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Math Help - Line through two points in field C X C

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    Line through two points in field C X C

    This should be simple but is giving me fits. Given two points in field C X C (C is field of complex numbers), how do you determine the equation of the line? For example, two points are (2,5) and (i,10).
    Last edited by Pointer; April 27th 2010 at 05:31 AM.
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    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by Pointer View Post
    This should be simple but is giving me fits. Given two points in field C X C (C is field of complex numbers), how do you determine the equation of the line? For example, two points are (2,5) and (i,10).
    Try thinking of \mathbb{C}^2 = \mathbb{C} \times \mathbb{C} as \mathbb{R}^4. You know what lines here look like, and elements in \mathbb{C}^2 look like (a+ib, c+id) but can be thought of as (a, b, c, d).
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    Or, just do it as you would in R^2: y= mx+ b but now m, b, x, and y can be complex numbers.

    Since (2, 5), x= 2 and y= 5 must satisfy the equation. 5= 2m+ b. Since (i, 10) is on the line, x= i and y= 10 must satisfy the equation. 10= im+ b.

    subtracting the first equation from the first eliminates b and gives 5= (i- 2)m. Then m= \frac{5}{i- 2}= \frac{5}{i- 2}\frac{-i- 2}{-i- 2} = \frac{10- 5i}{1+ 4}= 2- i. Now you can use either of those equations to solve for b.
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