Results 1 to 9 of 9

Math Help - help on basic properties of complex number

  1. #1
    Newbie
    Joined
    Aug 2010
    Posts
    4

    help on basic properties of complex number

    I got confused on the operation and representation of complex numbers. It is
    quite basic, and please help me understand. My question is as follows:

    (1) Since 1=exp(2*pi*i), it seems that, for any real number a, we may have

    exp(a*i)=exp((2*pi*i)*(a/(2*pi)))=(exp(2*pi*i))^(a/(2*pi))
    = 1^(a/(2*pi))
    =1

    (2) For any complex number z, since 1+ exp(2*pi*i), it seems that

    z^(1/4)=(1*z)^(1/4)=((exp(2*pi*i))^(1/4)*z^(1/4)
    = (exp((pi/2)*i))*z^(1/4)
    = i*z^(1/4)
    by the same token, since 1=exp(-2*pi*i), it seems that

    z*(1/4)=(-i)*z^(1/4)

    All those are confusingly wrong, and I wish to understand why. Many thanks for you kindness to help.
    Last edited by rcl333; August 31st 2010 at 11:45 AM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    Joined
    Aug 2009
    From
    Israel
    Posts
    976
    a^{b  c} = (a^b)^c is not generally true for complex numbers.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member
    Joined
    Mar 2008
    Posts
    934
    Thanks
    33
    Awards
    1
    Quote Originally Posted by rcl333 View Post
    I got confused on the operation and representation of complex numbers. It is
    quite basic, and please help me understand. My question is as follows:

    (1) Since 1=exp(2*pi*i), it seems that, for any real number a, we may have

    exp(a*i)=exp((2*pi*i)*(a/(2*pi)))=(exp(2*pi*i))^(a/(2*pi))
    = 1^(a/(2*pi))
    =1

    (2) For any complex number z, since 1+ exp(2*pi*i), it seems that

    z^(1/4)=(1*z)^(1/4)=((exp(2*pi*i))^(1/4)*z^(1/4)
    = (exp((pi/2)*i))*z^(1/4)
    = i*z^(1/4)
    by the same token, since 1=exp(-2*pi*i), it seems that

    z*(1/4)=(-i)*z^(1/4)

    All those are confusingly wrong, and I wish to understand why. Many thanks for you kindness to help.
    Hi rcl333,

    You confusion is due to the fact that some of the identities we take for granted about exponentiation of real numbers do not hold for complex numbers. In particular, it is not true that (e^a)^b = e^{ab}. Try reading the section on powers of complex numbers on this web page,

    Exponentiation - Wikipedia, the free encyclopedia

    especially the sub-section titled "Failure of Power and Logarithm Identities". Then if you are still confused, let us know.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Newbie
    Joined
    Aug 2010
    Posts
    4
    I am extremely grateful for your kind and clear explanation with the Wikipedia reference. Now I understand that this rule does not always work for complex numbers. Please help me understand also when can we use this rule? and when can not?, because in Wikipedia site with the subsection "roots of the unity", it used the expression that (exp(2*pi*i*(1/n)))^k=exp(2*pi*i*k/n) in terms of the rule (e^a)^b=e^(ab).

    Many thanks again.

    Quote Originally Posted by awkward View Post
    Hi rcl333,

    You confusion is due to the fact that some of the identities we take for granted about exponentiation of real numbers do not hold for complex numbers. In particular, it is not true that (e^a)^b = e^{ab}. Try reading the section on powers of complex numbers on this web page,

    Exponentiation - Wikipedia, the free encyclopedia

    especially the sub-section titled "Failure of Power and Logarithm Identities". Then if you are still confused, let us know.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Super Member
    Joined
    Mar 2008
    Posts
    934
    Thanks
    33
    Awards
    1
    The reason that

    (e^{2 \pi i (1/n)})^k = e^{2 \pi i k / n}

    is that, in this case, it doesn't matter which branch of the complex logarithm of e^{2 \pi i (1/n)} is chosen. The "obvious" choice is 2 \pi i (1/n), but 2 \pi i (1/n) + 2 m \pi i is also a valid choice for any integer m. (This is because e^{2 m \pi i} = 1.)

    But for any choice of the branch of the logarithm, we have
    e^{(2 \pi i (1/n) + 2 m \pi i) k} = e^{(2 \pi i k/n + 2 k m \pi i)} = e^{2 \pi i k/n} \cdot e^{2 k m \pi i} = e^{2 \pi i k/n} \cdot 1 = e^{2 \pi i k/n}
    because km is an integer.

    The reason this trick does not work in general for (e^a)^b is that bm is not, in general, an integer.
    Last edited by awkward; September 1st 2010 at 02:16 PM. Reason: typo
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Newbie
    Joined
    Aug 2010
    Posts
    4
    Thanks, it is very helpful and clear.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Newbie
    Joined
    Aug 2010
    Posts
    4

    help on followup question regarding basic properties of real/complex numbers

    I appreciate the help earlier on the exponent rules on complex numbers. Please forgive my naiveness to the following followup questions:

    Since for complex numbers, there are many branches of solutions for fractional power, how would we define which branch to choice, for example, if we have a function F(z)=a^z where a is a positive number (or even positive integer, this kind of function is basic in complex analysis), it shouldn't take multiple values, and which branch defines this function?

    In particular, even in the real number syatem, 1 and -1 all serve as the squared root of 1. What is the meaning of 1^(1/2)? Which value and when should be adopted?

    Thanks.
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Super Member
    Joined
    Aug 2009
    From
    Israel
    Posts
    976
    Principal value - Wikipedia, the free encyclopedia

    The meaning of \sqrt{x} for x \in \mathbb{R} is the positive number y such that y^2 = x.
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Super Member
    Joined
    Mar 2008
    Posts
    934
    Thanks
    33
    Awards
    1
    Quote Originally Posted by rcl333 View Post
    I appreciate the help earlier on the exponent rules on complex numbers. Please forgive my naiveness to the following followup questions:

    Since for complex numbers, there are many branches of solutions for fractional power, how would we define which branch to choice, for example, if we have a function F(z)=a^z where a is a positive number (or even positive integer, this kind of function is basic in complex analysis), it shouldn't take multiple values, and which branch defines this function?

    In particular, even in the real number syatem, 1 and -1 all serve as the squared root of 1. What is the meaning of 1^(1/2)? Which value and when should be adopted?

    Thanks.
    In the real numbers, the convention is that \sqrt{x} always denotes the positive root. The familiar identities like \sqrt{xy} = \sqrt{x} \cdot \sqrt{y} depend on this convention.

    In the complex numbers, no such convention works well in all cases, so if you need a single value of one of the multi-valued "functions", you must pick a branch of the function for that purpose. You may need to pick different branches for different problems. Fundamentally, however, one must simply be always aware of the fact that there are multiple possibilities for values of functions like the exponential and logarithmic functions.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Basic Properties of Number (Proof)
    Posted in the Algebra Forum
    Replies: 3
    Last Post: June 10th 2010, 08:24 PM
  2. Basic Diff EQ question: finding a function with given properties
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: September 9th 2009, 04:09 AM
  3. Five Properties: basic proofs
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: April 23rd 2009, 05:24 AM
  4. Expected Value and Variance (Basic Properties)
    Posted in the Advanced Statistics Forum
    Replies: 5
    Last Post: February 15th 2009, 07:45 AM
  5. complex number properties
    Posted in the Pre-Calculus Forum
    Replies: 2
    Last Post: April 15th 2008, 06:02 AM

Search Tags


/mathhelpforum @mathhelpforum