could have been -5 only if the multiplication rules allowed us to keep the products of two negative numbers as negative. So we cannot violate the multiplication rule. That is why we call the square root of -1 as 'imaginary' or not possible in real world. So we invent imaginary axis "i - axis" to represent these 'imaginary' roots or co-ordinates.

 

[matt grime]

"real world"? come off it, i is no more imaginary than e, pi or anything else in mathematics.

 

[Borogoves]

"real world"? come off it, i is no more imaginary than e, pi or anything else in mathematics.

So I assume you are a platonist then. But in any case, the real numbers, irrespective of how they are constructed, closely represent the real world, and that was the reason the system was adopted primarily.

 

[matt grime]

i am abslotuely not a platonist, you cannot deduce that from what i said and besides pplatonism has nothing to do with it: whether i think these things have an indepednet physical existence in soem other realm doesn't affect the fact that they have none in this.

i said i and e are equally "real". onotologically they are practically equivalent. complex numbers, being a divisoin structure on R^2 are equally as "real" as the real numbers since they onlyu require the reals, and the basic properties of sets for their definition. and i is incredibly useful in "the real world".

point out something in the "real" world that is e in a way that i cannot be expressed. labellign the reals real and the vomplexes imaginary speaks only of our psychology not our mathematics.

 

[Borogoves]

i am abslotuely not a platonist, you cannot deduce that from what i said. i said i and e are equally "real". onotologically they are practically equivalent. complex numbers, being a divisoin structure on R^2 are equally as "real" as the real numbers since they onlyu require the reals, and the basic properties of sets for their definition. and i is incredibly useful in "the real world". point out something in the "real" world that is e in a way that i cannot be expressed. labellign the reals real and the vomplexes imaginary speaks only of our psychology not our mathematics.

Yes I agree that the labelling of reals/complex is a misnomer.
It all depends on the definition of real or imaginary.

Exponential growth or decay is ubiquitous in the real world, whereas i is far from intuitive. The irony is that i, pi, and e are all captured beautifully in the Euler formula.

 

[matt grime]

but that has nothing to do with the mathematics of it.

 

[Borogoves]

but that has nothing to do with the mathematics of it.

What are you talking about ?

What do you personally mean by real or imaginary ?

Point out an application of i then ? What matters is that a mathematical model, describes the world even though it may not be perfect.

Do you study math to make discoveries in your chosen field or because you enjoy it as a subject in its own right ?

 

[toocool_sashi]

Lets go back to some electricity. V=iR Ohm's Law. i denotes current. Why isn't current usually denoted by "c" for current or "A" for ampere?? well "mathematicians" believe that current is denoted by i so as 2 show the importance of complex numbers in electrical engineering ;) u can find out from any elec engg abt the validity of this point...i wudnt know...im starting mechanical engineering course in abt 2 weeks :) cheers!

 

[lurflurf]

Lets go back to some electricity. V=iR Ohm's Law. i denotes current. Why isn't current usually denoted by "c" for current or "A" for ampere?? well "mathematicians" believe that current is denoted by i so as 2 show the importance of complex numbers in electrical engineering ;) u can find out from any elec engg abt the validity of this point...i wudnt know...im starting mechanical engineering course in abt 2 weeks :) cheers!

Is this a joke(there was a ;))? The symbol used to represent something is not important. The "i" in Ohm's law has nothing to do with complex numbers. To avoid confusion scientist and engineers commonly use j^2=-1 when confusion might arise with another symbol, such as when i is being used to represent current.

 

[meckano]

I've read somewhere that the square root of -1 is handy when it is divided by itself, leaving ofcourse, 1.
I think it was used to show that a ?muon? leaves the opposite side of a mountain as soon as it enters the mountain.

I prefer, at times, to see the square root of -1 and division by zero as meaning that it has left our 'real' world / ceases to exist... whatever.
In the muon case above, I would see it as leaving 'reality' as it hit the mountain, and another was created at the same time on the opposite side.
Similarly, an electron in a copper conductor does not flow, rather it bumps one which bumps the next. ~~~And the little one said, roll-over, roll-over...
- solong as that is still the believed scenario for an electron.

 

[ChrisHarvey]

I've just started reading "Visual Complex Analysis" by Tristan Needham. It is an amazing book that explains complex numbers in such an clear way from a geometric point of view. I covered complex numbers in a basic way in high school and was left asking myself the same questions being asked here. What exactly are they? What do they mean? They just seemed to be a mathematical curiosity. Having only just read the first few chapters I can honestly say I feel 'happy' with them and am seeing uses for them where I woudn't usually think of them. I would recommend the book to anyone. One thing that sticks in my mind so far is the geometric derivation of Euler's formula in chapter 1. Who would have thought that e^(i*theta) = cos(theta) + i*sin(theta) could seem obvious when thought of through the eyes of geometry (and when shown!)?

Visual Complex Analysis