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Thread: Imaginary numbers

  1. #1 Imaginary numbers 
    Forum Professor leohopkins's Avatar
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    What IS an imaginary number and and how is it used ?


    The hand of time rested on the half-hour mark, and all along that old front line of the English there came a whistling and a crying. The men of the first wave climbed up the parapets, in tumult, darkness, and the presence of death, and having done with all pleasant things, advanced across No Man's Land to begin the Battle of the Somme. - Poet John Masefield.

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  3. #2  
    Moderator Moderator AlexP's Avatar
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    an imaginary number is what you get when you find the square root of a negative number. for example, the square root of -1 is imaginary and is represented by "1i" The i is equivalent to "the square root of negative one". So the square root of -25 is 5i.


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  4. #3 Okay........ 
    Forum Professor leohopkins's Avatar
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    Erm.........I have just tried to input -5 and find out the square root of it using a calculator.....although I get an error. (Invalid input for function)

    Any ideas ?
    The hand of time rested on the half-hour mark, and all along that old front line of the English there came a whistling and a crying. The men of the first wave climbed up the parapets, in tumult, darkness, and the presence of death, and having done with all pleasant things, advanced across No Man's Land to begin the Battle of the Somme. - Poet John Masefield.

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  5. #4  
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    That's because your calculator (ie the bill gates sold you) - is crap - you'll need a scientific/engineering calculator which allows mathematics on imaginary numbers. It's usually marked r>p or p>r (polar rectangular coordinates).

    Your calculator can only give one answer to a sum, there are two roots though to any number ie 16=+/- 4^2.


    Your calculator lives in 'line land' - you need to get one that works in 'plane land' :wink:
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  6. #5  
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    no scientific calculator i've ever seen works on imaginary numbers.
    however the $200 (Australian Dollar) graphics calculator, which i hired out through my school and thus was saved from having to pay that rediculous amount of money, does the job reasonably well.

    but you can do calculations just fine with pen and paper.
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  7. #6  
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    A few sharp calculators do - the one i had in undergrad could handle complex arithmetic and some basic matrix algebra.

    Complex numbers are important as they are algebraic completion of the reals - or translated into normal language any polynomial with real coefficients has a complex zero.

    The actual construction of the numbers aint too difficult - but lets leave that for another post.

    Here is an easy example, consider the polynomial f(x) = x<sup>2</sup> + 1 which obviously does not have a real zero. The complex number i is defined to be that number such that f(i) = i<sup>2</sup> + 1 = 0. Now there is nothing too amazing there, the real interesting thing is that you only have to add extend the reals once to make them algebraicly complete!
    As is often the case with technical subjects we are presented with an unfortunate choice: an explanation that is accurate but incomprehensible, or comprehensible but wrong.
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    The three Scientific Calcs I have are the Casio FX7000G, Casio FX590, and FX992vb - all have it RP functions, - the 590 is getting on for 20 years old now.
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    I think i had the Sharp EL506WB (or what ever its predecessor was, i cant remember now) - was quite a handy little beast
    As is often the case with technical subjects we are presented with an unfortunate choice: an explanation that is accurate but incomprehensible, or comprehensible but wrong.
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  10. #9  
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    the TI-83(+) and i'm sure TI-84 can handle imaginaries. That's what's the standard issue is in my school.
    "There is a kind of lazy pleasure in useless and out-of-the-way erudition." -Jorge Luis Borges
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  11. #10  
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    Quote Originally Posted by Megabrain
    The three Scientific Calcs I have are the Casio FX7000G, Casio FX590, and FX992vb - all have it RP functions, - the 590 is getting on for 20 years old now.
    CASIO fx-82MS is pretty sub-standard then, don't know how old that is. thats standard issue in my school. the other scientific calculators i've seen must have been horribly old.
    we get issued with TI-84's if we need them, but i hardly ever used mine for dealing with complex numbers.
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  12. #11 maths 
    Forum Professor leohopkins's Avatar
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    Yes. to be honest im complete rubbish at maths. the most advanced mathematics i know would be very basic algebra. i.e x + 7 = 10 therefore x = 3.

    Thats it.

    Oh and the equivalent sign - i learned that just a month ago. still dont know how to use it though.

    I do have a particular dislike for mathematics though i guess its because i hate rounding up / rounding down numbers. the laws of nature dont subscribe to mathematics and therefore do not necessarily come in integers.
    The hand of time rested on the half-hour mark, and all along that old front line of the English there came a whistling and a crying. The men of the first wave climbed up the parapets, in tumult, darkness, and the presence of death, and having done with all pleasant things, advanced across No Man's Land to begin the Battle of the Somme. - Poet John Masefield.

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  13. #12  
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    I found me old calculator - was the Sharp EL-506V

    If i want a programmable calculator i would get the HP12C, if i wanted a graphics calculator i would get a laptop with matlab and mathematica installed on it lol.
    As is often the case with technical subjects we are presented with an unfortunate choice: an explanation that is accurate but incomprehensible, or comprehensible but wrong.
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  14. #13 Re: maths 
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    Quote Originally Posted by leohopkins
    Yes. to be honest im complete rubbish at maths. the most advanced mathematics i know would be very basic algebra. i.e x + 7 = 10 therefore x = 3.

    Thats it.

    Oh and the equivalent sign - i learned that just a month ago. still dont know how to use it though.

    I do have a particular dislike for mathematics though i guess its because i hate rounding up / rounding down numbers. the laws of nature dont subscribe to mathematics and therefore do not necessarily come in integers.
    Err... Very little in math deals with integers. For example, if x + 7.35 = 10.12, then x = 2.77. Also, number such as pi and e can't be expressed in any finite number of integer values. (Unlike 2.77 which equals 277/100.) Also, most modern research in physics (that I know of at least) says that the universe is quantized, meaning that in fact the laws of nature do come in integers.
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  15. #14  
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    Math is the exact opposite for me since through it a lot of nature can be understood. It was math alone through Dirac that showed a positron may exist even when that wasnt even the expected result. Just fell out of the equation as one of the answers. It was shown they existed a few years later by Carl Anderson.
    I've always wanted to know why. Just takes me a little longer than most, lol.
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  16. #15  
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    Quote Originally Posted by Chemboy
    an imaginary number is what you get when you find the square root of a negative number. for example, the square root of -1 is imaginary and is represented by "1i" The i is equivalent to "the square root of negative one". So the square root of -25 is 5i.
    Yeah, that's the school-book definition. I don't like it, and I'll tell you why.

    Suppose we define √-1 = i. We know that, in general, for positive real x, there are two solutions to √x - positive and negative, e.g. √4 = ±2. To be consistent, then, we would have to admit that √-1 = ±i, either that or to partition the reals into positive and negative, each with their own "rule for roots". This is not nice.

    So rather have have one rule for the roots of positive numbers and another for those of negative numbers, it would be far better to define the imaginary unit as i<sup>2</sup> = -1.

    The impact on calculations is, of course, negligible. But it is more hygienic!
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  17. #16  
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    imaginary numbers are fun
    (a+bi)(c+di) is the multiplication of 2 complex numbers, just use ordinary math to solve it with the addition of i²=-1

    Quaternion is even more fun
    they arent Commutativity in multiplication and division[/quote]
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  18. #17  
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    lets not forget the Octonions, those aren't even associative.
    yes there is fun for everyone when it comes to imaginary numbers.
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  19. #18  
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    yeah lets party
    I am zelos. Destroyer of planets, exterminator of life, conquerer of worlds. I have come to rule this uiniverse. And there is nothing u pathetic biengs can do to stop me

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  20. #19  
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    the real party is in finding a practical use for Octonions. seriously whats the deal with there existance?
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  21. #20  
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    and Quaternion . they are fun but for what reason except party makes do they serve?
    I am zelos. Destroyer of planets, exterminator of life, conquerer of worlds. I have come to rule this uiniverse. And there is nothing u pathetic biengs can do to stop me

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  22. #21  
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    and where does it end? next thing you know theres sednions and other hypernumbers... seems like they don't want the party to end.
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  23. #22  
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    nope, seems like it
    where did i put the party hats?
    I am zelos. Destroyer of planets, exterminator of life, conquerer of worlds. I have come to rule this uiniverse. And there is nothing u pathetic biengs can do to stop me

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  24. #23  
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    Quote Originally Posted by Zelos
    nope, seems like it
    where did i put the party hats?
    (1+3i)(1-3i) to your right
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  25. #24  
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    I know abou quaternions and what not but what's the principle of octonians, I've wondered myself will people ever stop making up numbers! We have an infinite number of them anyway!
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  26. #25  
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    Quote Originally Posted by Robbie
    I know about quaternions and what not
    Well, I don't. Please explain them (in detail, if you don't mind).
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  27. #26  
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    ok, let me try (im not doing maths in college so maybe someoe else could discuss it in more depth) but it was discovered by irish mathematician Wiliiam hamelton who famously thought of it while walking along the grand canal in Dublin one evening.

    i2 =j2 = k2 = -1
    so:
    ij = -ji = k
    jk = -kj = i
    ki = -ik = j

    Quaternions re non commutative, which is properly expresse in matrix form and there's loads of other interesting stuff about them which was not understood at the time of invention (and I still dont!) but mybe if someone could help out...
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  28. #27  
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    Quaternions are important in computer graphics since they can represent 3D rotations in a very nice way (IMHO, much nicer than any other way I've seen). Full details here.
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  29. #28  
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    Quote Originally Posted by Robbie

    i2 =j2 = k2 = -1
    so:
    ij = -ji = k
    jk = -kj = i
    ki = -ik = j
    I don't see the "so", bolded in yours. Anyway that much I can read anywhere, but thanks for the attempt - it's just that you came on like an expert.
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  30. #29  
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    his thing is a bit wrong the correct thing is
    i²=j²=k²=ijk=-1
    I am zelos. Destroyer of planets, exterminator of life, conquerer of worlds. I have come to rule this uiniverse. And there is nothing u pathetic biengs can do to stop me

    On the eighth day Zelos said: 'Let there be darkness,' and the light was never again seen.

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  31. #30  
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    Quote Originally Posted by MagiMaster
    Quaternions are important in computer graphics since they can represent 3D rotations in a very nice way (IMHO, much nicer than any other way I've seen). Full details here.
    well now i'll never look at computer games in the same way ever again.
    thanks for the link.
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  32. #31  
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    Quote Originally Posted by wallaby
    well now i'll never look at computer games in the same way ever again.
    thanks for the link.

    That's what I was thinking. Nice link btw.

    I've looked up quaternions numerous times but never got an example for their use, only equations and identities which isnt my strong suit. I have a better idea now to go with those rules.
    I've always wanted to know why. Just takes me a little longer than most, lol.
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  33. #32 Magic Squares 
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    Magic Squares.

    Almost like sudoko but a THOUSAND times better !!!
    The hand of time rested on the half-hour mark, and all along that old front line of the English there came a whistling and a crying. The men of the first wave climbed up the parapets, in tumult, darkness, and the presence of death, and having done with all pleasant things, advanced across No Man's Land to begin the Battle of the Somme. - Poet John Masefield.

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  34. #33  
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    Quote Originally Posted by Zelos
    his thing is a bit wrong the correct thing is
    i²=j²=k²=ijk=-1
    Um Zelos, your definition and the one given previously are equivalent - so how is his wrong?
    As is often the case with technical subjects we are presented with an unfortunate choice: an explanation that is accurate but incomprehensible, or comprehensible but wrong.
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