Geometry
Introduction to Euclidean geometry
None
Intro to lines, line segments, and rays
Let's get familiar with the difference between lines, line segments, and rays. Hint: a ray is somewhere between a line and a line segment!
Discussion and questions for this video
 What I want to do in this video is
 think about the difference between a line segment, a line,
 and a ray.
 And this is the pure geometrical versions of these things.
 And so, a line segment is actually probably
 what most of us associate with a line in our everyday lives.
 A line segment is something just like that.
 For lack of a better word, a straight line.
 But why we call it a segment is that it actually
 has a starting and a stopping point.
 So, most of the lines that we experience
 in our everyday reality are actually
 line segments when we think of it
 from a pure geometrical point of view.
 and I know I drew a little bit of a curve here,
 but this is supposed to be completely straight,
 but this is a line segment.
 The segment is based on the fact that it
 has an ending point and a starting point,
 or a starting point and an ending point.
 A line, if you're thinking about it
 in the pure geometric sense of a line, is essentially,
 it does not stop.
 It doesn't have a starting point and an ending point.
 It keeps going on forever in both directions.
 So a line would look like this.
 And to show that it keeps on going on
 forever in that direction right over there, we draw this arrow,
 and to keep showing that it goes on forever
 in kind of the down left direction,
 we draw this arrow right over here.
 So obviously, I've never encountered something
 that just keeps on going straight forever.
 But in math that's the neat thing about math
 we can think about these abstract notions.
 And so the mathematical purest geometric sense of a line
 is this straight thing that goes on forever.
 Now, a ray is something in between.
 A ray has a well defined starting point.
 So that's its starting point, but then
 it just keeps on going on forever.
 So the ray might start over here,
 but then it just keeps on going.
 So that right over there is a ray.
 Now, with that out of the way, let's actually
 try to do the Khan Academy module
 on recognizing the difference between line segments, lines,
 and rays.
 And I think you'll find it pretty straightforward based
 on our little classification right over here.
 So, let me get the module going.
 Where did I put it?
 There you go.
 All right.
 So what is this thing right over here?
 Well, it has two arrows on both ends,
 so it's implying that it goes on forever.
 So this is going to be a line.
 Let's check our answer.
 Yeah, it's a line.
 Now it's taking some time, oh, correct, next question.
 All right, now what about this thing?
 Well, once again, arrows on both sides.
 It means that this thing is going
 to go on forever in both directions.
 So once again, it is a line.
 Fair enough.
 Let's do another one.
 Here we have one arrow, so it goes on forever
 in this direction, but it has a welldefined starting point.
 So it starts there, and then goes on forever.
 And if you remember, that's what a ray is.
 One starting point, but goes on forever.
 Or one way to think about it, goes on forever
 in only one direction.
 So that is a ray.
 So let's do another question.
 This right over here, you have a starting point and an ending
 point, or you could call this the start point and the ending
 point, but it doesn't go on forever in either direction.
 So this right over here is a line segment.
 There you go.
 So hopefully that gives you enough to work
 your way through this module.
 And you might notice, when I did this module right here,
 there is no video.
 And that's exactly what this video is.
 It's the video for this module.
Be specific, and indicate a time in the video:
At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?

Have something that's not a question about this content? 
This discussion area is not meant for answering homework questions.
would an infinite line and an infinite ray be equally long? That's my question.
Yes. It's a tricky concept because it feels like an infinitely long ray is only half as long as an infinitely long line. But technically, half of infinity is still infinity.
no, look at set theory as an example. if there is a set that extends infinitely to all the positive numbers, and then there is a set that extends infinitely in both directions, with negative numbers and positive numbers, they are not equal set, because even though both are infinite, you cannot match up each element os the positive set with each element of the negative set. In other words, for every centimeter of the ray, there would be twice as many centimeter of line, therefore the line is longer
The definition of something that is infinite, is that it never ends. As someone else said, half of something that is infinite is still infinite. Think of it this way: If you have something that never ends,no matter how many times you divide it, it will still go on forever. You cannot simply put it into "real life" examples relating to us here on Earth.
Quite an interesting discussion here. Are we are even measuring it in reality at all, or as solely a mathematical concept? In the real physical world, I would think it depends on if space itself is curved or flat. I don't think curved space could be infinite. But I'm not an astrophysicist, just offering my two cents.
@Stuart Wickard, you cannot compare infinity=infinity/2 TO 1=1/2, because as one is an actual or definite number, infinity is more of a concept or an estimate, and nobody knows what a half of it is. that's why infinity divided by two is still an infinite number(infinity)
A line is an entirely different concept as a ray. It cannot be answered fully no matter what you do, but if necessary, you should think about the following. The answer depends on whether you are using Euclidian or nonEuclidian geometry. It further depends on whether you are using practical or theoretical science. In Euclidian geometry, the ray would not meet its starting point from the other side like you would if you went around the world. Therefor, in practical science the line would be longer than the ray. However, since infinite means going on forever, the ray would go on forever and so would the line, they would be the same length in theoretical science! In nonEuclidian geometry, the ray would basically wrap around the universe like you might wrap around the world. Therefor, in that case it would meet itself from the other side, and so pass through the same points as the line would, so in practical science they would be the same length. But in theoretical science, the line would cover twice the universe's length by the time the ray would cover the universe's length once, so in theoretical science, the line would be longer. If you see this question on a math test, writing down that both are true or neither are true or even that it is impossible to answer would be good, but you should explain why too. Does this help?
不相同，直线与射线是2个概念的，射线有起始点
They would, because only one side of a ray goes on to infinity, double infinity (which is what happens with a line) is still the same as normal infinity. So yes, a ray and a line going on forever would be the same length as each other.
there are different kinds of infinity. Read about the mathematician Cantor, and the concept of Aleph.
no because a line does not stop a ray starts from one point,but a line just goes on forever
it is a tricky concept.it feels like an infinitely long ray is only half as long as an infinitely long line. But technically, half of infinity is still infinity.
The ray is equally long on one way and on the other way it stops or ends so it shorter. Over all though they both go to infinity so the are equal if u count both ways.
Yes, Because if you think hard about it you will find out that they both keep on going forever and will never stop so they will always be the same
I would say not because the line goes infinite in both directions and a ray would go in one direction,so if you then stop it the line will be longer.Hope this helped!:D
It's like the theory of the universe being infinite. If you assume this to be true, and try to calculate the percentage of the space in the universe that we occupy, you couldn't really come up with any answer but 0. If the space is infinite, there is no "total" to use as a reference. In the same way, any two infinite lines are going to be of the same length, regardless of being a line or a ray. Neither has a terminating point for comparison.
https://www.youtube.com/watch?v=elvOZm0d4H0&list=TLhJ7eRVKZoBI watch this video
Yes. There's no biggest infinity.
A line or a halfray going to infiny, they are all going to infiny.
"some infinitys are bigger than others." = wrong!
You can compare infinity tending towards  or tending towards +, but you can't compare 2 infinities, they are considered as equal. There is halfray, but no halfinfinity!
A line or a halfray going to infiny, they are all going to infiny.
"some infinitys are bigger than others." = wrong!
You can compare infinity tending towards  or tending towards +, but you can't compare 2 infinities, they are considered as equal. There is halfray, but no halfinfinity!
Well, half of infinity is still infinity, but a smaller infinity. This is actually discussed in calculus.
I believe so, since only line SEGMENTS stop at any piont of time.
In my view (thinking of this as a philosophical question, rather than a mathematical one) infinite is the designation we give to things that are immeasurable. Infinite means "not finite" or "not measurable". Thus neither is knowingly bigger or smaller than the other because they have no measurable length or size. Both in essence have no distance, but the real distinction between the two is that a ray extends or "moves" (for lack of a better term) in one direction, while a line extends in two directions (i.e. movement to the right only as opposed to left and right simultaneously). Infinity is not truly a number, but a concept of something being limitless.
Think of infinity as a way to express an evergrowing number. Even if you take half of that number away, it is still evergrowing. Therefore, a ray and a line are the same length.
Infinity and infinity = infinity no matter what.... so what Gabe said is right they cannot be diffrent lenghts even tho one starts and one is twice the size... (This part refers to a movie skip if you haven't seen it Unless you wanna under stand it better>) Think of it as in transformers... That big cube... it was still the same power even when it was bigger or smaller, that is the same thing with infinity... (END OF MOVIE REFERENCE) As in the example Its the smaller or bigger thing such as ray and line that is equal. Even tho they are diffrent numbers that don't come out right... In infinity multiplicationadditionsubtractiondevision with any number would still come out to infinity so infinity would never ever not be equal to itself.
How does half of infinity equal infinity? half of 1 isn't 1. Although, it is an infinite number
YES on one side
Yess because it is an infinity line
Yes something can go forever LIFE
it's infinite and you can stop when ever you want,same for both.
It depends. If the ray's starting point is let's say, 2 inches parallel to the line, it will be 2 inches shorter forever.
I agree with that. 1 is not 1/2, but technically, there are levels of infinity, so 1/2 of infinity is still a lower level than one infinity that is twice its size, and their lengths are variables, so it is impossible to find out if they are equally long, and it s extremely likely, maybe even certain that they are both the same length, so I think that the hypothetical answer from me is no, for the question you asked to me.
Infinite means without end. Neither a line nor a ray have an ending point.
I think ray never be infinity. because it has starting end.but infinity doesn't have start and end points.
yes, but a ray COULD be longer than a line, I mean why not?
even though it is viewed as 1/2 is smaller than a line, both are infinite; half of infinity is still infinity
some infinities are bigger than other infinities according to vi hart of khan academy
https://www.khanacademy.org/math/recreationalmath/vihart/infinity/v/proofinfinities
https://www.khanacademy.org/math/recreationalmath/vihart/infinity/v/proofinfinities
they both go forever, so they are the same.
Infinite numbers can't be operated on, so even though it seems like lines are double as long as rays, 2 Infiniti is equal to Infiniti. Therefore a line and a ray are the same length, Infiniti
yes because they both go on forever.
some infinities are bigger than other infinities.
"some infinities are bigger than other infinities"
Well not exactly but it seems that a ray would be only half the length of a line because rays extend in only 1 direction.
Yes. It would both be the same. It would be the same because a ray is an infinite line. Also, a line is also an infinite line. Infinity stretches on forever, so they are equally the same length.
no. Both directions makes it longer than one
Here's a scenario I posted in a comment section: setting up a winding toy to start walking from one end point, and two winding toys back to back at another point to begin walking, all at the same time, and after however much time has passed of your choosing, measuring how far the one traveled vs. the line of the two when ending this ridiculous race of the winding toys. That would make for two line segments and not a ray and line though, at a measurable distance from one point to another.. so I would have to agree with many here on how neither beats the other one out a line extending infinitely both directions, and a ray extending infinitely in one direction to be equal in length in its concept.
ifinity is INFITE
No, that is not possible. The ray continues going on only in one direction, whereas the line goes on in both directions. The line's length is infinite, but the ray's length is infinite minus the space leftover, where the ray can not reach.
This is a difficult question to answer. technically, ray=line. But wait! double infinity.... that's strange!
Neither the ray nor the line truly has a length. Both are infinite quantities. When comparing infinities, you need to ask whether a onetoone mapping is possible or not. You can easily describe a onetoone mapping between a ray and a line, therefore the infinities are of the same size.
yes and no since a line is going on both ways and a ray is only one way but they both go on for ever
A few things to keep in mind ...
1) A Line, Line Segment, or Ray are tools _used to help make measurments of the real world_. They are not real themselves, you cannot pick up a Line, Line Segment, or Ray and put it in a bucket and walk around with it. In themselves they are theoretical, ultimately derived from real world measurments, and therefore use to assist in real world measurement....but still they exist only as an idea.
2) Measurement of the real world always have errors in them, they are ___neverever_ perfect. Perfect is for theory only.
3) *Finite* means having limits measurable, countable, calculable limts. All measurements are subject to error _and can be quantified_.
4) *Infinite* means something that is _not_ finite. Infinite _has no limits_, there is no definite number (quantification) that can be used to count and express the measurement of infinity *because infinity does not have the quality of being countable*. If it is not countable, it is not measurable, and not calculable. *No quantity can be assigned to infinity*. Keep in mind the use of the words _quality_ and _quantity_
5) If you consider the mathemetics of Limits in the context of calculus, it is defined as approaching a number, but not actually touching it, there is not definite quantity, only a relative approximation (even if it supersuper close, it is only an approximation).

Keeping the above in mind, when you ask if a ray and a line are equally long, the question itself is flawed.
Infinity does not have the quality of being measurable. *The use of the word _LONG_ is not applicable to the use of the word _INFINITE_ because there is no quantity to infinity*. You cannot measure the immeasurable.
This question has similar problems to answering "*Could you tell me the quantity of number values that exist between the number 1 and the number 2*" ? in that you are working with a continuum and trying to assign discrete values to it, and then trying to declare that you have stated all or most of them, at best you can only start from having none of them, and then have some of them.
Take some time to separate the ideas of point 1) and 2) above from points 3) and 4). There are differences between what it is, how you can describe it, and what you can do with it.

With respect to point 5) above, the question you posed would have calculus component to it if you asked to measure the distance travelled by an object moving away from a point in a straight line at a certain speed (Like a ray), as compared to the distance travelled by 2 objects moving in opposite directions from a point, each at a certain speed (Line a line). Can you see how a Ray, and Line are tools for measuring the real world?
Since the question was not constrained by a qualifier such as speed, it is incomplete and therefore also flawed. If it did have a speed component to some objects moving in some direction, you could get a length or distance answer for any time span from the starting point, whereever and whenever that starting point might be.
1) A Line, Line Segment, or Ray are tools _used to help make measurments of the real world_. They are not real themselves, you cannot pick up a Line, Line Segment, or Ray and put it in a bucket and walk around with it. In themselves they are theoretical, ultimately derived from real world measurments, and therefore use to assist in real world measurement....but still they exist only as an idea.
2) Measurement of the real world always have errors in them, they are ___neverever_ perfect. Perfect is for theory only.
3) *Finite* means having limits measurable, countable, calculable limts. All measurements are subject to error _and can be quantified_.
4) *Infinite* means something that is _not_ finite. Infinite _has no limits_, there is no definite number (quantification) that can be used to count and express the measurement of infinity *because infinity does not have the quality of being countable*. If it is not countable, it is not measurable, and not calculable. *No quantity can be assigned to infinity*. Keep in mind the use of the words _quality_ and _quantity_
5) If you consider the mathemetics of Limits in the context of calculus, it is defined as approaching a number, but not actually touching it, there is not definite quantity, only a relative approximation (even if it supersuper close, it is only an approximation).

Keeping the above in mind, when you ask if a ray and a line are equally long, the question itself is flawed.
Infinity does not have the quality of being measurable. *The use of the word _LONG_ is not applicable to the use of the word _INFINITE_ because there is no quantity to infinity*. You cannot measure the immeasurable.
This question has similar problems to answering "*Could you tell me the quantity of number values that exist between the number 1 and the number 2*" ? in that you are working with a continuum and trying to assign discrete values to it, and then trying to declare that you have stated all or most of them, at best you can only start from having none of them, and then have some of them.
Take some time to separate the ideas of point 1) and 2) above from points 3) and 4). There are differences between what it is, how you can describe it, and what you can do with it.

With respect to point 5) above, the question you posed would have calculus component to it if you asked to measure the distance travelled by an object moving away from a point in a straight line at a certain speed (Like a ray), as compared to the distance travelled by 2 objects moving in opposite directions from a point, each at a certain speed (Line a line). Can you see how a Ray, and Line are tools for measuring the real world?
Since the question was not constrained by a qualifier such as speed, it is incomplete and therefore also flawed. If it did have a speed component to some objects moving in some direction, you could get a length or distance answer for any time span from the starting point, whereever and whenever that starting point might be.
I say no, because a *ray* is going forever but in _one direction_ but for *line* it will go in _two ways forever_. So if you stop both at a certain point the ray will be two times shorter than the line because the line was going in two directions at once and the ray was going in one direction at once, so I would say no, the *ray* will not be as long as the *line*.
But if they never stop, it would be *infinite*, so there are _2 answers_, depending to the _*timings*_ (If they stop or go infinitely)._
P.S.The ones in *bold* are the one we are talking about.
P.S.S. So I say no, if it were yes than...
More
Example:
```
Line Segment
●●
Ray
>
Line
<>
```
That is my example.
So that us why I think *Lines* are longer than *Ray*, and this is the main reason, "_The *Lines* go in two different way(s) at the same time, but *Rays* go in one way(s)_".
P.S.S.S. Tell me if _you_ do not get it.
But if they never stop, it would be *infinite*, so there are _2 answers_, depending to the _*timings*_ (If they stop or go infinitely)._
P.S.The ones in *bold* are the one we are talking about.
P.S.S. So I say no, if it were yes than...
More
Example:
```
Line Segment
●●
Ray
>
Line
<>
```
That is my example.
So that us why I think *Lines* are longer than *Ray*, and this is the main reason, "_The *Lines* go in two different way(s) at the same time, but *Rays* go in one way(s)_".
P.S.S.S. Tell me if _you_ do not get it.
Yes the infinite line and ray would be the same because of how no matter what the lines go on for ever that is why on the end you make an arrow to show that it goes on forever. Also because that the degrees stay the same the small area at the bottom were the two lines meet do not change only the lines length. Now if you connect the lines and one is 20 cm long and another ray line is 5 cm long and you connect the lines the area will be different for both.
Well actually the line would be longer,because some infinity s are bigger than others.
Ex.:parts of whole numbers such as 1/8 1/3 1/4 and many others.
Ex.:parts of whole numbers such as 1/8 1/3 1/4 and many others.
well I guess it would be just as long on one side and not on the other
Alguns infinitos são maiores que outros...(John Green)
a ray ends at one point and continues the other side but if both were going the same speed the line would be doubling the size by both sides
just watch the video from 0:00 3:38 and learn stuff from it :3
but a ray has to end
a line can be as long as you want
a line can be as long as you want
yes they would both be equal
Well no because a ray spreads out infinitely in only one direction when a line spreads out in both directions therefore if you were to stop them from expanding the line would be about twice as long as the ray.
they are both infinite, so yes but infinity has no length so they couldn't be measured. even if a ray has only 1 side going to infinity, half of infinity is still infinity (or a quarter, or a eighth or even infinity for that matter)
i agree with the people who said that the ray only goes in one direction while the line goes in two opposite directions
no because a line is infinite in both directions but a ray is infinite in one direction
No. Since a line goes on forever in both directions a ray only goes in one direction. It cannot be parallel.
There are different sets of infinity, so no. A ray is not as long as a line.
no and yes. no because a ray extends in only one direction while the other point is stationary.but a line extends endlessly in both directions .so a ray it will be 1/2 of a line.
yes is because both of them are infinite and no body can measure it.so we cannot say its exact distance.
yes is because both of them are infinite and no body can measure it.so we cannot say its exact distance.
Another way to think about it would be, if you cut an infinity sign in half, you get a circle, which is still infinite but in a different way.
Take a look at this video. It talks about infinities and BIGGER infinities.
https://www.youtube.com/watch?v=elvOZm0d4H0
It talks about how some infinities are larger than others. For example, their are infinities in diagonals of fractions. It's hard to explain. Just check out the video.
https://www.youtube.com/watch?v=elvOZm0d4H0
It talks about how some infinities are larger than others. For example, their are infinities in diagonals of fractions. It's hard to explain. Just check out the video.
I think that both the ray and the line are the same length. If you think about it, there doesn't necessarily need to be negative and positive numbers denoting every single point on the line and ray. That case would only be on a plane. If you have any more questions, please reply below. Hope this helps!
Like nickspoon said, it doesn't matter if it is a ray or a line because infinity is the concept of things that never end. so basically, they would be the same never ending length.
the line would be longer. as some one said ealier set theory.
if you were to count 1 , 2, 3 , 4 ....
and someone else counted 1 , 1.5 ,2, 2.5 ,3 , 3.5 ...
the second one would have more
if you were to count 1 , 2, 3 , 4 ....
and someone else counted 1 , 1.5 ,2, 2.5 ,3 , 3.5 ...
the second one would have more
You know, the line would be longer, because it goes into negative, and a ray does not! Well, if it starts negative, it does not go infinity down
Yes and no, because "equal" has little or no meaning when you're talking about infinity.
I believe that you just blew my mind...
No because a ray only stretches in one way a line stretches in both.
I guess no because a line doesn't have an end or starting point and a ray does so the line must be longer
No because an infinite line is going to go forever in both directions and a infinite ray will go forever in only one direction.
No because an infinite line is going on forever in two directions, but a infinite ray is going on forever in one direction
An infinite ray is one half of an infinite line. So that means an infinite line isn't equal to an infinite ray.
No because infinite line has no starting or ending point where an infinite ray has an starting point.
No because a line is different from a ray
no, because a ray only expands in one directions unlike a line which expands in both directions.
It's actually simple 1 DOES NOT = 1/2
here:
1≠ 1/2
here:
1≠ 1/2
So then what is the point of infinite ray and infinite line?
A ray begins at a point and then continues on in one direction like the rays of the sun. An infinite line goes off in both directions. It has no beginning and no ending.... much like a highway has many points you can stop along the way but seems to continue on and on in both directions.
lines go on for ever nut rays have starter points and then go on forever
a ray has one vertex but the line doesnt have a vertex from the both sides it keeps on going from each side.
ome oh waneo my omg
;)
;)
To get an geometric figur because geometry is made of points,line segments and strait lines
An infinite ray has a starting point, goes off in one direction with no end. An infinite line has no starting or ending point and goes in two directions.
considering that, as someone said, half of infinity is still infinity, you would never be able to get to the end represented as an arrow whereas it would actually be possibly to reach the point represented as a dot
A ray is limited to one direction, and goes on infinitely in that certain direction. On the other hand, a line is not limited to anything concerning the directions on a 2D plane. (You can only go left, right, up, and down on a 2D plane). The point is very controversial. I would personally say that this is just pure geometry. It doesn't need to be practical. We do this just for the sake of understanding. For example, there isn't anything that actually goes on forever in any direction in real life. That is, unless it is light.
A line doesn't have a point but a ray has a starting point
An infinite line has no start or nor end. a ray has a start, but goes on forever.
a ray doesn't have a point,it has a thing that looks like this on the end: > . a line doesn't have a point too,only a line segment
the main difference between the two is that a ray actually has a starting point, though it doesn't necessarily have an end point, whereas a line has neither a starting nor and end point
What is a segment?
A segment(when talking about a line) is only part of a line. Like he said, a line goes on forever but a line segment doesn't. ♥
A line segment is a line in which it has a starting point and an ending point.
Have you watched the video? Let us know which part you didn't understand.
part of a line as it does not go on forever
A segment is simply part of a line. Think about a road being the line as it stretches on in two directions. The segment is a part of that road with defined endings, say between two stores, or between a store and any other object on the road.
A line segment is basically a line that stops at both ends.
This is a line. <>
This is a line segment. oo
This is a line. <>
This is a line segment. oo
You can look up what a 'segment' is on Google...
segment= oo
starts and stops not neverending
starts and stops not neverending
the portion of a line between any two points
A line segment ends on both sides
Some thing that is divided into two parts.
Yes, even in everyday language a segment is just a "part of".
a segment directly means a part
so a line segment is a prat of a line
so a line segment is a prat of a line
a line segment is part of a line
He explains it in this video: https://www.khanacademy.org/math/geometry/segmentsandangles/intro_euclid/v/languageandnotationofbasicgeometry
Basically, a segment is a line with two endpoints. It doesn't go on forever like a line, and a segment can be part of an infinite line.
Basically, a segment is a line with two endpoints. It doesn't go on forever like a line, and a segment can be part of an infinite line.
You mean a line segment a line segment is a line that has a starting point and an ending point.
agrreed!
A segment is nothing but the part of a line.a line extends infinetly whereas a line segment ends at its end points & has finite length.
A part of a line
A line segment is two points on a line on each side.
look im not a girl! and i doin this for fun ! plus i am kurtzed1, and i ahead of my grade!!!!!
So regarding the infinite line an infinite ray concept I have a question. If a ray goes on forever it would be regarded as infinite. However a ray has a starting point unlike the line. The line is truly infinite because it goes on forever in both directions. The ray is infinite but only in one direction so wouldn't that mean that it isn't truly infinite? At some point along a infinite ray you get to starting point, at some point along an infinite line you never get to a starting point. That line has an always will exist, but the ray has a beginning.
I like this question because I believe that it invokes a deeper question about what "infinite" actually is. Before I get into my big rant about the concept of "infinite", I want to say that I believe that ray should be considered as truly infinite. Maybe my big rant will give some insight as to why I believe that.
To begin my rant, I shall compare rays and lines to number lines.
A ray is like the number line, but back when you only knew about nonnegative numbers. It started at 0, and it started counting (1, 2, 3, 4, ...). We know that this process of counting can literally take forever as the number can always keep getting larger no matter how far you've gotten. So we can say that these numbers count on to infinity.
A line is like the number line after you learned about negative numbers. So, we still have the positive infinity, but the numbers can also count backwards from 0 (1, 2, 3, 4, ...). Similar argument as before, we could take this back to negative infinity.
So, there are an infinite number of numbers greater than zero, and there are an infinite number of numbers less than zero. And there are thus an infinite number of numbers altogether. So to relate this back to rays and lines. The "positive ray" of the number line is infinite. The "negative ray" of the number line is infinite. And the entire "number line" is also infinite.
Since the number line is made up of 2 rays, does this mean that the line is twice as infinite as a ray? And if so, what does that even mean?
There is also another perspective on this concept of "infinite". I could even talk about a line segment. On the number line, we could look at only the segment of the number line between the numbers 1 and 2. We also learn at some point that there are numbers with fractional components. Between 1 and 2, there are a few we can identify that have a single decimal place after the decimal point (1.1, 1.2, 1.3, 1.4, ... all the way to 1.9). But there are still more in between those numbers, such as in between 1.3 and 1.4, there are 1.31, 1.32, 1.33, 1.34, ... up to 1.39. And this is happening between other points as well. As we do this, we are identifying numbers that are closer and closer to each other, but no matter how close they get, there are always still more points in between. And it shows no sign of slowing down. This process can be done forever, and we end up saying that there are an infinite number of numbers between 1 and 2. This is like saying that the number of points on a line segment is infinite.
A number line has an infinite number of numbers on it. We could say that in between each pair of consecutive numbers, there is a line segment. And we can conclude that there are in infinite number of line segments on this line. If each line segment has a number of points equal to infinity, and there are an infinite number of line segments, then the number of points on a line is one infinity multiplied by another infinity.
So I've just identified different ways of saying just how large the infiniteness of a line is. So what does this tell us about what infinity is? It is not just some unimaginably large number. "infinte" is a word used to describe a process that can never be finished. There will always be more steps in the process to complete no matter how much progress you have made. Counting is a process that will never end. No matter where you stop, there is always further that you could count.
To begin my rant, I shall compare rays and lines to number lines.
A ray is like the number line, but back when you only knew about nonnegative numbers. It started at 0, and it started counting (1, 2, 3, 4, ...). We know that this process of counting can literally take forever as the number can always keep getting larger no matter how far you've gotten. So we can say that these numbers count on to infinity.
A line is like the number line after you learned about negative numbers. So, we still have the positive infinity, but the numbers can also count backwards from 0 (1, 2, 3, 4, ...). Similar argument as before, we could take this back to negative infinity.
So, there are an infinite number of numbers greater than zero, and there are an infinite number of numbers less than zero. And there are thus an infinite number of numbers altogether. So to relate this back to rays and lines. The "positive ray" of the number line is infinite. The "negative ray" of the number line is infinite. And the entire "number line" is also infinite.
Since the number line is made up of 2 rays, does this mean that the line is twice as infinite as a ray? And if so, what does that even mean?
There is also another perspective on this concept of "infinite". I could even talk about a line segment. On the number line, we could look at only the segment of the number line between the numbers 1 and 2. We also learn at some point that there are numbers with fractional components. Between 1 and 2, there are a few we can identify that have a single decimal place after the decimal point (1.1, 1.2, 1.3, 1.4, ... all the way to 1.9). But there are still more in between those numbers, such as in between 1.3 and 1.4, there are 1.31, 1.32, 1.33, 1.34, ... up to 1.39. And this is happening between other points as well. As we do this, we are identifying numbers that are closer and closer to each other, but no matter how close they get, there are always still more points in between. And it shows no sign of slowing down. This process can be done forever, and we end up saying that there are an infinite number of numbers between 1 and 2. This is like saying that the number of points on a line segment is infinite.
A number line has an infinite number of numbers on it. We could say that in between each pair of consecutive numbers, there is a line segment. And we can conclude that there are in infinite number of line segments on this line. If each line segment has a number of points equal to infinity, and there are an infinite number of line segments, then the number of points on a line is one infinity multiplied by another infinity.
So I've just identified different ways of saying just how large the infiniteness of a line is. So what does this tell us about what infinity is? It is not just some unimaginably large number. "infinte" is a word used to describe a process that can never be finished. There will always be more steps in the process to complete no matter how much progress you have made. Counting is a process that will never end. No matter where you stop, there is always further that you could count.
Zeno's Paradox, right there.
Although, I'm just egomaniacal to suggest if not a solution, a path to one.
Consider that math is generally considered to be a selfconsistent, 'truth' of the universe. As we smash particles together at CERN, and as math advances, we tend to see that the internal consistency of math closely mirrors the consistency of every physical phenomenon we can observe.
This may seem like an odd tangent, but assume the universe contains some arbitrarily high, but *specific* and finite number of particles. Matter and energy cannot be created or destroyed, but merely conserved... so whatever number of particles, or whatever energy potential the universe formed with, it will keep indefinitely.
Now another brief tangent, you can (it's believed) not travel faster than light. The faster your ship moves, the more energy it takes to continue accelerating, until you approach light speed, at which point you'd need infinite energy to accelerate further. Importantly, this should also be able to be expressed as: "you'd need the total energy of the universe, to accelerate further".
What has this got to do with your rant, Ethan? The spacecraft requires the total energy potential of the universe, simply travel faster than the speed of light of that same universe. Why? because the spacecraft all the particles and energy and mass that make it up are *part of the system* that is The Universe.
Part of the system. That's the important part.
The human brain is also part of the system of The Universe. As is a computer, or any other device capable of calculating numbers. To calculate those numbers, absolutely *requires* the manipulation of part of the universe. Your computer processor manipulates trillions of electrons to calculate numbers. So does your brain. If you think about it, your imagination isn't "imaginary" in the sense that most people make it mean... your imagination is the result of your brain firing off electrons from one neuron to the next.
Why is THAT important? I contend that, if your brain (or mine, or a computer) began to actually calculate an infinity in anything other than a symbolic sense, it would require increasingly infinite energy and mass resources to draw from. Now if the universe possesses a finite (but absurdly high) level of energy and mass, *that full amount* would be required to be manipulated by the computer to actually calculate it. The universe is nothing but bits and bytes, really.

How does this apply to your example of infinite numbers between 1 and 2?
I contend that there's a limit to that, just the same. The number of numbers you can have between 1 and 2 cannot be calculated to exceed what's allowable given the specific energy potential of the universe. The calculating device could find the answer, but it would have to vaporize the entire universe just to do so.
Basically, I think that "infinity" doesn't exist in the traditional "nonstop", or "forever" sense, I think that "infinity" is a very specific number... it has a very specific correlation to the total energy/mass potential that makes up our universe, and to power a device capable of fully calculating that number would require that exact level of energy. The only way to calculate the universe is, thus, by destroying it. A computer or brain cannot run a calculation without manipulating some amount of the universe as fuel, after all.
There's an Asimov short story titled The Last Question, and it my post started to remind me of it. Read it, it's awesome.
Although, I'm just egomaniacal to suggest if not a solution, a path to one.
Consider that math is generally considered to be a selfconsistent, 'truth' of the universe. As we smash particles together at CERN, and as math advances, we tend to see that the internal consistency of math closely mirrors the consistency of every physical phenomenon we can observe.
This may seem like an odd tangent, but assume the universe contains some arbitrarily high, but *specific* and finite number of particles. Matter and energy cannot be created or destroyed, but merely conserved... so whatever number of particles, or whatever energy potential the universe formed with, it will keep indefinitely.
Now another brief tangent, you can (it's believed) not travel faster than light. The faster your ship moves, the more energy it takes to continue accelerating, until you approach light speed, at which point you'd need infinite energy to accelerate further. Importantly, this should also be able to be expressed as: "you'd need the total energy of the universe, to accelerate further".
What has this got to do with your rant, Ethan? The spacecraft requires the total energy potential of the universe, simply travel faster than the speed of light of that same universe. Why? because the spacecraft all the particles and energy and mass that make it up are *part of the system* that is The Universe.
Part of the system. That's the important part.
The human brain is also part of the system of The Universe. As is a computer, or any other device capable of calculating numbers. To calculate those numbers, absolutely *requires* the manipulation of part of the universe. Your computer processor manipulates trillions of electrons to calculate numbers. So does your brain. If you think about it, your imagination isn't "imaginary" in the sense that most people make it mean... your imagination is the result of your brain firing off electrons from one neuron to the next.
Why is THAT important? I contend that, if your brain (or mine, or a computer) began to actually calculate an infinity in anything other than a symbolic sense, it would require increasingly infinite energy and mass resources to draw from. Now if the universe possesses a finite (but absurdly high) level of energy and mass, *that full amount* would be required to be manipulated by the computer to actually calculate it. The universe is nothing but bits and bytes, really.

How does this apply to your example of infinite numbers between 1 and 2?
I contend that there's a limit to that, just the same. The number of numbers you can have between 1 and 2 cannot be calculated to exceed what's allowable given the specific energy potential of the universe. The calculating device could find the answer, but it would have to vaporize the entire universe just to do so.
Basically, I think that "infinity" doesn't exist in the traditional "nonstop", or "forever" sense, I think that "infinity" is a very specific number... it has a very specific correlation to the total energy/mass potential that makes up our universe, and to power a device capable of fully calculating that number would require that exact level of energy. The only way to calculate the universe is, thus, by destroying it. A computer or brain cannot run a calculation without manipulating some amount of the universe as fuel, after all.
There's an Asimov short story titled The Last Question, and it my post started to remind me of it. Read it, it's awesome.
Actually a ray is infinite , as well as a line, because it has the one end point not two, like a line segment, and goes on forever and comes back so even with or with out the infinite in the title it will be the same
can a line also be a ray?
Lines cannot be rays.
Line: Extends forever in two directions
Ray: Extends forever in one direction and has one endpoint
Line Segment: Has two endpoints and doesn't extend forever
Line: Extends forever in two directions
Ray: Extends forever in one direction and has one endpoint
Line Segment: Has two endpoints and doesn't extend forever
No, a line cannot be a ray, but a ray can be inside of a line. Look here.
A B
<00>
There are actually two rays in line AB, ray AB and ray BA (Starting at A and going through B, or starting at B and going through A.). I hope you find this helpful. ;)
A B
<00>
There are actually two rays in line AB, ray AB and ray BA (Starting at A and going through B, or starting at B and going through A.). I hope you find this helpful. ;)
No a line cannot be a ray:
A line goes on forever in two directions
And a ray goes on forever in one direction and stays on one point.
Therefore a ray by definition cannot be a line.
A line goes on forever in two directions
And a ray goes on forever in one direction and stays on one point.
Therefore a ray by definition cannot be a line.
no, rays have a point in which they start but lines don't.
No,a line cannot be a ray because lines don't have endpoints and continue in both directions while rays have one endpoint and continue in one direction.
No because a line extends in both directions but a ray extends in only one direction
We'll yes it can because lines and rays both have end points and go strait
blah people you guys will become millionaires
Would time be described as a line or a ray?
Or can it even be described that way?
Are there similar properties to ideas with more dimensions?
Or can it even be described that way?
Are there similar properties to ideas with more dimensions?
That is an interesting thought. To answer these question we would need to know if time had a start as well if time will have an end.
Assuming that time can be described this way, I believe that time could be represented as a ray. I'm assuming that time started 13.7 billion years ago when the big bang occurred. I am also assuming that the universe will never end which is the prevailing theory as of now, but some scientist disagree and argue that the universe could have an end.
Things can get very confusing when you take into effect the theory of relativity which states that time is not the same for everyone. If someone is moving then time is passing by slower than someone not moving! This means if i move for a little and then stand still, my experience of time will be completely different from someone standing still the entire time! (The difference in time would only be recognizable when moving at a large fraction of the speed of light which is about 300,000,000 meters per second or 186,000 miles per second so if I move the time dilation will be so small no one will recognize it but it will have occurred!) This means if I drew a time ray mine would be different than the other person not moving!
Things get very interesting if we consider extra dimensions. We live in a universe with 3 spatial dimensions and 1 time dimension. Since time is a dimension what if there was multiple time dimensions in another universe? If time had 2 dimensions time would have to be represented as an plane! This place could have an infinite edge and an noninfinite edge! What if time was 3 dimensional! Now I'm all excited! It could be any object that is three dimensional. We could have a universe with its time, if graphed, to look like anything.
Just remember Leo that even the most simplest of question can have the most mindblowing answers.
”I don't know anything, but I do know that everything is interesting if you go into it deeply enough.”
Richard Feynman
Assuming that time can be described this way, I believe that time could be represented as a ray. I'm assuming that time started 13.7 billion years ago when the big bang occurred. I am also assuming that the universe will never end which is the prevailing theory as of now, but some scientist disagree and argue that the universe could have an end.
Things can get very confusing when you take into effect the theory of relativity which states that time is not the same for everyone. If someone is moving then time is passing by slower than someone not moving! This means if i move for a little and then stand still, my experience of time will be completely different from someone standing still the entire time! (The difference in time would only be recognizable when moving at a large fraction of the speed of light which is about 300,000,000 meters per second or 186,000 miles per second so if I move the time dilation will be so small no one will recognize it but it will have occurred!) This means if I drew a time ray mine would be different than the other person not moving!
Things get very interesting if we consider extra dimensions. We live in a universe with 3 spatial dimensions and 1 time dimension. Since time is a dimension what if there was multiple time dimensions in another universe? If time had 2 dimensions time would have to be represented as an plane! This place could have an infinite edge and an noninfinite edge! What if time was 3 dimensional! Now I'm all excited! It could be any object that is three dimensional. We could have a universe with its time, if graphed, to look like anything.
Just remember Leo that even the most simplest of question can have the most mindblowing answers.
”I don't know anything, but I do know that everything is interesting if you go into it deeply enough.”
Richard Feynman
Time goes the same no mater what exept using Einstienes thiery of relativity traveling at the speed of light (you can't go faster) or in space
At 1:34, what does Sal mean by "abstract notions?"
*Abstract Notions*: Things that different/exotic, and don't usually appear in everyday life. Abstract notions are mainly used in math.
In this video, the abstract notion is the line: it continues forever
Hope that helps!
In this video, the abstract notion is the line: it continues forever
Hope that helps!
what are skew lines
In other words, you can only have skew lines in 3 or more dimensions, because in 2D plane geometry, lines are either parallel or intersecting, but not both and not neither.
Skew lines are lines that do not intersect but are not parallel.
They are not on the same level or plane for this to happen like one line goes over another like a bridge over a road but it is neither intersecting nor parallel.
They are not on the same level or plane for this to happen like one line goes over another like a bridge over a road but it is neither intersecting nor parallel.
I think that is cool
they are not parallel but they also never touch.
If it does not start with a point, then it is not a ray. it is a line.
Ray o>
Line <>
Ray o>
Line <>
They are not parallel and they never meet
what is the difference between a ray segment and a line?
A ray has one endpoint and travels in one direction, while a line is the total opposite of a line segment, having no endpoints and travelling in 2 directions, going into infinity.
A ray segment is when you have a defined starting and you go off in any direction and then have a defined stopping point.A line has no defined starting and no defined ending,so it goes on forever.I hope this helps!
A ray has a point on one end of the line, but on the other end there is no ending. It continues forever in one direction. A segment has one end point on one side, and one on the other. In other words, it doesn't extend forever in any direction. A line has no end points. The line continues forever in both directions. Hope this helps!
a ray goes on in one direction,a line never stops, and a line segment has two ends
A ray segment has somewhere to stop. A line goes in both directions and does not end.
I think so because a ray comes with an end point
A Ray goes on forever in one direction, a line goes on forever in both
A line does not end.A ray segment has a stopping point
do rays relate to line segments?
it does, because you can the the infinite line in half
I think it doesn't. that's my opion
I don't think so! Rays only have one end point, line segments have two end points in which do not go in either direction, and Rays on the other hand go on forever in only one direction because Rays only have one end point!
Rays do relate with line segments. Line segments have at two points in which go on forever in both directions. Rays also have two points, but one point goes forever in a direction and the other point has an endpoint and doesn't extend forever.
If you start with a line segment, say segment AB, you can define ray AB (which has an endpoint at point A) as the union of segment AB and all points that are beyond point B and collinear with segment AB.
If there is only one point defined in a ray, could you define it? If so, how?
A ●────────>
A ●────────>
You can't with just one point. You would need a second point somewhere on the way to show what direction it is going in.
you can define it with 2 points (one being the ending point) OR with a point (the ending) and a inclination. otherwise it's impossible.
if don't have the ending point you also need a length (or a distance) from one point to the ending point.
if don't have the ending point you also need a length (or a distance) from one point to the ending point.
the definition are 2 points Ex. AB with an arrow on top.
No you just name it Ray A or Ray X
You need a second point
What is the diferense between a line segement and a line and ray?
A line segment has two endpoints.
A ray has one endpoint — the other end goes off to infinity.
A line has no endpoints — both sides go off to infinity.
Hope this helps!
A ray has one endpoint — the other end goes off to infinity.
A line has no endpoints — both sides go off to infinity.
Hope this helps!
Line segments have 2 endpoints and don't go on. Lines have no endpoints and go on forever in 2 directions. Rays have one endpoint and the other end goes on forever(in one direction).
if a line goes on forever what would be a line in the real world?
The world and universe itself curve, so on Earth a straight line forever just circles the world endlessly, like the moon. Straight lines go on forever in terms like that as a way to explain things and because there isn't a way to make a straight line that you need to consider the curvature of the earth in everyday life. (There are a few manmade objects that would, Great wall of china for example)
Great question Prince Jack. In the "real world," especially in astronomy and cosmology, we have to use NonEuclidean geometry to measure distances because gravity creates curvatures in spacetime.
at 1:49 in the video, it said that a ray had a well defined starting point? What did they mean? Did they mean the little arrow on the end?
the little arrow tells us that it can extend in that direction where that arrow points to infinity. However, the dot/ point on the opposite end of the ray (or on the opposite of the arrow ) indicates the starting point. It is fixed and cannot be moved/ extended.
A line is a series of points which keep growing from both ends. A line segment is a portion of line with two end points. A ray is a line which has got one end point. The open end point is infinite.
Thanks
Thanks
What is an exact position called ??
Can a ray be named after a single point(origin) or is it necessary that it should pass through another point, also if a ray is passing through three points or more
for example : a ray starting at point A passing through point B further through point C and the ray is named as Ray AB can it also be called Ray BC?
for example : a ray starting at point A passing through point B further through point C and the ray is named as Ray AB can it also be called Ray BC?
A ray needs to have two points. Either name is acceptable.
Both seem okay, since a ray needs an end point and an arrow on the other end. Both fit that description.
if a a ray is infinte but with a starting point does it never come back onto itself? essntialy then turning itself into a line?
this concept is more the fact that it would continuously move through infinitesimal space and time and is a universal concept. you don't see the beginning or end. a ray is a point in witch a line is separated based on intended direction. because spacetime is a three dimensional plane a ray can travel anywhere on this plane as long as it doesn't curve
No because as he said in the video at 2:57 and other parts it goes IN ONE DIRECTION. Imagine a flat infinity.
Well that cant happen because a ray goes on in one direction for ever, and a line goes on for 2 directions forever.
A ray is straight, so it will never come back to itself. However, it is not a line. A line "expands" forever both directions ( <> ), but a ray only "expands" in one direction ( •> ). I hope this explanation helps.
Well, I sort of disagree. It's exactly the fact that a ray is straight that means it WILL come back to itself, since spacetime is 3dimensional.
Is a vector similar to a ray?
They are really only similar in notation. A ray is infinite, while a vector is more like a line segment with a directional component.
I suppose you could think of it that way. A ray is a line that continues infinitely in exactly one direction. A vector is a measurement of distance that includes how far something traveled, and in which direction. Both rays and vectors specify a particular direction, but while rays go in that direction forever, vectors stop at some point.
would an infinite line and an infinite ray be equally long?
Technically yes, because they both have an infinite distance.
Infinite = Infinite
Infinite = Infinite
Where are there lines in real life? Doesn't everything have an end?
All of math is an abstract concept. It does not exist in the real world. We designed math to be highly useful in the real world, but it is just an abstract concept.
Lines do not exist any more than the number 2 exists in the real world. They are both exceedingly useful concepts for interacting with reality, but they do not literally exist.
Lines do not exist any more than the number 2 exists in the real world. They are both exceedingly useful concepts for interacting with reality, but they do not literally exist.
YES, Keith is right! Suppose, there is a POINT, but where is it? How can u draw a point? It's smaller than u can imagine! Now, many points creates line by coming one after another, then a line also doesn't exist i.e. it's thinner than u could imagine! Now, many lines coming side wise forms a surface. But, when a line is thinner than u can think, then a surface should also has a cross sectional area (thickness) less than u can think!
Now, some surfaces joining downwards forms a 3D object and IT IS the only thing which can exist. Points, lines, surfaces are all abstract mathematical concepts, they are rather building blocks or units of 3D objects which have volume :)
Now, some surfaces joining downwards forms a 3D object and IT IS the only thing which can exist. Points, lines, surfaces are all abstract mathematical concepts, they are rather building blocks or units of 3D objects which have volume :)
At 2:33 they say that the figure is a line. But two endpoints are given, so it doesn't make sense for it to be a line and not a line segment.
It's useful to focus on the arrows  at both ends  that tells us, "YES" it can be a line.
(In the video around 1:00 minute, when Sal sketched a line... I think he should sketch two points along the line, however, he was busy emphasizing those arrows at each end to show it is a line.)
It is good that you saw that it seems to represent a line segment, because the two points there, show a segment within the line. (If the question was, can this represent a LINE SEGMENT? it would be right to say, YES.)
A LINE is defined by two points (where any point has a location) and the line does NOT STOP at either point, so two "arrows" in the sketch always means "this is a line."
Two points A and C and the sketch extends with lines and arrows away from both of two points, can correctly be identified as line segment AC, ray AC, ray CA, and line AC (or line CA).
(In the video around 1:00 minute, when Sal sketched a line... I think he should sketch two points along the line, however, he was busy emphasizing those arrows at each end to show it is a line.)
It is good that you saw that it seems to represent a line segment, because the two points there, show a segment within the line. (If the question was, can this represent a LINE SEGMENT? it would be right to say, YES.)
A LINE is defined by two points (where any point has a location) and the line does NOT STOP at either point, so two "arrows" in the sketch always means "this is a line."
Two points A and C and the sketch extends with lines and arrows away from both of two points, can correctly be identified as line segment AC, ray AC, ray CA, and line AC (or line CA).
Sal says that a line basically goes on forever. So would you be able to count the equator as a line?
You can, but not in the same way. The equator is a great circle of a sphere (assuming an idealized earth). In Euclidean (or flat space) geometry, this great circle would simply be a circle. However, if you allow yourself to deviate from the Euclidean parallel postulate, you can construct another geometry called spherical geometry in which a great circle is legitimately considered a line.
That's right, because it has no endpoints
Would two lines that are coincident (identical lines) have infinite intersection? I know that two distinct lines intersect at one or no points. But two coincident lines?
Yes, they have infinite intersections.