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 :)

 

[CellCoree]

I can provide a response to this concept. In mathematics, complex numbers are represented by a combination of a real number and an imaginary number, usually written as a + bi, where a is the real part and bi is the imaginary part. The imaginary part, represented by the letter i, is the square root of -1. This allows us to work with numbers that cannot be represented on a traditional number line.

Now, when we consider parallel lines in the context of complex numbers, we are actually referring to lines in the complex plane. This is a two-dimensional plane, where the horizontal axis represents the real numbers and the vertical axis represents the imaginary numbers. In this plane, parallel lines can be represented by equations of the form z = a + bi, where a and b are constants and z is a complex number.

So, what does it mean for parallel lines to meet at infinity in this context? It means that as we move along these parallel lines in the complex plane, they will eventually converge at a single point, which is known as the point at infinity. This point is not a specific value, but rather a concept that represents the idea of infinite distance. In other words, as we continue to move along these parallel lines, they will never truly intersect, but they will get closer and closer to each other until they are essentially at the same point, which we can think of as the point at infinity.

In terms of a graph, we can represent this concept by plotting parallel lines on the complex plane and observing how they approach the point at infinity as we move along them. This idea is important in fields such as complex analysis and projective geometry, where we use the concept of the point at infinity to solve complex problems and better understand the behavior of complex numbers.

In conclusion, the statement "parallel lines meet at infinity" in the context of complex numbers means that as we move along these lines in the complex plane, they will approach a single point at infinite distance. This concept has important applications in mathematics and helps us better understand the behavior of complex numbers.