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The imaginary Unit

  1. Dec 11, 2013 #1
    When i is defined by an equation which has 2 solutions, how does it make sense to do algebra with complex numbers?
     
  2. jcsd
  3. Dec 11, 2013 #2
    How is i defined by an equation which has 2 solutions?

    Oh, I understand what you mean. I don't think it matters, because of the definition, everything true for i will be true for -i too. It becomes purely a notational issue, we have two solutions, and they are additive inverses, just pick a symbol for one and put a "-" in front of the other to show that it is the additive inverse. It doesn't matter.
     
    Last edited: Dec 11, 2013
  4. Dec 12, 2013 #3
    I feel kind of stupid, but I don't understand this argument that if everything is true for i it will be for -i too. Could you give some examples? I feel it's kind of the same as choosing a right or lefthanded coordinate system but then again I have not understood that either.
     
  5. Dec 12, 2013 #4

    HallsofIvy

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    It doesn't! We don't define "i" by "[itex]i^2= -1[/itex]"- although elementary treatments may introduce it that way.

    Rather, we define the complex numbers as pairs of 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). That way (a, 0)+ (c, 0)= (a+ c, 0) and (a, 0)(c, 0)= (ac, 0) sp we can associate the "complex number" (a, 0) with the real number, a. Also, if we define i= (0, 1) then we have [itex]i^2= (0, 1)(0, 1)= (0(0)- 1(1), 0(1)+ 1(0))= (-1, 0)[/itex] so that "[itex]i^2= -1[/itex]". We can then say (a, b)= (a, 0)+ (0, b)= a(1, 0)+ b(0, 1)= a(1)+ b(i)= a+ bi.

    The point is that the way a mathematical concept is introduced to the student is not necessarily the way it is formally defined by mathematicians.
     
  6. Dec 12, 2013 #5
    Okay so maybe you can help me understand where it goes wrong in the following (which motivated this thread for a start):

    1. 1/i = sqrt(1)/sqrt(-1)
    2. 1/i * i/i = sqrt(1/-1)
    3. i / -1 = sqrt(-1)
    4. -i = i

    What is the problem by simply defining i=sqrt(-1) and where does it go wrong in the above when doing so?
     
  7. Dec 12, 2013 #6

    Mentallic

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    The problem is going from line 1 to 2,

    [tex]\frac{\sqrt{a}}{\sqrt{b}}=\sqrt{\frac{a}{b}}[/tex]

    Is a rule that applies only to positive a and b values.
     
  8. Dec 12, 2013 #7
    OK, I think it will become obvious to you.

    3i + i = 4i
    3(-i) + (-i) = 4(-i)

    For example, or even
    e^(-i(pi)) = -1

    Where i/-i is only referenced once.

    If we make the 'basis' for the 1 dimensional vector space -i instead of i, the change is only superficial.

    Essentially, I believe that -i and i may be regarded as equals (not that they are equal to one another, but that they are on the same footing.)

    It doesn't matter which one is which, as long as both exist. I don't believe you can say ANYTHING that differentiates I from -i... but I may be wrong.
     
  9. Dec 12, 2013 #8
    I see. But what exactly is the difference then between defining i=(0,1) and i=√(-1). Is the only problem that the last one could make you mistakingly think that you can apply usual rules for square roots? And noting that this is not the case, are the 2 definitions the equivalent?
     
  10. Dec 12, 2013 #9
    All you've done is arbitrary chosen (0,1) to be i instead of (0, -1). If (0,1) = i, then (0,-1) = -i, and then we're back where we started: -what's the difference?-
     
    Last edited: Dec 12, 2013
  11. Dec 12, 2013 #10
  12. Dec 12, 2013 #11

    rbj

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    i'm in agreement, Hall, with 1Mile. originally the concept of "imaginary" numbers and then "complex" numbers did come from, i believe, the solution to quadratics (or higher-order polynomials) set to 0 when there were no "real" numbers (the set of rational and irrational numbers that are ordered on the "real" number line) that satisfy those equations.

    saying that

    [tex] (a, b) + (c, d) = (a+c, b+d) [/tex]

    makes sense just from a normed, linear space POV. but when you say that

    [tex] (a, b) \times (c, d) = (ac-bd, ad+bc) [/tex]

    as a definition, we can just as well say that is motivated by or a result of the definition:

    [tex] i^2 = -1 [/tex] .

    i might recommend the OP to take a look at the wikipedia article on it, especially the 2nd section. i think that explains it pretty well without just requiring the OP to accept [itex] (a, b) \times (c, d) = (ac-bd, ad+bc) [/itex] as a definition for how complex numbers multiply.
     
  13. Dec 12, 2013 #12

    pwsnafu

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    This is not true. It comes from the the Tartagalia-Cardano formula, i.e. the cubic formula. As you know cubics always have a real solution, however from time to time the formula generates solutions with square roots of negative numbers. For example ##x^3 - 15x = 4## produces
    ##x = \sqrt[3]{2 + \sqrt{-121}} + \sqrt[3]{2 - \sqrt{-121}}##
    but this equals 4.

    Another example is in 1702, when Leibniz demonstrated to Hyugens that
    ##\sqrt6 = \sqrt{1+\sqrt{-3}}+\sqrt{1-\sqrt{-3}}##
    who exclaimed "defies all human understanding".

    All of this predates the notion of complex numbers.
     
  14. Dec 12, 2013 #13

    rbj

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    there's a reason i parenthetically inserted "or higher-order polynomials".

    but, from a pedagogical POV, i don't understand why it's necessary to go above quadratics to introduce someone conceptually to imaginary and complex numbers. you don't need to go to cubic or quartic to see the need for imaginary and complex numbers.
     
  15. Dec 12, 2013 #14

    pwsnafu

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    You wrote "set to 0 when there were no "real" numbers (the set of rational and irrational numbers that are ordered on the "real" number line) that satisfy those equations." That part is wrong. It's the real solutions that caused the discovery. Back then ##x^2 + 1 = 0## simply had no solutions.

    But you weren't making a point of pedagogy. You were making a claim about the history of mathematics.
     
  16. Dec 12, 2013 #15

    rbj

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    no. you cannot represent me nor what i was saying. you cannot even presume to.

    it was purely a point of pedagogy from the beginning. as best as i can tell, you were making an historical point. but i was covering my arse anyway.
     
  17. Dec 12, 2013 #16

    pwsnafu

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    What? You wrote
    Explain please how that is not about history. You explicitly used the phrase "originally...did come from". How am I supposed to parse that as a pedagogical point?

    Look, if you are claiming that teaching complex numbers is easier through quadratics, then I agree with you. But that was not what you wrote in the first paragraph.
     
    Last edited: Dec 12, 2013
  18. Dec 12, 2013 #17
    Folks is this arguing helping?
     
  19. Dec 12, 2013 #18

    pwsnafu

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    Yeah. I'm done. Moving on.

    About 1Mile's point about i and -i being equals.
    As Halls pointed out complex numbers are formally defined as pairs, with ##i = (0,1)##. This space of numbers is of course ##\mathbb{C}##.

    Let's create a new number system. I'll define ##j = (0,-1)## and call this space ##\mathbb{D}## which is composed of numbers ##\alpha + \beta j##.

    What 1Mile is saying that if I make a statement (theorem) that is true on ##\mathbb{C}##, there is an equivalent statement that is true on ##\mathbb{D}##. And we can use conjugation to write it down.
     
  20. Dec 12, 2013 #19

    rbj

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    they are qualitatively equivalent (both have equal claim to squaring to -1), but not quantitatively, since they are not zero and are negatives of each other.

    [itex]\mathbb{C}[/itex] is not [itex]\mathbb{R}^2[/itex]. Halls was complete when he stated the rules of addition and multiplication of these pairs. my point is that the "formally defined as pairs" is not the original meaning (in case you're wondering, i mean the original pedagogical meaning), but can be made equivalent as long as you define the rules of addition (trivial, about the only way you can) and multiplication (less trivial, there are other ways to define multiplication of 2-vectors which are not compatible with complex numbers). i really don't see any point to it or advantage over the common pedagogy (geez, i have to be careful to remind snafu, lest the words be misconstrued).
     
  21. Dec 13, 2013 #20
    Getting all argumentative is getting us nowhere, rbj. I think all of pwsnafu's replies have been very accurate.
     
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