Line through two points in field C X C

**Pointer** · Apr 27th 2010, 05:06 AM

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).

**Swlabr** · Apr 27th 2010, 07:12 AM

Originally Posted by Pointer

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 $(a, b, c, d)$ .

**HallsofIvy** · Apr 27th 2010, 07:42 AM

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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