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    Depends on context. In writing, \mathbb{C} \backslash \mathbb{R} would probably be the way to go to avoid ambiguity. Verbally you'd probally do the same: "C minus R".
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    (Original post by Glutamic Acid)
    Hmm, but are there numbers outside of the complex set? (I should know but I, err, don't.)
    Yes and no. There are other number systems: Quaternions, Octonians... but that's slightly different

    You would really need to start by telling me what you mean by "numbers"
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    (Original post by SimonM)
    Yes and no. There are other number systems: Quaternions, Octonians... but that's slightly different

    You would really need to start by telling me what you mean by "numbers"
    Quaternions was the system I was thinking about, but neglected to mention them as I don't really know what they're about. Come to think of it, I'm not sure I'd define a "number".
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    (Original post by Glutamic Acid)
    Quaternions was the system I was thinking about, but neglected to mention them as I don't really know what they're about. Come to think of it, I'm not sure I'd define a "number".
    Well, there are plenty of "types" of numbers, and since "numbers" are a useless concept without at least some kind of function defined on them. (Even if that function is something as simple as a "successor" function which gives our numbers the purpose to tell us how many cows we have)

    That said, you can't ask if there are any numbers outside the complex set, if you wont tell me what counts!
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    It seems to me that anything which admits a ring structure which can be extended to a division algebra would cover everything up to quaternions, but also would omit certain types of modular arithmetic. On the other hand, if we say anything constructed by extending or modifying the natural numbers is a "number system", we'd be forced to accept things like vectors and matrices as numbers. Hmmm. This is an interesting problem.
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    I was just about to say how you really need to define what is meant by a number i.e why would we call a complex number a "number" but not the spiral matrices which are equivalent to them.
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    (Original post by Erdős)
     \mathbb{H} ? (I have no experience of Quaternions).
    Yep. (Which I discovered from Wikipedia earlier.)
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    (Original post by toasted-lion)
    You do maths while out for a meal?
    He's picking up my bad (?) habits.
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    This is a hard problem.

    Solve

    x+2=3

    My method

    x+2=3

    (x+2)\times0= 3\times 0

    x\times 0 + 2 \times 0= 0

    x\times 0 = 0

    x=\frac{0}{0}

    x=0^0

    x=1 using the fact that a^0=1 for all a.

    Check
    1+2=3=3
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    if you were infront of me now I would probably pull my pants down and dickslap you Simplicity

    Spoiler:
    Show
    joke
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    simplicity :love: - you coming to the maths meet?
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    (Original post by Simplicity)
    This is a hard problem.

    Solve

    x+2=3

    My method

    x+2=3

    (x+2)\times0= 3\times 0

    x\times 0 + 2 \times 0= 0

    x\times 0 = 0

    x=\frac{0}{0}

    x=0^0

    x=1 using the fact that a^0=1 for all a.

    Check
    1+2=3=3
    Alternative:

    Spoiler:
    Show

    x + 2 = 3 \implies x = 1
    \implies (x)(1) = (1)
    \implies (x) = (1)^{-1}(1) = (1)

    There may be a quicker method using augmented matrices.


    Edit: I should have postmultiplied my inverse matrix, so my method is invalid.
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    How would you actually solve the equation using a matrix?

    Also, how you solve a equation like
    x^2-y^2=d
    and
    x+y=e
    lets say e=1 and d=2

    As I got a similar question in FP4 and couldn't anwser it as it was not in the books or the other pasts papers, I really don't want to read the uni book I got on matrices to find out.
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    they have a tinge of simulataneous equations
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    (Original post by Hathlan)
    He's picking up my bad (?) habits.
    I don't know about bad, I was thinking more unusual. "I'll have the soup of the day, but will you give me a minute to solve this cubic first?" I just can't imagine it.

    "Anything else with your meal madam, salt and vinegar perhaps?"
    "No thank you, just a protractor"

    No no, definitely can't imagine it.
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    (Original post by toasted-lion)
    I don't know about bad, I was thinking more unusual. "I'll have the soup of the day, but will you give me a minute to solve this cubic first?" I just can't imagine it.

    "Anything else with your meal madam, salt and vinegar perhaps?"
    "No thank you, just a protractor"

    No no, definitely can't imagine it.
    I usually have my own protractor if I suspect I'm going to need one. (Heehee).
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    (Original post by Hathlan)
    I usually have my own protractor if I suspect I'm going to need one. (Heehee).
    I had a feeling this was coming.
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    Here's something interesting I discovered a few days ago. (But it's probably in the lecture notes I haven't revised...)

    Suppose you have a quadric surface S described by a quadratic form: Let S = \{ \mathbf{x} \in \mathbb{R}^3 : \mathbf{x}^T \mathbf{A} \mathbf{x} = k \}, where \mathbf{A} is a constant symmetric matrix and k is a constant. Show that \mathbf{A}\mathbf{x} is normal to S at every point \mathbf{x} \in S.

    Spoiler:
    Show
    Hint: Consider the level sets of the quadratic form \mathbf{x}^T \mathbf{A} \mathbf{x}.
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    Zhen, surely you just take the grad of it and it drops out in 1 line?
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    Yes, but it was interesting nonetheless. I didn't expect
    1. that it was so simple, and
    2. that it would give an interpretation of A as something other than a matrix of coefficients.

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