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Loci for z - curly bracket use

  1. Oct 24, 2009 #1
    Hello,

    I'm just unsure about the use of curly brackets in expressing loci of complex number z,
    to which I've been introduced in our calculus class:
    - I can't find elaborations on this on-line :confused:
    - No material I can find in our text books(???)

    I do know it is used to express the locus for a moving point z on complex axes, looks something like this:
    {z: .....}

    following the colons is the expression I presume to be the same as the usual modulus expression which I am used to when working out loci. Sometimes there are equal / unequal signs inside the curlies etc.(??) (to give area locus...?) (e.g. the modulus < 2 giving area bound by circle, not including circle itself of course)

    I am familiar with replacing z with x + yi, manipulating, expanding (squaring modulus) and those matters concerning loci, but I don't understand exactly how these curly brackets are used.

    If anyone knows, could please explain to me? That would be very kind, thanks in advance :smile:

    Cheers,
    Yotam :smile:
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Oct 24, 2009 #2

    CompuChip

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    You mean you are having trouble with the notation {z: |z| < 2} ?

    In that case it is good you ask, because this is important in mathematics in general. In fact I think it is very bad that you haven't learned about sets as one of the very first things.

    So here's a (hopefully turning out to be) brief explanation. In mathematics, we work with sets, which can be seen simply as a collection of objects grouped together. They are denoted with curly brackets, for example:
    your study group's ages: { 14, 15, 16, 17, 18 }
    primary colors: { red, green, blue }
    people in your family: { mother, father, elder sister, younger sister, grandpa, grandma, aunt }
    But they can also be infinitely large, for example:
    positive integers: { 1, 2, 3, 4, .... }
    all fractions: {1, 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5, ... }

    Some of these sets are given a standard name, for example
    N (also [itex]\mathbb{N}[/itex]): the set {0, 1, 2, 3, 4, ...} (sometimes excluding 0) of all natural numbers
    Z (also [itex]\mathbb{Z}[/itex]): the set {..., -3, -2, -1, 0, 1, 2, 3, ...} of integers
    Q (also [itex]\mathbb{Q}[/itex]): the set of all fractions
    R (also [itex]\mathbb{R}[/itex]): the set of all real numbers
    C (also [itex]\mathbb{C}[/itex]): the set of all complex numbers.

    For finite sets, we can simply list all the elements, but for finite sets this is more tricky. For example, the set of all non-negative even integers strictly smaller than 10 can be simply given by {0, 2, 4, 6, 8}. However, the set of all non-negative even integers cannot be written in this way. Of course, we can write {0, 2, 4, 6, 8, 10, ...} and anyone will understand what you mean by the dots, but it's not really mathematically rigorous (for example, you might also have attempted to write down {0, 2, 6, 8, 10, 20, 40, 80, 100, 200, 400, 800, 1000, ....} instead of {0, 2, 4, 6, 8, 10, 12, 14, 16, 18, ...} for all we know).
    So we have invented another notation, which allows us to define a set like this:
    [tex] \{ x \in S \mid P \}[/tex]
    Here, S is itself a set and P is some property (which might be given in mathematical notation or simply in words). This should be read as: the set of all x in the set S, which (additionally) satisfy the property P.
    For example, I could define the set of all non-negative even integers in a number of ways as
    [tex] \{ x \in \mathbb{N} \mid x \text{ is even} \} [/tex] or [tex] \{ x \in \mathbb{N} \mid x = 2k \text{ for some } k \in \mathbb{N} \} [/tex]
    [tex] \{ x \in \mathbb{Z} \mid x \text{ is non-negative and even} \} [/tex]
    [tex] \{ x \in \mathbb{Z} \mid x > 0 \text{ and } x \text{ is even} \} [/tex]
    [tex] \{ x \in \mathbb{R} \mid x \text{ is a non-negative even integer } \} [/tex]
    ...

    Similarly, I can define for example the set of complex numbers as
    [tex] \mathbb{C} = \{ x + i y \mid x, y \in \mathbb{R} \} [/tex]
    by which I mean: the set C is defined by all numbers of the form x + i y, for which x and y are real numbers (i.e. elements of the set R).
    The right half of the complex plane, is then for example
    [tex] \mathbb{C}_+ = \{ x + i y \in \mathbb{C} \mid x > 0 \}[/tex]

    Is this clear so far?
     
  4. Oct 25, 2009 #3
    wh-ha :smile: very clear, thank-you!
    So this is essentially (with regard to some complex number z) collecting a SET of points...? which is the locus (did I get this right?) and the condition is what you pointed out as the expression following the vertical bar {...|condition}

    So... where does the z followed by colons coming into this? are the colons the same thing as vertical bar? i.e. {z: ...} same as {z| ...} ?
    so something as simple as asking for locus like this:
    - find locus etc. etc. for ....
    |z| = 3
    x^2 + y^2 = 9
    which I DO understand well (circle radius 3 centre at origin), can be written also like this?
    {z: |z| = 3}??? as simple as that?
    so, again, is z an element(?) of some set(?) and |z| = 3 the property / condition to be satisfied by the set of points that we are trying to collect (all points of distance 3 from origin i.e. circle)?
    So what is the 'standard' notation of loci with curled brackets (i.e. expressing as sets) when concerning complex numbers / complex number plane? the colons...?
    Thank-you again for the fantastic explanation :smile:
    Yotam
     
  5. Oct 25, 2009 #4

    Office_Shredder

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    Looks like you've pretty much nailed it. There is no meaningful difference between a colon or a vertical bar, just different notation different people use
     
  6. Oct 26, 2009 #5
    So it doesn't make a difference... okay thank-you!
    I would guess though now that I know, colons are better in this complex number context, so as not to confuse with modulus signs! :smile: :smile: :smile:

    Thank-you both misters CompuChip and Office Shredder for help / explanations / confirmations
    Cheers :smile:
     
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