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    What is the difference between real constant and complex constant?

    For example, 3 is a real constant, but is it also a complex constant? Can you give some examples of complex constant that are not real constant?

    Thanks in advance !
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    Any real number is also a complex number. Every complex number is of the form z = x + i y, where x and y are real numbers, and i = \sqrt{-1}.
    x is the real part of z and y is the imaginary part.

    So any complex number with y nonzero is not a real number. Examples of complex constants that are not real: 3 + 2 i, -5 + 19 i, or just i itself.
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    (Original post by liewchuan)
    Any real number is also a complex number. Every complex number is of the form z = x + i y, where x and y are real numbers, and i = \sqrt{-1}.
    x is the real part of z and y is the imaginary part.

    So any complex number with y nonzero is not a real number. Examples of complex constants that are not real: 3 + 2 i, -5 + 19 i, or just i itself.
    Thanks but now I have one more question.

    Does that mean that every complex number is a (complex) constant? If not, can you give some examples of complex number that are not complex constant?

    Thanks !
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    Every real number is also a complex number - it just has an imaginary part that is zero.

    When you consider the complex numbers, some of them are also real numbers (those with imaginary part zero) but others are not real such as 3+4i.

    An important special case to remember is 0, which is real, imaginary and complex. This is because it can be expressed as 0, 0i or 0+0i
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    All numbers are constants aren't they.

    In an equation you might have an unknown value such as x or z which can take on many values in which case it's called a variable. If it can take on complex values it is called a complex variable. An constant C is not really a variable but merely a place-holder for "any particular value". That maybe is a subtle difference (if any).
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    (Original post by ttoby)
    Every real number is also a complex number - it just has an imaginary part that is zero.

    When you consider the complex numbers, some of them are also real numbers (those with imaginary part zero) but others are not real such as 3+4i.

    An important special case to remember is 0, which is real, imaginary and complex. This is because it can be expressed as 0, 0i or 0+0i
    I understand that, but what I was asking is whether you mind giving me some examples of complex number that are not (complex constant)?

    For example, why is 3 + 2i a complex constant? Is there any complex number that is not complex constant ? Thanks
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    (Original post by melbss01)
    Thanks but now I have one more question.

    Does that mean that every complex number is a (complex) constant? If not, can you give some examples of complex number that are not complex constant?

    Thanks !
    We say a number is a constant if it never changes. For example the formula for the area of a circle is \pi r^2. Here pi and 2 are constants as they never change. However the radius r can change so that is not a constant.

    An example of a complex constant is i which never changes.

    If you just write down some complex number then that automatically makes it a constant since it's not going to change if you're only referring to one specific complex number.

    However if you have an expression containing a variable (such as z) that can take a variety of complex numbers then z is not a constant. For example I could write z^2 where z is complex but not a constant. You could, if you wanted, plug in a value for z and calculate z squared. But writing z^2 on its own doesn't tell you that z has to be any particular value.
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    (Original post by melbss01)
    I understand that, but what I was asking is whether you mind giving me some examples of complex number that are not (complex constant)?
    I wrote that post before seeing the follow up posts, hence I didn't answer your later question. I've now seen it so look at my post above for an answer.
    For example, why is 3 + 2i a complex constant? Is there any complex number that is not complex constant ? Thanks
    3+2i is a complex constant because it's complex and because it's a constant. The reason it's constant is because we've specified that it has to take on one single value (3+2i).
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    (Original post by ttoby)
    I wrote that post before seeing the follow up posts, hence I didn't answer your later question. I've now seen it so look at my post above for an answer.
    3+2i is a complex constant because it's complex and because it's a constant. The reason it's constant is because we've specified that it has to take on one single value (3+2i).
    Thanks! By the way, how do we differentiate 3 + 2i ? I have just learned the subject and I am kind of confused. Is the answer 0 ?
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    (Original post by melbss01)
    Thanks! By the way, how do we differentiate 3 + 2i ? I have just learned the subject and I am kind of confused. Is the answer 0 ?
    Yes. The formal reason for this is:

    If we take a function f:\mathbb{C}\rightarrow\mathbb{C  } (i.e. a function which takes complex functions as input then outputs complex numbers)

    defined as f(z)=3+2i

    Then the derivative f'(z) is defined as f'(z)=\lim_{h\rightarrow0}\dfrac  {f(z+h)-f(z)}{h} = \lim_{h\rightarrow0}\dfrac{(3+2i  )-(3+2i)}{h} = \lim_{h\rightarrow0}0 = 0
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    (Original post by ttoby)
    Yes. The formal reason for this is:

    If we take a function f:\mathbb{C}\rightarrow\mathbb{C  } (i.e. a function which takes complex functions as input then outputs complex numbers)

    defined as f(z)=3+2i

    Then the derivative f'(z) is defined as f'(z)=\lim_{h\rightarrow0}\dfrac  {f(z+h)-f(z)}{h} = \lim_{h\rightarrow0}\dfrac{(3+2i  )-(3+2i)}{h} = \lim_{h\rightarrow0}0 = 0
    If z = x + iy,

    Is the partial derivative of z w.r.t x is 1?
    Is the partial derivative of z w.r.t y is i?

    Thanks.
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    (Original post by melbss01)
    If z = x + iy,

    Is the partial derivative of z w.r.t x is 1?
    Is the partial derivative of z w.r.t y is i?

    Thanks.
    If you're considering this as a function from \mathbb{R}^2 to \mathbb{C} then I'm pretty sure you can say that. However if you're at university then avoid using that result unless you've been taught about partial derivatives involving complex numbers.
 
 
 
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