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Thread: Square root of negative 1

  1. #1 Square root of negative 1 
    Forum Professor marcusclayman's Avatar
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    How is this used in a mathematical formula? How should it be thought of, to understand it best? More importantly, is there a way to put it in a calculator? Mine just gives me errors whenever I try to use it.


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    Forum Ph.D. Heinsbergrelatz's Avatar
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    im not a complex analysis professional, but what you are talking about here is the imaginary unit(its not a real number), so really if you type square root one, the calculator will give you nothing, as it cant be computed. there is a separate symbol for that value, which is "i', there should be the letter "i" in your calculator unless your calculator is one of those which only has numbers and plus minus... on it.

    the imaginary unit is under complex numbers, and its first encountered by students when handling with roots of quadratic equations with the use of quadratic formula, when b^2-4ac<0. Really the imaginary number allows mathematicians to extend the real number line to the complex system, which has many many applications one of which is; electrical circuits, impedance etc..

    Dr.Rocket, Guitarist or other experienced members in mathematics will enlighten you more in detail.

    http://mathworld.wolfram.com/ImaginaryUnit.html-check it out.


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    I've been trying to understand i for a long time, I still don't really get it.
    But here's the Wiki on it; http://en.wikipedia.org/wiki/Imaginary_unit
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    Quote Originally Posted by GiantEvil
    I've been trying to understand i for a long time, I still don't really get it.
    But here's the Wiki on it; http://en.wikipedia.org/wiki/Imaginary_unit
    The complex numbers are just with vector addition and a multiplication rule . Then .
     

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    Quote Originally Posted by DrRocket
    Quote Originally Posted by GiantEvil
    I've been trying to understand i for a long time, I still don't really get it.
    But here's the Wiki on it; http://en.wikipedia.org/wiki/Imaginary_unit
    The complex numbers are just with vector addition and a multiplication rule . Then .
    Thank's Doc. That actually help's a bunch.
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    Well, without wishing to appear rude, DrRocket's explanation had me tugging my beard for a while. Of course he is correct, but perhaps a little opaque.

    Try this instead. Consider the polynomial . One says that the "zeros" of this polynomial are those values of for which is true.

    First note that there are implied integer coefficients in this polynomial - in this case 1.

    So a number is defined to be an algebraic number iff it is a zero of some polynomial with integer coefficients

    Now there is a theorem, called the Fundamental Theorem of Algebra, that states that any polynomial of degree has at most zeros and moreover that . (The proof is hard!)

    So we seek at least one algebraic number such that . In fact there are 2, which is the usual case for degree 2 polynomials, as indeed for those of higher degree.

    So one defines the objects as the 2 zeros for our polynomial. Since these quite obviously not real numbers (in the usual sense of the word) our is called the "imaginary unit". (Notice that )

    Further, one says that, given the field of real numbers, there is an extension of this field by our imaginary unit such that .

    Now it is not hard to show (using the field axioms on ) that is also a field, whose elements must be of the form

    And so (finally!!) one says that he complex numbers are the algebraic completion of the reals, which quite simply means that there is no polynomial whose zeros cannot be found in or its subfield (usually both)

    OMG, I had intended to add clarity to this thread - I now see I have done the opposite. Ho hum
     

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    Thank you Guitarist for your excellent exposition as well.
    The only clear explanation of I can think of is , but it's not an explanation at all. That's like saying "a cow goes moo".
    I was some of the mud that got to sit up and look around.
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  9. #8  
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    Quote Originally Posted by Guitarist
    Now there is a theorem, called the Fundamental Theorem of Algebra, that states that any polynomial of degree has at most zeros and moreover that . (The proof is hard!)
    That theorem is based on calculus on the complex numbers, so you need to already know about the complex number field in order to invoke it.
     

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    Quote Originally Posted by GiantEvil
    Thank you Guitarist for your excellent exposition as well.
    The only clear explanation of I can think of is , but it's not an explanation at all. That's like saying "a cow goes moo".
    There is a difference.

    "A cow goes moo." is a property of cows.

    i=√(-1) is a definition.
     

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    indeed, it is a definition.[/tex]
     

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    It's very useful in AC vector math. However, to keep from confusing "i" with "I" for current, it's designated "j."

    See this:

    Polar and rectangular notation

    Complex number arithmetic
     

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    Quote Originally Posted by marcusclayman View Post
    How is this used in a mathematical formula? How should it be thought of, to understand it best? More importantly, is there a way to put it in a calculator? Mine just gives me errors whenever I try to use it.
    The square root of a number, is a smaller number which when multiplied by itself, produced the original number.
    For example, 3 x 3 = 9, where 3 is one of the square roots of 9.
    But note that -3 x -3 = 9, so -3 is also a square root of 9.

    You may notice that squaring both a positive root (eg. 3), and a negative root (eg. -3), always produces positive numbers. That means it is not possible to square any real number, and have a negative result. eg. R x R and (-R x -R) can never be less than 0, where R is the square root of a number.

    But some mathematical problems would find it useful if we could have a square root of a negative number, such as SQRT(-1). Since there are no real numbers that could be a valid root, mathematicians make up a new set of imaginary numbers, such that SQRT(-1) = i, where "i" is the imaginary "number" representing the square root of -1.
    Hence i x i = -1 (by definition)

    We can also combine these imaginary numbers with real number to produce so-called complex numbers, such as (2 + i).

    Your calculator can only handle imaginary numbers such as "i" , if it has been designed to supports imaginary and complex numbers.
    Ian Tresman
     

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    One problem: square roots aren't always smaller. The square root of 0.5 is about 0.707. Also, complex numbers can't be ordered like that anyway.
     

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    Forum Freshman iantresman's Avatar
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    Quote Originally Posted by MagiMaster View Post
    One problem: square roots aren't always smaller. The square root of 0.5 is about 0.707. Also, complex numbers can't be ordered like that anyway.
    Mea culpa, good point, my mistake.

    Quote Originally Posted by MagiMaster View Post
    complex numbers can't be ordered like that
    Not sure what you mean here. Do you mean order by magnitude?
    Last edited by iantresman; July 21st, 2011 at 05:14 AM. Reason: Additional point
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  16. #15  
    Forum Radioactive Isotope MagiMaster's Avatar
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    You can order their magnitudes, since those are real numbers, but complex numbers can't be ordered. You can't say that 1+i < 2-i (see here).
    Last edited by MagiMaster; July 21st, 2011 at 09:00 PM.
     

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    [funny that the SCIENTIFIC mode does NOT have SQRT and standard mode DOES....
    sometimes it's just better not to think about what choices MS makes for
    us...
    mathematical problems would find it useful if wehave a square root of a negative number, such as sqrt(-1). Since there are no real numbers that could be a valid root, mathematicians make up a new set of imaginary numbers, such that sqrt(-1) = i, where "i" is the imaginary "number" representing the square root of -1.
    Hence i x i = -1 (by definition)
    Calculator can only handle imaginary numbers such as "i" , if it has been designed to supports imaginary and complex numbers.
    You can clear your doubts on online calculator sites where square root calculator is used and also online tutors are also available to clear your doubts.
     

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    Brassica oleracea Strange's Avatar
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    There was quite a good (non-technical) discussion of this on the BBC recently: BBC - BBC Radio 4 Programmes - In Our Time, Imaginary Numbers
     

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    I remember being uncomfortable with the arithmetic until I ran across the intuitive grounding of i as the next step in adding rotations in space to the capabilities -

    multiplication by 1 (the positive reals) being a direction or orientation (a quantity "right/left" as usually depicted), no rotation, just headed out

    multiplication by negative 1 the first rotation (since it's in 1 dimension it's just a flip or reversal of whichever direction you were headed, the only "rotation" possible),

    multiplication by i the second kind of rotation (which can be multiplied by 1, flipped by -1, etc) - usually depicted as a counterclockwise turn to up/down.

    That picture is the same as Rocket's vector algebra explication, with a different "feel" that somehow clarified things for me.

    (btw: if you run into a kid who is baffled by the multiplication rules for negative numbers, sometimes they catch on to the "flip" idea - the way it's taught in most schools using different directions for negative and positive pictured via a thermometer or the number line or the like, creates bafflement in some kids when they hit multiplication).
     

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    Here is my attempt at a gentle intro to what rocket and guitarist have been talking about in this thread. Hopefully it's not too slanted or just wrong.

    A complex number is an ordered pair of real numbers. I find it useful to think in terms of an analogy with the x-y coordinate system, just remeber it's only an analogy not part of the definition.

    So: an ordered pair of real numbers in an x-y coordinate system gives you the coordinates of a point, e.g x=2 y=3 is written (2,3). The fact that (2,3) is ordered is very important because the order tells you whether your looking at the x coordinate or y coordinate. There is information, a description, in the order the numbers are written.

    Likewise, there is information in the order a complex number si written. The complex number (a,b) identifies a second set of real numbers with the usual set, so the entire usual set can be represented by (a,0), where the component 'a' can be any real number. The other component, which i've set at zero for the moment, is the imaginary component.

    The main thing to grasp is that the complex numbers of the form (a,0) are for practical purposes the same thing as the corresponding real numbers. So you can see in the same way that the field of rational numbers Q exist within the field of real numbers R, likewise R exists within the field of complex numbers C. C is an expansion of the real field.

    By definition, i=(0,1)

    Then by the definitions of multiplication for complex numbers (Let x and y be two complex numbers, x=(a,b) y=(c,d). Define: xy=(ac-bd, ad+bc) );

    i^2=(0,1)(0,1)
    =(-1,0)
    =-1

    And that's the definition you usually see; i^2=-1.
    Last edited by CMR80606; October 22nd, 2011 at 04:40 PM.
     

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    Untitled Document

    Imaginary Numbers are not Real - the Geometric Algebra of Spacetime

    Abstract: This paper contains a tutorial introduction to the ideas of geometric algebra, concentrating on its physical applications. We show how the definition of a `geometric product' of vectors in 2- and 3-dimensional space provides precise geometrical interpretations of the imaginary numbers often used in conventional methods. Reflections and rotations are analysed in terms of bilinear spinor transformations, and are then related to the theory of analytic functions and their natural extension in more than two dimensions (monogenics). Physics is greatly facilitated by the use of Hestenes' spacetime algebra, which automatically incorporates the geometric structure of spacetime. This is demonstrated by examples from electromagnetism. In the course of this purely classical exposition many surprising results are obtained - results which are usually thought to belong to the preserve of quantum theory. We conclude that geometric algebra is the most powerful and general language available for the development of mathematical physics.
     

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    If I may, imaginary numbers, in not being real, provide a type of "wiring outside the board we operate on" approach to formulation. They are great for "imaginary" situations, knots not easily understood using real numbers. They are not absolutely unreal, because as we know i^2 = -1. Thus in "real" applications, they are super-suited to "symmetry" problems. I think, what physics and mathematics has thus far done with imaginary numbers, complex numbers, is not all there is to other possible applications of "i". So although we become charmed by the algorithms we have set "i" to, this should not limit our further development of more potential uses for the idea of "i".
     

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    Quote Originally Posted by marcusclayman View Post
    How is this used in a mathematical formula? How should it be thought of, to understand it best? More importantly, is there a way to put it in a calculator? Mine just gives me errors whenever I try to use it.
    Here is an easy to use square root calculator that can handle imaginary and complex numbers. All you have to do is type in a number.
    Last edited by Guitarist; May 6th, 2012 at 12:35 PM.
     

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    Is this the same user that kept posting links to calculator sites? Also reviving old, old threads?

    Even though this is spam (and Wolfram Alpha remains supreme), I find the tool useful for its immediate result and versatility among complex numbers. Still, it's spam!
     

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    Agreed.

    Thread closed

    PS Possible spam link deleted


    PPS Any member here is of course welcome to ask questions about imaginary/complex numbers
    Last edited by Guitarist; May 6th, 2012 at 12:38 PM.
     

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