Complex numbers - parallel lines meet at infinity ? What does it mean?

[Femme_physics]

Complex numbers - "parallel lines meet at infinity"? What does it mean?

We started learning about complex numbers last week. One of the first things my teacher said was that "We learned that parallel lines never meet. But as it turns out, they meet at infinity."

I'm willing to accept it (sorta...even though it's rather bewildering). But, I mainly want to know how does that mathmatically relate to complex numbers? Where in complex numbers does it show that parallel lines meet at infinity? Is there a graph that shows parallel line meeting in a complex numbers chart, or something?

 

[HallsofIvy]

Just as you can "extend" the real numbers by including $+\infty$ and $-\infty$ at each end, so you can extend the complex numbers by adding a "circle" of different "infinities". In that (very extended) system, you can think of parallel lines as intersecting "at infinity" since each direction corresponds to a specific "infinity".

That's not just true for the complex numbers- its more of a geometric property of the two dimensional plane. Just as the usual laws of arithmetic do not apply to the "extended real numbers", so they do not apply to the "extended complex numbers".

 

[Mute]

The idea that parallel lines meet at infinity can be roughly motivated in the following way: suppose you have two non-parallel lines that meet at a point. You fix the points at which the lines cross the axes, and then pull on the intersection point and stretch the lines. As you pull that point further and further the slopes of the lines start looking more and more parallel. If you pull the original intersection point an infinite distance away from its starting point, the two lines are parallel.

As for how it relates to complex numbers, I'm not entirely sure where your professor wants to make the connection. My best guess is that he will introduce the concept of the Riemann sphere - a mapping of the complex plane onto the sphere, in which all points at infinity get mapped to the north pole. So, any lines parallel in the complex plane will meet at the north pole of the Riemann sphere.

 

[Femme_physics]

First off-- thanks for the replies.

It makes sense. I kinda want to ask "what IS infinity?" but I fear that is one of those greatly debated theories. It's hard to wrap your head even around this word! "Infinity"... I imagine...something that doesn't end...but you say...that this something that doesn't end, really kinda curls around or allows for a meeting point of...lines...things... this is really getting deep.

As for how it relates to complex numbers, I'm not entirely sure where your professor wants to make the connection. My best guess is that he will introduce the concept of the Riemann sphere - a mapping of the complex plane onto the sphere, in which all points at infinity get mapped to the north pole. So, any lines parallel in the complex plane will meet at the north pole of the Riemann sphere.

Well, my teacher (BA) didn't specifically make any relation to math, which is why I'm asking. I'm guessing though Riemann sphere is indeed the answer -- thanks! I'll look into it :)