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Hi, I m trying to find out, what is imaginary unit/number. i^2

  1. Jul 1, 2012 #1
    Hi,I m trying to find out,what is imaginary unit/number.How to imagine it.I was reading many articles and analogies,but I still have some questions to get better idea what is it.So I m going to write some points,that are mystic for me.
    My question is maybe more philosophic than mathematic but:What was the impuls to invite imaginary number.Was it so,that one day mathematicians made a decision to make number ,,i"?From out of blue,or how?It seems to me,like mathematicians can define everything,what they want.But if negative times negative gives positive.And positive times positive gives us positive,then it is impossible(but we know that it is not true,but pretends we :) )and enough.We cannot do with x^2=-1 nothing.And that s the fact.But clever mathematicians said:,,Well come on,make a little convention and i^2 will be equal to -1."So it seems,like mathematians adjust math,how they need.----So I hope,that you understand where I m going.....
    Thanks a lot for patient
     
  2. jcsd
  3. Jul 1, 2012 #2
    Re: i^2

    Well the wikipedia article has a good bit on the history and motivation behind i, but briefly, with i it becomes possible to solve huge numbers of polynomials. If complex numbers (which stem from i) did not exist, many polynomials would have less roots than then mathematically should.

    Also, i can be used to find real number solutions to problems where the numbers would not be possible to find without the use of i.

    http://en.wikipedia.org/wiki/Complex_number#History
     
  4. Jul 1, 2012 #3
    Re: i^2

    Yes, mathematicians adjust math how they need it. For example, I'm working on a paper about a situation in which the mathematician wanted to factor a polynomial into linear factors. Well they don't in the reals so he turned to complex numbers. Then he got a more general case and complex wasn't even enough, so he went to "hypercomplex" numbers. Seems silly sometimes, but the thing is, when you can deconstruct something using these strange notions of
    complex" or even "hypercomplex" numbers, the main point is usually that you are preserving the information and you can get back to it after manipulating it. Like a bridge to get some information from point a to point b and nothing was lost in the process.
     
  5. Jul 1, 2012 #4

    HallsofIvy

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    Re: i^2

    The reason to do it- and, in fact, the impetus to define imaginary numbers at all, dates back to Cardano's "cubic formula" which involves finding square roots, then combining them with other calculations. There exist cubic equations which can be shown, by graphing, for example, to have three real roots, but such that the square roots in the cubic formula are of negative numbers. The imaginary parts cancel but you need them to apply Cardano's formula.
     
  6. Jul 1, 2012 #5

    haruspex

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    Re: i^2

    It's important to understand that complex numbers (and all sorts of other extensions and contractions of the concept of number), and operations on them, are only useful because they can be handled in a consistent way. We can define addition etc. for them in ways that are consistent with reals.
    Similarly, log(z) can be defined for complex z in such a way that it gives the 'right' value when z happens to be real, and the usual ways of manipulating the log function don't give rise to inconsistencies either. If this were not the case, we could have complex numbers but the log function could not be applied to them.
    So it is not true that mathematicians can create arbitrary extensions. A classic is infinity; it is not a number in the sense that you can apply all the usual arithmetic operations to it in a consistent way. E.g. infinity minus infinity cannot be consistently defined. There are various uses for 'infinity', but they mean rather different concepts according to context: as a shorthand in limit processes, in transfinite arithmetic, in projective geometry...
     
  7. Jul 1, 2012 #6
    Re: i^2

    There's actually a very simple geometric visualization that i2 = -1.

    First think of the real line, running left to right. Zero in the middle, the positive reals to the right, the negative reals to the left. The integers are marked with little tick marks ... 1, 2, 3, ... to the right, -1, -2, -3, ... going left.

    Imagine the arrow starting from 0 and pointing right to 2. In other words the directed line segment from 0 to 2. It has a length of 2 and a direction of right. You can even think of it as a vector, since it has both a magnitude and direction.

    What happens if we multiply this vector by, say, the number 5? We get a vector pointing right with magnitude 10. So we can think of multiplication by 5 as a "stretch the length by 5" operator.

    What if we multiply that vector by 5 again? We get the arrow from 0 to 50. In other words the "stretch by 5" operator applied twice in a row gives you the "stretch by 25" operator; or, 52 = 25.

    So there's a correspondence between our idea of multiplying numbers, on the one hand; and composing stretching operators.

    But it's boring to just stretch right-pointing horizontal arrows. What if we try to flip the direction? Well, that corresponds to multiplication by -1. So if I have the arrow from 0 to 5 and I multiply it by -1, that has the effect of reflecting the original arrow in the origin.

    And as before, the usual arithmetic works. Stretching to the right by 6 and flipping left gives you -6; or, (-1)(6) =6.

    By the way, what happens if we start with the arrow from 0 to -1 pointing left; and multiply it by -1? We get the arrow from 0 to 1 pointing right. This gives a visualization of the fact that

    (-1)(-1) = 1.

    We noticed earlier that 5 x 5 = 25 as stretch operators. We can ask ourselves the question, is there some geometric operation, let's call it i, such that i2 = -1?

    It's pretty clear that no possible combination of stretches and flips on the real line is ever going to have that property. But ....

    What if we go to two dimensions?

    That is, let i be a counterclockwise rotation of an arrow through and angle of pi/2, or 90 degrees. So if we start from the arrow from 0 to 5 and we apply i to it, we get an arrow from the origin with length 5, pointing up. Or from the origin to the point (0,5) on the x-y plane, if you know about the 2-dimensional Cartesian plane.

    And what if I then do that twice? I get back to -5 on the real line.

    Or if I'd started with 1, we'd have i2 = -1.

    From now now whenever you see the symbol i referring to the so-called "imaginary unit," you can think to yourself: That's just a 90-degree counterclockwise rotation of the plane. If I did it twice I'd be back to -1. Simple!
     
    Last edited: Jul 1, 2012
  8. Jul 2, 2012 #7

    arildno

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    Re: i^2

    And what if you CREATE something that "looks like" the multiplication you are used to, but doesn't actually use as its ELEMENTS "numbers" you are used to, but some other types of structures you would like to call some other type of "numbers"?

    Basically, multiplication between numbers YOU are used to follows the rules you assign to them.
    But, does that mean mathematicians should ONLY be interested in those structures called numbers that can be visualized as lying along a single line?
    Couldn't we make "numbers" into points in the plane, instead?
    Essentially, as others have said, the complex numbers are just that, "numbers-as-points-on-a-plane", rather than the traditional "numbers-as-points-on-a-line".
    And for complex numbers, the mathematical operation MOSt similar to "multiplication", DOES allow for i*i=-1
     
  9. Jul 2, 2012 #8
    Re: i^2

    Bringing us to my favorite voicemail joke ever; I'm sorry the number you have dialed is imaginary.please rotate by ninety degrees and try again
     
  10. Jul 3, 2012 #9
    Re: i^2

    Thanks a lot,you are super men.So mathematicians invited new dimension randomly?And could number has higher dimensions than two?For example something like 3D and so on....Is it possible?If not,why?Theoreticaly....
    Thanks a lot
     
  11. Jul 3, 2012 #10
    Re: i^2

    One can get various sorts of numbers from various other sorts by constructing extensions of them. The most interesting extensions are those that make various operations closed.

    An operation op is closed over set S if op(any members of S) is always in S.


    For numbers, one starts with the natural or counting numbers: 0, 1, 2, 3, ... One can define them with 0 and a successor function; Peano's axioms describe how 0 and that function behave. With Peano's axioms, one can define addition, multiplication, and exponentiation, and one can show that they have their familiar properties.

    The cardinalities (numbers of members) of finite sets can be shown to follow Peano's axioms, and the set-theoretic definitions of addition, multiplication, and exponentiation can also be related to their Peano-axiom definitions.


    The natural numbers are closed under addition and multiplication, but not under subtraction or division. The subtraction-closure extension of the natural numbers is the integers, and the division-closure extension of the integers is the rational numbers.

    Algebraic closure is closure of the polynomial-equation-solution operation. When a set of numbers is algebraically closed, every polynomial equation with coefficients in that set has all solutions in that set. The algebraic-closure extension of the rational numbers is the complex algebraic numbers.

    Cauchy closure is closure of the Cauchy-sequence operation. When a set of numbers is Cauchy-closed, every Cauchy sequence whose members are all in that set has a limit in that set. The Cauchy-closure extension of the rational numbers is the real numbers.

    Combining algebraic closure and Cauchy closure gives the most general complex numbers. "Complex real numbers"?


    Complex numbers can be realized as pairs of real numbers obeying
    (a0,a1) + (b0,b0) = (a0+a1, b0+b1)
    (a0,a1) * (b0,b1) = (a0*b0 - a1*b1, a0*b1 + a1*b0)
    One can define other multiplication-like operations on ordered lists like matrix multiplication, quaternion multiplication, etc.
     
    Last edited: Jul 3, 2012
  12. Jul 3, 2012 #11
    Re: i^2

    Likewise, one can get the integers from the natural numbers by setting up pairs of them:

    Subtraction:
    a1+x = a0 -> x = a0 - a1 = (a0,a1)
    Non-identical ones can be equal:
    (a0,a1) = (b0,b1) if a0+b1 = a1+b0
    Equality to natural number:
    (a,0) = a
    Addition and multiplication:
    (a0,a1) + (b0,b1) = (a0+b0, a1+b1)
    (a0,a1) * (b0,b1) = (a0*b0 + a1*b1, a0*b1 + a1*b0)

    One can get the rational numbers from the integers in a similar fashion:

    Division:
    a1*x = a0 -> x = a0/a1 = (a0,a1)
    Non-identical equality:
    (a0,a1) = (b0,b1) if a0*b1 = a1*b0
    Addition and multiplication:
    (a0,a1) + (b0,b1) = (a0*b1 + a1*b0, a1*b1)
    (a0,a1) * (b0,b1) = (a0*b0, a1*b1)


    Cauchy sequence: {a(1), a(2), ..., a(n), ...} where for every eps > 0, there is some N such that |a(n1) - a(n2)| < eps for all n1, n2 > N.

    If it has a limit a(lim), then for every eps > 0, there is some N such that |a(n) - a(lim)| < eps for all n > N.

    A simple example of Cauchy non-closure. The Cauchy sequence {1, 1/2, 1/4, 1/8, 1/16, ...} has limit 0, but 0 does not have the form 2non-positive integer.

    Adding trailing digits to decimal or other place-system representations gives a Cauchy sequence of rational numbers:
    {1, 1.4, 1.41, 1.414, 1.4142, ...} has limit sqrt(2)
     
  13. Jul 3, 2012 #12

    haruspex

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    Re: i^2

    Extensions cannot be made at whim. As I mentioned, the key requirement is to be able to extend operations in a consistent way. From reals you can extend into complex numbers and quaternions, but I'm not aware of any further possible extensions of that sort.
    You can raise the dimensions at your pleasure by going into vector spaces, but you have to give up the multiplication operation, so it's not a true extension.
    Many other variants (in the sense that they resemble numbers in some way) are also not extensions, since something is removed: groups, rings, finite fields...
     
  14. Jul 3, 2012 #13
    Re: i^2

    You can extend it into eight dimension (the octonions), but you lose associativity in the process (just as quaternions lose the commutativity of the complex numbers, and the complex numbers lose the "total-ordered-fieldedness" of the reals). All four (the reals, the complex numbers, the quaternions, and the octonions) are normed division algebras over the reals; it's a rather famous theorem that, up to isomorphism, only these four are possible.

    The word "number" doesn't mean anything in mathematics. Some structures can be three-dimensional and some can't; the complex numbers are just a two dimensional structure that extend the reals in certain, useful ways.
     
  15. Jul 3, 2012 #14
    Re: i^2

    Those are "division algebras", because they have a well-defined division operation. For every a and nonzero b in it, there is one x such that a = b*x and one y such that a = y*b.

    Division algebra - Wikipedia

    Real-number-like division algebras only have dimensions 1, 2, 4, 8.

    The real numbers, the complex numbers, the quaternions, and the octonions are their classic examples. All these can be constructed with the
    Cayley–Dickson construction - Wikipedia
    • Complex numbers lose ordering, self-conjugacy.
    • Quaternions lose commutativity.
    • Octonions lose associativity, though they are alternating (a*a*b, a*b*a, b*a*a are associative).
    • Sedenions, the next ones, lose normed multiplication, the division-algebra property, and alternativity. However, they are still power-associative: am*an = am+n.
    • All the higher ones are like the sedenions.

    You can do matrix multiplication on n*n matrices, but a general matrix ring will have divisors of zero and nonzero non-invertible matrices.

    Zero divisors:
    {{0,a},{0,0}} . {{0,0},{b,0}} = {{0,0},{0,0}}
    where neither a nor b is 0. Neither of the matrices on the left is invertible.
     
  16. Jul 5, 2012 #15
    Re: i^2

    Thanks as I can see it is more complicated.OK,and if imaginary number is only a adjusting of math,why was so long time,until they discovered it.It stupid question,I know,but,if we something adjusting to us,we cannot find it so long,because we make laws,as we want.A child could make this idea:,,Well,let s define a new number i^2.It equals to -1."
    I dont know,if you understand my question,but I hope yes.
     
  17. Jul 5, 2012 #16

    HallsofIvy

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    Re: i^2

    They didn't need it until then! One can, in fact, make up whatever mathematical systems one wants with what ever properties one wants. Whether it is worth the trouble of doing that and whether other people will use it depends upon how useful it is.
     
  18. Jul 5, 2012 #17

    haruspex

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    Re: i^2

    I think you're still missing the point that it is not that simple. Try defining a new number Ω such that Ω * 0 = 1. You cannot do it in a consistent manner. It is no small matter to demonstrate that a new entity or operation can be included in the number system consistently.
    There was also thousands of years of tradition to overturn. The Greeks struggled to accept irrational numbers. The west was slow to accept the notion of 0 as a number. These things are always so much simpler in hindsight.
     
  19. Jul 5, 2012 #18
    Re: i^2

    That's not really how it works. The complex numbers are a field with operations defined in a specific way that just so happens to contain elements that square to -1. They are the algebraic extension of the real numbers, and they are not something that could be constructed by a five year old.
     
  20. Jul 5, 2012 #19

    HallsofIvy

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    Re: i^2

    One of the difficulties with just saying "i^2= -1" is that, rather quickly, one can show that for any number, r, there exist two numbers x such that x^2= -r. Since there are two complex numbers whose square is -1, which one is meant by "i"?

    One way to avoid that difficulty is to define the complex numbers as pairs real numbers, (a, b), with addition defined by (a, b)+ (c, d)= (a+ c, b+ d) and multiplication defined by (a, b)(c, d)= (ac- bd, ad+ bc).
     
  21. Jul 5, 2012 #20
    Re: i^2

    Complex numbers arise naturally in the geometry of the 2D plane--much more naturally, in my mind, than their historical origins in the study of polynomials. The polar form of complex numbers and how that corresponds to the real coordinates of a vector in a 2D space is significant. One should not take away that there is a unique number [itex]i[/itex] that squares to -1 and is the generator of rotations in a plane. Rather, there is an object that has such properties, and beyond 2D geometries, there are many such objects. In 3D, there are at least 3 such objects that all square to -1, and this has long been known in the guise of quaternions. In relativity, you have objects that square to +1 yet are not 1 themselves. None of this needs to be put in on an ad hoc basis, though. It all comes naturally from the study of geometry and vector spaces.
     
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