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Math Help - Complex Numbers basics problem

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    Complex Numbers basics problem

    hi... there has been a question which always crosses my mind whenever i see complex numbers...
    we say..
    |x| is defined as squareroot(x^2)
    so,
    why do we not write |i| as i... (squareroot(i^2)=i)? (note that 'i' means iota)
    now u'll say that modulus function has also been defined as distance of the argument(i, in this case) from the origin..., which u say is i unit from the origin.. but why can't be the distance of a point from the origin be 'i'(iota) units ???? whats wrong with it ???

    let me frame an example here..
    u draw a graph with Y-axis scale as 1 unit = 3... then in reference to this graph, do we write |3|=1 (as distance from origin=1 unit) ????? the answer is a no.... then why cant we treat iota in the same way?? here also we are defining 1 unit of y-axis as 'i'....
    Last edited by pranjvas; February 19th 2013 at 09:00 AM.
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  2. #2
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    Re: Complex Numbers basics problem

    Quote Originally Posted by pranjvas View Post
    hi... there has been a question which always crosses my mind whenever i see complex numbers...
    we say..
    |x| is defined as squareroot(x^2)
    so,
    why do we not write |i| as i... (squareroot(i^2)=i)? (note that 'i' means iota)
    now u'll say that modulus function has also been defined as distance of the argument(i, in this case) from the origin..., which u say is i unit from the origin.. but why can't be the distance of a point from the origin be 'i'(iota) units ???? whats wrong with it ???
    What's wrong with it is that "distance" is a positive real number (have you ever seen a meter stick or ruler with "i" on it). In order to get that we define the absolute value of a complex number as |z|= \sqrt{z\overline{z}} where \overline{z} is the "complex conjugate" of z. In particular, the complex conjugate of i is -i so that |i|= \sqrt{i(-i)}= \sqrt{1}= 1.

    let me frame an example here..
    u draw a graph with Y-axis scale as i unit = 3... then in reference to this graph, do we write |3|=1 (as distance from origin=1 unit) ????? the answer is a no.... then why cant we treat iota in the same way?? here also we are defining 1 unit of y-axis as 'i'....
    I don't understand what you are trying to say here. What does "i unit= 3" mean? It sounds like you are just trying to redefine "i" to be "3i". You can't do that and have the same number system as before. If you start with "non-sense", labeling "3" as a "1", leads to non-sense: |3|= 1!

    (This reminds me of a joke Abraham Lincoln used to tell: Question: if you call a lambs tail a leg, how many legs does it have? Answer: four- calling a tail a leg doesn't make it one! In mathematics we are pretty free to "call" things what we want. But once you have labled a point on the y axis "3" you cannot then also label it "i".)
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    Re: Complex Numbers basics problem

    Quote Originally Posted by HallsofIvy View Post
    What's wrong with it is that "distance" is a positive real number (have you ever seen a meter stick or ruler with "i" on it). In order to get that we define the absolute value of a complex number as |z|= \sqrt{z\overline{z}} where \overline{z} is the "complex conjugate" of z. In particular, the complex conjugate of i is -i so that |i|= \sqrt{i(-i)}= \sqrt{1}= 1.


    I don't understand what you are trying to say here. What does "i unit= 3" mean? It sounds like you are just trying to redefine "i" to be "3i". You can't do that and have the same number system as before. If you start with "non-sense", labeling "3" as a "1", leads to non-sense: |3|= 1!

    (This reminds me of a joke Abraham Lincoln used to tell: Question: if you call a lambs tail a leg, how many legs does it have? Answer: four- calling a tail a leg doesn't make it one! In mathematics we are pretty free to "call" things what we want. But once you have labled a point on the y axis "3" you cannot then also label it "i".)
    i m sorry, it was a typing mistake.. what i meant was "1 unit = 3"... and not "i unit = 3"
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