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Math Help - Complex number verification.

  1. #1
    Super Member Showcase_22's Avatar
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    Complex number verification.

    These are the questions that are leaving me slightly puzzled:

    Which of the following are true?

    i) Every real number is a complex number.
    ii) The square of an imaginary number is a negative real number.
    iii)there is a complex number which is both real and imaginary.

    My second question is: "for which positive integral values of n is i^n real?


    This is what I put:

    i)I put false because every real number can be expressed as a+0i=a. The complex part does not exist.

    ii) (a+bi)^2=a^2+2abi-b^2=a^2-b^2+2abi

    Using that I put false because an imaginary part (2ab) exists.

    iii) For this I put false. a+bi was my example of a complex number. Since both parts have to exist together to make it a complex number, a number cannot be both real and complex.

    For the second question I did [tex](-1)^ \frac{n}{2} [\MATH]. Therefore n has to be even. This doesn't seem quite right since "values" implies I either have to write a number or an inequality.

    It seemed a little fishy to get three false answers for each other the questions. I was hoping someone out there could verify that my answers are (hopefully!) correct.

    Thanks in advance!
    Last edited by Showcase_22; September 30th 2008 at 09:00 AM.
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  2. #2
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    i) is true as a + 0i is a complex number
    ii) is also true, an imaginary number has no real part, so your squaring ai
    iii) I am unsure about this, if 0 counts then yes otherwise it is false.

    EDIT: Sorry i misread the second question, yes your right n must be even.

    Bobak
    Last edited by bobak; September 30th 2008 at 09:24 AM. Reason: misread question
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  3. #3
    Moo
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    Hello
    Quote Originally Posted by Showcase_22 View Post
    These are the questions that are leaving me slightly puzzled:

    Which of the following are true?

    i) Every real number is a complex number.
    ii) The square of an imaginary number is a negative real number.
    iii)there is a complex number which is both real and imaginary.

    My second question is: "for which positive integral values of n is i^n real?


    This is what I put:

    i)I put false because every real number can be expressed as a+0i=a. The complex part does not exist.
    Actually, the complex part does exist and you even wrote it.

    A complex number is in the form a+ib where a and b can be any real number (including 0)

    ii)(a+bi)^2=a^2+2abi-b^2=a^2-b^2+2abi

    Using that I put false because an imaginary part (2ab) exists.
    Note the difference between imaginary number and complex number. An imaginary is in the form ai, whereas a complex number is in the form a+bi.
    A real number is defined as a complex number having its imaginary part equal to 0.
    An imaginary number is defined as a complex number having its real part equal to 0.

    iii) For this I put false. a+bi was my example of a complex number. Since both parts have to exist together to make it a complex number, a number cannot be both real and complex.
    0 is a real and an imaginary number.


    For the second question I did (-1)^ \frac{n}{2} . Therefore n has to be even. This doesn't seem quite right since "values" implies I either have to write a number or an inequality.
    Ok, I didn't see the second question...
    Look the pattern...
    i^1=i
    i^{2}=-1
    i^{3}=-i
    i^4=1

    It seemed a little fishy to get three false answers for each other the questions. I was hoping someone out there could verify that my answers are (hopefully!) correct.
    yah cap'tain !
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  4. #4
    Super Member Showcase_22's Avatar
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    really? a+0i is complex????? You learn something new every day!

    oh yeah! part ii) woulod be like that. I was using a "complex" number. :s

    iii) I am unsure about this, if 0 counts then yes otherwise it is false.

    Are you thinking of 0+0i? I don't actually know if that's real or imaginary.....i'm a little shaky on this (must read up!)


    The second question is false.
    umm, that wasn't a true or false question.

    The question was:

    My second question is: "for which positive integral values of n is i^n real"?

    and I put even numbers. It just doesn't seem right....
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  5. #5
    Super Member Showcase_22's Avatar
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    Hang on a second, let's just summarise what we've deduced so far:

    i) Every real number is a complex number.


    This is true because a complex number can be expressed as a+bi and b is allowed to be 0.

    ii) The square of an imaginary number is a negative real number.


    This is also true because (bi)^2=b^2i^2=-b^2. This ( -b^2) is always negative since b^2 is always postive.

    iii)there is a complex number which is both real and imaginary.


    This is true since, quoting Moo, "0 is a real and imaginary number". Thanks Moo!

    With regards to question 2:

    My second question is: "for which positive integral values of n is i^n real?


    The solution is n must be an even number (take a nice glance at Moo's pattern, thanks yet again).

    Ps. I started writing this post after I noticed Moo and Bobak had edited some of theirs so there might not be any point in this. I guess it helps anyone else reading this thread!

    yah cap'tain !
    Ready the sails!! weigh the anchor!

    I could have been a pirate, or a financier but then I thought "why not try a more challenging career?......."
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  6. #6
    Moo
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    Quote Originally Posted by Showcase_22 View Post
    Hang on a second, let's just summarise what we've deduced so far:



    This is true because a complex number can be expressed as a+bi and b is allowed to be 0.



    This is also true because (bi)^2=b^2i^2=-b^2. This ( -b^2) is always negative since b^2 is always postive.



    This is true since, quoting Moo, "0 is a real and imaginary number". Thanks Moo!

    With regards to question 2:



    The solution is n must be an even number (take a nice glance at Moo's pattern, thanks yet again).

    Ps. I started writing this post after I noticed Moo and Bobak had edited some of theirs so there might not be any point in this. I guess it helps anyone else reading this thread!



    Ready the sails!! weigh the anchor!

    I could have been a pirate, or a financier but then I thought "why not try a more challenging career?......."
    Perfect (even the song lol ! though i don't know it )

    But because I'm picky, I would add the definitions of complex, real and imaginary numbers (in this order), because it's the key of the solutions, especially the third
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  7. #7
    Super Member Showcase_22's Avatar
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    song is from Muppets treasure island!! Every lucky child has seen it, You must be one of the unlucky ones!

    Okay then. I'll hit google for the definitions.
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