zero

For any SPECIFIC definition of "point", the longer line has twice as many points. For the abstract "infinitely tiny" point, the answer is undefined. Another example of how abstractions can produce bizarre twists in logic! :)

Here's the proof that the lines have the same amount of points. It's quite counterintuitive - like lots of maths that talks about infinities. (You'll probably have to click to see it)

Bugs Bunny: Would you like to shoot me now or wait 'til you get home? Daffy Duck: Shoot him now! Shoot him now! Bugs Bunny: You keep outta this! He doesn't have to shoot you now! Daffy Duck: He does SO have to shoot me now! [to Elmer] Daffy Duck: I demand that you shoot me now! [Elmer raises his gun. As Daffy sticks his tongue out at Bugs, he is shot] Daffy Duck: Let'th run through that again. Bugs Bunny: Okay. [neutral toned] Bugs Bunny: Wouldja like to shoot me now or wait till ya get home. Daffy Duck: [neutral toned] Shoot him now, shoot him now. Bugs Bunny: [neutral toned] You keep outta dis, he doesn't hafta shoot you now. Daffy Duck: [with expression] HA! THAT'TH IT! HOLD IT RIGHT THERE! [to audience] Daffy Duck: Pronoun trouble. [to Bugs] Daffy Duck: It'th not "He doethn't have to shoot [pointing to Bugs] Daffy Duck: *you* now." It'th "He doethn't have to shoot [pointing to himself] Daffy Duck: *me* now." [with anger] Daffy Duck: Well, *I* thay he *does* have to shoot me now! [to Elmer] Daffy Duck: THO SHOOT ME NOW! [Elmer shoots him] Bugs Bunny: [Daffy stops short at Bugs] Yais? Daffy Duck: [Daffy puts himself back into position] Ohhhhhhh, no you don't. Not agian. Thorry. Thith time we'll try it from the other end. [to Elmer] Daffy Duck: Look, you're a hunter, right? Elmer Fudd: Wight. Daffy Duck: And thith ith Rabbit Theathon, right? Elmer Fudd: Wight. Bugs Bunny: And if he was a rabbit, what would you do? Daffy Duck: Yeah, if you're tho thmart, if I wath a rabbit, what *would* you do? Elmer Fudd: Well, I'd... Daffy Duck: [Elmer points his rifle at Daffy] Not again! [BANG! Bill falls down and Daffy puts it on his mouth again. To Bugs] Daffy Duck: Ha ha. Very funny. Ha ha ha ha. Daffy Duck: Now's your chance, "Hawkeye!" Shoot him! SHOOT HIM! Bugs Bunny: He's got me dead to rights, doc. Would you like to shoot him now or wait 'till you get home? Daffy Duck: Oh no you don't. Not THIS time! [to Elmer] Daffy Duck: Wait until you get home! [Hiding in Bugs' burrow] Bugs Bunny: Go and take a peak up an' see if he's still around Daffy Duck: Right-O! [Daffy looks out the hole, gunshot heard; Daffy comes back down] Bugs Bunny: Is he still there? Daffy Duck: [dazed] Still lurking about! Bugs Bunny: I know! You go up an' act as a decoy an' lure 'im away. Daffy Duck: No more for me, thanks! I'm drivin'! [faints] Bugs Bunny: Ah, well; like they say, never send a duck to do a rabbit's job. Daffy Duck: [to Bugs in drag] Out of sheer honesty, I demand that you tell him who you are! Well? Haven't you anything to say? Out of sheer honesty? Huh? Bugs Bunny: [to Elmer, in a women's voice] Yes. I would just love a duck dinner. [Kisses Elmer, who then shoots Daffy in an amorous daze] Daffy Duck: Awfully unsporting of me, I know, but what the hey, I gotta have some fun. [pause] Daffy Duck: And besides, it's really duck season.

Either zero or infinity, take your pick.

two

None, both have a beginning and end point.

Based on the way the question is asked, there could be any number of "correct" answers, or perhaps "none". deemikay, I think gets the essence of it, but I think Stableboy has a valid point as well. deemikay's approach, however, is the only one that can produce a valid numeric result. When the question is asked "How many more points are there on the longer line", there are a couple of ways one could attack the problem. Clearly there are twice as many points in the longer line, (lim[c(long line)/c(short line)]=2 where c(line) is the number of line segments of size dx that can fit in line and the limit is taken as dx approaches 0) however, this only gives us a ratio, and any attempt to describe this as a number will meet with failure. Since this gives us no answer to the question that is asked, the only thing one can do mathematically beyond pointing at the ratio, which is useful in practical terms for any discretization of the line segments, is to see if one is talking about the same kind of infinity in both cases. An engineer, I think would care about the ratio, but a mathematician would be more prone to consider the infinities. Upon examination, the infinities are the same. That is each point in the long line can be mapped to a point in the short line and vice versa. Thus, if you imagined each point taking its partner in the opposite line and skipping happily off to nirvana, then you would have no points left. Hence, you get a tidy numeric result, zero, which is the only answer that really fits the question.

1) Both segments have the same number of points, and it is also the number of point of a whole line: "Curiously, the number of points in a line segment is equal to that in an entire one-dimensional space (a line), and also to the number of points in an -dimensional space, as first recognized by Georg Cantor." Source: http://mathworld.wolfram.com/LineSegment.html 2) "In 1874, Cantor began looking for a 1–to–1 correspondence between the points of the unit square and the points of a unit line segment. In an 1877 letter to Dedekind, Cantor proved a far stronger result: there exists a 1–to–1 correspondence between the points on the unit line segment and all of the points in a p–dimensional space. About this discovery Cantor famously wrote to Dedekind: "Je le vois, mais je ne le crois pas!" ("I see it, but I don't believe it!")[23] The result that he found so astonishing has implications for geometry and the notion of dimension. In 1878, Cantor submitted another paper to Crelle's Journal, which again displeased Kronecker. Cantor wanted to withdraw the paper, but Dedekind persuaded him not to do so; moreover, Weierstrass supported its publication. Nevertheless, Cantor never again submitted anything to Crelle. This paper made precise the notion of a 1–to–1 correspondence, and defined denumerable sets as sets which can be put into a 1–to–1 correspondence with the natural numbers. Cantor introduces the notion of "power" (a term he took from Jakob Steiner) or "equivalence" of sets; two sets are equivalent (have the same power) if there exists a 1–to–1 correspondence between them." Source: http://en.wikipedia.org/wiki/Georg_Cantor 3) Here the full proof: "Theorem 12. The number of points on a finite line segment is the same as the number of points on an infinite ray. Until I can draw a GIF for this theorem, imagine a large letter "Z". Label the point at the upper left, C, that at the upper right, D, that at the lower left A. Imagine that the bottom bar continues infinitely to the right, and that point B is somewhere along it to the right of A. Proof. Let segment CD be parallel to ray AB. From point C we can draw a line through any point of AD, except D itself, and the line we draw will intersect AB somewhere; in this way we can pair any point on AD with exactly one point on AB. Conversely, from point C we can draw a line through any point of the ray AB, and the line we draw will intersect AD somewhere; in this way, we can pair any point on AB with exactly one point on AD. But this means the points on AD, minus D itself, and those on AB can be put into one-to-one correspondence. Hence they contain the same number of points. We have yet to say how many points are on AD (minus D itself) or the infinite ray AB, but we know it will be some infinite cardinality. Now since any infinite cardinal plus one equals the original infinite cardinal (Theorem 7), we may add back the point D, which we omitted above, without changing the cardinality of the set of points on the segment. Theorem 13. The number of points on a finite line segment is the same as the number of points on an infinite line. The proof is a simple variation on the proof for Theorem 12. Imagine the mirror image of the figure used in Theorem 12 (a "backward Z"), with a new point, D', to the left of C, and a new point, B', to the left of A. We would then prove that two line segments (AD and AD') together contain the same number of points as the infinite line. But the two line segments together make one longer, though still finite segment. Therefore, the number of points in a finite segment equals the number of points on an infinite line. " Source: http://www.earlham.edu/~peters/writing/infapp.htm

Well, even if you have a line with infinitely many numbers or just have 4 for example, the longer one will have 8, so 8/4=2. So in this case it will have 4 more which equals the shorter one. Absolute value is there for some reason, don't you think? So the answer "equals the shorter one" which can have "infinite values". Good Luck!. (Source: Graduate Math Student).