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Thread: The imaginary unit

  1. #1 The imaginary unit 
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    Is it possible to put the imaginary unit (i^2= -1) under a square root or in a denominator? Also, what algebraical functions are possible under such conditions?
    I am sorry for asking a question that is probably simple, but in my school we have just begun with the imaginary unit, and there are simply no such informations in my book, nor have I been able to find it on the internet. (Perhaps the problem is that English is not my first language.)


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  3. #2  
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    The imaginary unit (i) can be treated arithmetically like any other number. It has two square roots {(1+i)/√2)}, like any other number. Dividing by i is the same as multiplying by -i.


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    Thanks a lot. I haven't really understood it all, but it was helpful. Thank you.
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  5. #4  
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    One way to calculate the quantities you asked for is to rewrite it as exponentials and use the rules for calculating those. You may find this formulas very useful.



    this gives us that which in turn can be used to calculate the square root.

    Using Taylor series and other mathematical methods you can also give a meaningful definition for trigonometric functions or logarithms of complex numbers, but that is probably outside the scope of the first courses in complex variables.[/tex]
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  6. #5  
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    This is highly beyond my understanding, but I am grateful for you trying to explain that one to me...
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  7. #6  
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    Just to add;

    you also can manipulate these imaginary units to give simpler algebraic forms of certain functions. Of course this is done through the use of Euler's formula.


    e.g.

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    "Mathematicians stand on each other's shoulders."- Carl Friedrich Gauss


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  8. #7  
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    Note to other replies. It is obvious that the question was posed by someone just beginning mathematics beyond arithmetic. Trig functions and exponentials come much later.
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  9. #8  
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    In Sweden we learn trigonometry and exponentials before we learn complex numbers so I was taking a chance some concepts would be familiar. If not then it will not be very useful I am afraid.
    "Complexity is stupidity disguised as intellect."
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  10. #9  
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    We are learning complex numbers before trigonometry and exponentials, but I do have basic knowledge of trigonometry which I required for the physics competition.
    I have a bit of problem understanding what does x and e represent.
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    Fair enough, and I agree with mathman.

    Although the OP has been answered, there has been very little in the way of explanation. Speaking now as an ordinary member, I think it is poor practise to answer a question which, to some of my fellow members, seems elementary without some sort of explanation. Let's see if I can rectify this.....

    Let's first assume that exists and is a number. Now any number that we are likely to encounter in a normal lifetime can be written in the form (some can't, but forget them), where are real numbers, so write

    so that . Expanding we have that

    , which is nonsense UNLESS

    1) and

    2) .

    Condition (1) implies that . Just for illustration, suppose that this implies , though it need not. Question to the OPer -why?

    So for the square root any positive number will do: say (although of course the second equality is totally redundant!).

    But condition (2) must simultaneously be satisfied, so we require that , and one such number that fulfils both requirements is

    . (Notice there is some guess-work involved here. This almost always the case for extraction of square roots by hand. Even Newtons method, which usually works, involves informed guesses) Let's try this guess....

    .

    So under my assumption of a single root, and because there is a common denominator can be written compactly as

    , but since for any , we will have that as others have said.

    The OPer might like to show why this holds when They also might like to show why my (our) guess is the only correct one.
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  12. #11  
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    Quote Originally Posted by Sindrato
    We are learning complex numbers before trigonometry and exponentials, but I do have basic knowledge of trigonometry which I required for the physics competition.
    I have a bit of problem understanding what does x and e represent.
    If so I wouldn't worry much about about the x and the e. They will get an explanation when the time is right.

    I didn't even think of the possibility of math being teached in significantly different orders in different school systems and countries. Guess I learned something new today.

    Anyway, you asked whether the imaginary unit could be used in algebraical functions. For most functions the answer is yes, but the definitions for some of the operations may not be intuitive. Most probably you will not go much further than multiplication and division of complex numbers as a start. But you will probably encounter complex numbers again when you have studied some trigonometry and exponential functions.
    "Complexity is stupidity disguised as intellect."
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  13. #12  
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    On the topic of the tangent; Guitarist, what kind of number is there that can't be expressed in terms of the complex numbers? I thought every number conceivable was representable by
    Wise men speak because they have something to say; Fools, because they have to say something.
    -Plato

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  14. #13  
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    Guitarist, many thanks. I have understood completely, although after reading it twice. Out of the subject question, what does the OP means? Anyway, I will presume that the OP-er means me, and will try to answer your questions. The first question is quite clear, and is if
    than x must equal y, or minus y so that the result would be .
    As for the second question, the equation holds when because the . And the guess is the only one correct because in the end there must be and is the only such case. (Can you please tell me how do you write the plus/minus sign, fraction, and the square root?)
    However, there is one thing I find unclear in this explanation. If than shouldn't it be ?

    EDIT: forget the last question, I forgot that
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  15. #14 Re: The imaginary unit 
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    Quote Originally Posted by Sindrato
    Is it possible to put the imaginary unit (i^2= -1) under a square root or in a denominator? Also, what algebraical functions are possible under such conditions?
    I am sorry for asking a question that is probably simple, but in my school we have just begun with the imaginary unit, and there are simply no such informations in my book, nor have I been able to find it on the internet. (Perhaps the problem is that English is not my first language.)
    Any complex number will have two square roots, and i is not exception.

    However, unlike postive real numbers there is no apriori way to define what you mean by "THE square root" of a complex number. This is because the definition of fractional exponents, and indeed any non-integer exponent is ambiguous for complex numbers and requires that one first "pick a branch of the logarithm". This notion is probably rather foreign to you and requires a bit more advanced mathematics than what you are likely to have yet seen. But you will get a taste for it when you study "DeMoivre's theorem" a bit later in your study of complex numbers.

    Suffice it to say that you will need to be a bit more careful with algebraic manipulations of complex numbers than you have had to be thus far with real numbers, but it is well worth the price of the additional care and the complex numbers have a rich structure that you come to appreciate later on.
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  16. #15  
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    Quote Originally Posted by Sindrato
    what does the OP means? Anyway, I will presume that the OP-er means me, and will try to answer your questions.
    Oh sorry. OP means "original post" And OP-er means original poster (Your command of English is so good I had not realized you are not a native speaker)
    And the guess is the only one correct because in the end there must be and is the only such case.
    You have not shown why this is so, but don't worry - assertions of "uniqueness" are notoriously hard to prove. But as DrRocket says, has 2 roots, according to something called the Fundamental Theorem of Algebra, that is a polynomial of degree has at most distinct roots; I suggest you don't think about this too hard for now

    Can you please tell me how do you write the plus/minus sign, fraction, and the square root?
    Well there is a "sticky" at the top of this page, about it.....

    LaTex is a computer program, and has some functionalities built in. These include operators and operations, some but not all. Where these are built- in, use the back-slash "\" and the permitted code with argument in curly braces

    So that \sqrt{x} = , \int{x} = , and fractions likewise: \frac{1}{2} = and \pm =
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  17. #16  
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    Quote Originally Posted by Guitarist
    You have not shown why this is so, but don't worry - assertions of "uniqueness" are notoriously hard to prove. But as DrRocket says, has 2 roots, according to something called the Fundamental Theorem of Algebra, that is a polynomial of degree has at most distinct roots; I suggest you don't think about this too hard for now
    For the simple case of roots you don't need anything as sophisticated as the fundamental theorem of algebra, or equivalently the algebraic completeness of the complex numbers.

    DeMoivre's theorem is sufficient and that can be understood geometrically in terms of multiplication of complex numbers and wee bit of simple trigonometry -- looking at complex numbers in "polar coordinates" and dealing with the notion of the "argument" thereof.

    The point remains, however, that there is no unambiguous way to select among the available roots and no definition of what is meant by THE square root (or any other root) of a complex number in general.
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