A Number Line for Infinite Numbers

Added: 2026-06-04

[00:00:01]What? What is this?

[00:00:02]>> The concept of a number line which maps numbers onto a line. Just as numbers continue forever, the number line also continues forever.

[00:00:12]>> But what happens at the end?

[00:00:14]>> That is an interesting question. What happens at the end of the number line?

[00:00:18]Let us think about it for a moment.

[00:00:21]Imagine moving along the number line from the origin in the positive direction. That is the situation we will consider. Let us say this dotted line represents a very long distance.

[00:00:32]>> A very long distance. I kind of get it but also kind of don't. But for now, got it.

[00:00:40]>> Now, does the number line have an end or is there a world beyond it? In other words, is there an outside of the number line? Sundaman, what do you think?

[00:00:51]>> H when I think about it calmly, something feels strange. But what exactly is strange?

[00:00:58]>> I understand that feeling. Normally the number line has no end. No matter how far you move along the number line, what you find there is just a large number and even larger numbers continue beyond it. And the number line continues forever.

[00:01:15]>> That is true. Then maybe it really is meaningless to think about the outside of the number line.

[00:01:22]>> No, it is still too early to give up.

[00:01:24]Let us access a hint.

[00:01:26]>> Access.

[00:01:30]No. What? What? What just happened?

[00:01:32]Where are we?

[00:01:34]>> Sundamon, look at this.

[00:01:36]>> This is a number line, but something looks strange.

[00:01:40]>> This is interesting. On something that looks like a number line, positive infinity and negative infinity are drawn, and the line continues even beyond them.

[00:01:51]>> That can't be right. What in the world does this mean? There should only be finite numbers on the number line, right?

[00:01:57]>> From the usual point of view, yes. But this number line is different. This is perhaps we should call it a number line for infinite numbers. Well, using the symbol infinity here may be a little problematic, but let us set that aside for now.

[00:02:13]>> A number line for infinite numbers. But wait a second. Where is the number line we know so well? It looks like zero is here at least.

[00:02:23]>> Good observation. This number line has an origin just like the usual number line. That means if we zoom in near the origin, we can think of the familiar number line as appearing there.

[00:02:36]Conversely, outside the number line, a number line for infinite numbers spreads out. That is one way to think about it.

[00:02:45]By the way, using the mathematical concept of hyper real numbers, we can describe this idea of more rigorously. I am sure you have many questions now, but let us move on for the moment.

[00:02:57]>> This is kind of hard. You said to zoom in near the origin, but how much do we need to zoom in?

[00:03:03]>> Of course, we zoom in infinitely.

[00:03:06]>> What h this is getting even more mysterious. Can we think about it more simply? The image I have is if there is something called infinity on the number line, it feels like it should be far to the right of the origin.

[00:03:21]>> I see. Depending on the interpretation, that is not wrong. But the question is what this dotted line means. So let us combine the two ideas.

[00:03:32]>> I don't really understand but please go ahead.

[00:03:35]>> First I will draw a number line for infinite numbers. On this number line, if we zoom in infinitely near the origin, the ordinary number line appears.

[00:03:45]>> That is the one from before.

[00:03:47]>> Conversely, if we zoom out the ordinary number line infinitely, it looks like the part near the origin on the number line for infinite numbers. Then on the upper number line, we move from the origin to infinity. Now, if we zoom in infinitely again near infinity, it looks like this. This idea of zooming in or out infinitely is intuitive. So I will explain it in a more rigorous form later.

[00:04:14]>> Please do. Still this has turned into something amazing. Let me calm down and think. Okay. First looking at the lower number line. It seems like if we keep moving forward on the number line, we can reach infinity. But we need to be careful here. First, no matter how far you go on the usual number line, you cannot leave the world of ordinary numbers.

[00:04:37]>> By ordinary numbers, you mean what are usually called real numbers, >> right? The dotted line here does not simply represent a long distance. It represents movement from the origin to infinity on the number line for infinite numbers. So, we can finally reach infinity. Good for you. Good for me. No, wait a second. I have many questions.

[00:05:02]But the thing that bothers me most is, is it really okay that the line continues beyond infinity? Also, what is infinity + 1? Isn't infinity larger than any number? Infinity + 1 is larger than infinity. That feels strange. Infinity should be larger than any number. But now, an even larger number has appeared.

[00:05:26]>> Oh, dare. You noticed it. Indeed, this notation has some problems. I explained this in a past video, but roughly speaking, we can solve this problem by representing infinity as a sequence. Did we talk about that before?

[00:05:40]>> For example, let us represent the sequence 1 2 3 and so on by omega. The terms of this sequence become larger as we move forward. It has no upper bound.

[00:05:52]So, it can eventually become larger than any given number. G can become larger than any given number. That does sound like infinity.

[00:06:02]>> That is exactly the idea. Here, let us think of this sequence itself as representing an infinite number. Of course, there are many other sequences that can represent infinite numbers. So, please note that this is only one kind of infinity.

[00:06:17]>> H I see. Then let us treat the number one as the same as the sequence whose every term is one. Now we will consider adding two sequences. Then omega + 1 becomes the sequence 2 3 4 and so on.

[00:06:34]Here we are adding the sequences term by term.

[00:06:37]>> Let's see 1 + 1 is 2 2 + 1 is 3. I see it really works that way.

[00:06:46]>> And when we compare omega and omega + 1 each term of omega + 1 is larger by one.

[00:06:54]Therefore, it seems natural to think that omega + 1 is larger than omega.

[00:06:59]>> Boom. Yes.

[00:07:01]>> By the way, this method of constructing a world of numbers that includes infinite numbers is a simplified version of a construction of the hyperreal numbers.

[00:07:11]>> Hyperreal numbers. That sounds amazing.

[00:07:15]>> Now, let us rethink the number line for infinite numbers using this omega. This represents the number line for infinite numbers from before using omega.

[00:07:26]>> The infinity symbol has been replaced by omega. In this way of thinking, there is not just one infinite number. So it is not good to represent them all with the single symbol infinity. Returning to the topic here we are arranging sequences as if they were on a number line. So please note that this is not a number line in the usual sense.

[00:07:48]Omega is not a real number. It is the sequence 1 2 3 and so on which represents an infinite number. Also minus omega is obtained by putting a minus sign on each term of omega. That is the sequence -1 -2 -3 and so on. The terms of this sequence becomes smaller as we move forward. So we can say that it represents a negative infinite number.

[00:08:16]>> Boom. I see >> also 2 omega is obtained by doubling each term of omega that is the sequence 2 4 6 and so on.

[00:08:26]>> 2 omega is an infinite number larger than omega.

[00:08:30]>> Also we treat the number zero as the same as the sequence whose every term is zero.

[00:08:36]>> I see when I heard number line for infinite numbers it sounded kind of suspicious. But if we think about it using sequences, it is not that strange.

[00:08:47]>> By the way, just to add a note here, we are only showing some representative points. In reality, the values are not just isolated points. For example, we can also consider things like omega / 2.

[00:09:01]Got it?

[00:09:05]If we write the sequence 1 2 3 and so on as omega then omega represents one infinite number. For now I understand that there is such a way of thinking but one question comes to mind.

[00:09:19]>> What is it?

[00:09:20]>> That is what is omega squared? Can we think of omega squared as an infinite number larger than omega?

[00:09:28]>> Squaring an infinite number that is interesting. The idea is similar to what we have done so far. We think of omega squared as the sequence made by squaring each term of the sequence omega. Let's see 1 2 3 2. So it is the sequence 1 4 9 and so on. Exactly. Where would this omega squar be located on the number line? At least it seems much larger than omega.

[00:09:57]Could it be on the number line for infinite numbers expressed using omega?

[00:10:02]If we keep going farther and farther, will we find omega squared? H that is also interesting. Let's apply the ideas we have used so far. First, we will consider a number line whose unit is omega. One infinite number.

[00:10:18]>> Having omega as the unit means one unit on the scale is omega. Right?

[00:10:23]>> That's correct. If we zoom in infinitely near the origin of this number line, we obtain the real number line. Here one unit on the scale is one. And in fact, we can also think about something even higher than the number line whose unit is omega.

[00:10:40]>> What does that mean? A number line whose unit is omega squared.

[00:10:46]>> Exactly right. Here, one unit on the scale is omega squared. By the way, for the slide layout, one unit on the scale is drawn widely. Got it? On this higher level number line for infinite numbers, if we zoom in near the origin, we obtain the lower level number line for infinite numbers whose unit is omega. This time we zoomed in near the origin. But next, if we zoom in infinitely near omega squared, we obtain a number line centered at omega squar whose unit is omega. Hm, my head is getting all tangled up.

[00:11:22]>> Well, in short, no matter how far you move on the number line whose unit is omega, you cannot reach omega squared.

[00:11:31]In other words, this dotted line does not simply represent a long distance. It represents movement from the origin to omega squared on a higher level number line for infinite numbers.

[00:11:42]>> That is a huge scale.

[00:11:44]>> It does feel mysterious >> come to think of it. So far, we've only been going upward.

[00:11:52]What happens if we go downward? What happens if we zoom in infinitely near the origin of the usual number line?

[00:11:58]Good observation. In fact, by doing that, we obtain what we might call a number line for infinite decimal numbers. Let us think of one unit on the scale as 1 / omega.

[00:12:11]>> 1 / omega. Um, what does that mean?

[00:12:14]Since omega is infinite, 1 / omega means dividing one by an infinite number. So 1 / omega is an infinite decimal. Roughly speaking, yes, we call the sequence 1 2 3 4 and so on omega. Then 1 / omega can be understood as the sequence made by taking the reciprocal of each term 1 1/2 and so on. the terms of this sequence get closer to zero as we move forward.

[00:12:45]So 1 / omega can be thought of as representing a positive infinite decimal. So that is what it means.

[00:12:52]Looking at it this way, even the usual number line starts to feel mysterious.

[00:12:57]By the way, so far we have been casually saying zoom in infinitely and zoom out infinitely. But that means changing the unit of the scale like this, right?

[00:13:08]>> Yes. Zooming in infinitely and zooming out infinitely are metaphorical expressions. Please note that the usual number line contains only real numbers.

[00:13:18]So there is no point called an infinite decimal near the origin. That said, intuitively we can think of these number lines as viewing a world of numbers that includes infinities and infinite decimals at different scales and with different units. In that sense, it would be too much to say that they are completely independent.

[00:13:38]On the other hand, roughly speaking, we can think of infinite and infinite decimal numbers as having something like layers, right? These numbers live in different layers. However, even if we call them layers, we need to remember that there are also things between them.

[00:13:55]For example, between 1 and omega, there are infinite numbers in different layers such as the log of omega and the square root of omega.

[00:14:04]>> Wow, that is amazing. These are only examples and there are many other kinds of infinities and infant decimals. So keep that in mind. This time our discussion was based on a simplified version of hyperreal numbers. But there are many ways to think about infinities and infinite decimals. We have explained many of them on this channel. So I will put links in the description.

[00:14:28]>> Um did we have videos like that? Thanks for watching. If you'd like to support this channel, consider becoming a member. In this video, we talk about the purpose of our membership. Check it out if you're interested. Well then, take care everyone. See you later.