In a recent OP I looked at a discovery by mathematician Norman Wildberger, who found a general method for generating power series solutions of polynomial equations of any degree. Wildberger has an interesting, extremely unconventional and (in my opinion) flawed philosophy of mathematics, which among other things denies the existence of irrational numbers. Here he explains why √2 doesn’t exist, at least not in the way that mainstream mathematicians thinks it does:
There’s lots to criticize about this, but I’ll save it for the comments.
No, you are not. You are jumping to conclusions.
He possibly does not agree that the rational numbers in that interval constitute a set.
This is how I explained to my intro calc students what it means to say sqrt 2 is irrational in respect to the right triangle.
Take the side of 1 on the triangle and divide it into as many equal parts as you wish. No matter what size of part you create, a whole number of those parts will not completely cover the hypotenuse. There is no unit that is compatible with both the side and the hypotenuse.
For what it’s worth, this is pretty much what the ancient proof that sqrt 2 is not a rational number a/b shows.
Inconceivable.
aleta,
Thanks, that clears it up for me.
Intro calc is about my level :: The sad thing is that I attended a really good maths course when I started my job in the oil industry, but that is so long ago that I don’t remember much of it. After that, in my daily work, I never had a need for it so it has gone really rusty.
keiths:
Neil:
Well, he would certainly reject what you and I think of as the set of all rational numbers in that interval. We regard it as an infinite set, and he denies the existence of those (and of infinities in general). However, he obviously knows that there are rational numbers in that interval. If there are rational numbers there, but not infinitely many, then there are finitely many. He accepts the existence of finite sets, so why wouldn’t he regard the finitely many rationals in that interval as a set?
If he thinks that set is finite, then he is missing some of the rationals. Infinitely many. No finite set can encompass all of the rationals in the interval [0,1].
Even if he had some bizarre and idiosyncratic definition of ‘set’ that excluded the rationals in that interval, he would still run into the problem. Instead of a ‘set’, we can call it a ‘collection’, a ‘bunch’, a ‘shitload’, whatever. A finite shitload of rational numbers between 0 and 1 cannot include all of the rational numbers in that interval. If there are missing numbers, then there are missing “dots” on the number line.
I should add that the missing dots (corresponding to missing rational numbers) aren’t caused solely by Wildberger’s aversion to infinite sets. Some of them are caused by his demand that numbers be representable using the finite resources available to us in our universe.
Think of a rational number whose finite decimal expansion has so many digits that it can’t possibly be represented within our universe. Such a number doesn’t exist, according to Wildberger. So it isn’t just irrationals that don’t exist, by that criterion. An infinite shitload of rationals are nonexistent, too.
You are assuming the law of the excluded middle. But intuitionists do not accept that law.
I guess my question is, does the number line include irrational numbers.
To Flint: Yes, except for Wildbergerians. Are you doubting that? See my example above about the hypotenuse of an isosceles triangle with side equal 1.
Neil,
If you’re suggesting that Wildberger believes in sets/collections/bunches/shitloads that are neither finite nor infinite, I’d like to see some evidence of that.
In any case, it doesn’t matter for the point I’m making.You don’t need to affirm that the set/collection/bunch/shitload of rational numbers in [0,1] is finite (in Wildberger’s view). The fact that it’s not infinite is enough to show that some rational numbers — some “dots” on the number line — are missing.
If set A is infinite and set B contains every element of A, then B is infinite. In order for B not to be infinite, it is necessary for it to exclude some elements of A. Infinitely many, in fact.
So if A is the set of all rationals in [0,1] (which you and I know to be infinite) yet Wildberger thinks that a non-infinite set B contains all of the rationals in that interval, it is necessary that B excludes infinitely many elements of A. Those are the missing rational numbers — the missing “dots” on the number line.
Flint:
Yes. Every point on the number line corresponds to a number, and those can be either rational or irrational. Aleta’s example shows how you can locate the point corresponding to √2.
This way of talking reflect your view of mathematics — and, for that matter, it reflects my view. But that is not how intuitionists see things.
My point is only that you shouldn’t be attributing beliefs to WIldberger that might not even fit his way of conceptualizing mathematics.
In the video, Wildberger maintains that there are actually three versions of √2 which he refers to as “applied”, “algebraic”, and “analytic”. According to him, the first two are fine, but the third one — which is the one that most people use — is illegitimate.
I’ll start with the “applied” √2. On the whiteboard, Wildberger writes:
He claims that this is the version of √2 that is used by applied mathematicians, engineers, and scientists, “even though they may not be completely conscious” of doing so.
Well, no. Most applied mathematicians, engineers and scientists are well aware of the difference between an approximation and the thing being approximated. I’m an engineer, and during my long career I have never encountered a colleague who had trouble understanding that.
When Wildberger writes “√2 = r” and then claims that r is a rational number, his statement is false. The Pythagoreans proved it, and Wildberger cites their proof in his video. √2 ≈ r can be true, but √2 = r cannot.
Second, by Wildberger’s definition, there are actually infinitely many different values of √2. All that’s required is that r^2 be approximately 2, where “approximately 2” means “falls within some specified interval around 2”. No matter how small the interval, there are infinitely many rational values of r such that r^2 falls within it. Worse still, the size of that interval depends on the application. So
and
can both be true.
If I derived the following expression during a calculation…
…I wouldn’t hesitate to simplify it to just ‘3’. But under Wildberger’s scheme, I’d have to consider the possibility that the value of √2 in the numerator was different from the value of √2 in the denominator and that the quotient might therefore not be equal to 3.
It’s a mess. Far better to treat √2 like any other number. It has one and only one value, just like the number 3 has one and only one value. It occupies a single point on the number line, just like 3 does. If we need to do a numerical calculation involving √2, we can approximate it, just like we can approximate the value of Avogadro’s number (if we choose not to carry all of the digits). Approximation is fine, but we should never lose sight of the fact that approximations are only approximations. To say that “√2 = r”, as Wildberger does, is simply false. To borrow Jock’s phrase, it is “an invitation to error”.
keiths:
Neil:
As I mentioned earlier, Wildberger is obviously a finitist, but it’s not clear that he’s an intuitionist, since he denies the existence of real numbers.
Second, the question isn’t whether Wildberger thinks there are missing numbers. The question is whether there are missing numbers, and you and I seem to agree that there are. We know that whatever non-infinite set of rational numbers Wildberger might have in mind, there are other rational numbers in [0,1] that are not included in his set.
This is a significant point. When the discussion began, I thought that Wildberger was merely denying the existence of the irrationals. Now I realize that he’s denying the existence of infinitely many rationals, too. His number line is tattered and torn. It looks like someone blasted it with a shotgun.
And my counterpoint is that we should base our assessments of Wildberger’s beliefs on what he says and writes, and on the implications of what he says and writes, just as we would do with anyone else. No one including Wildberger states all of their beliefs explicitly, and in any case most of us don’t have time to read everything he’s ever written and watch every video he’s ever made. When we lack any counterevidence, it’s reasonable to assume that Wildberger’s position on an issue aligns with that of mainstream mathematicians.
Here’s an extreme example to demonstrate the point. I think that 57 + 1 = 58, and I attribute that view to Wildberger, too. Has he ever stated that explicitly? Not that I know of. Is it possible that he denies it? Yes. He could, for example, be a secret member of some demented neo-Pythagorean cult that denies the existence of the number 58. That seems unlikely, though. I think he believes that 57 + 1 = 58.
Am I “jumping to conclusions”? Is it wrong for me to attribute that belief to him without solid proof that it “fits his way of conceptualizing mathematics”? No, and no. My attribution is reasonable. For similar reasons, I maintain that it’s reasonable to attribute to him the belief that since (in his view) there aren’t infinitely many rationals in the interval [0,1], the number must be finite.
If you find evidence that he doesn’t hold that belief, great. Let me know and I’ll revise my position.
To reiterate, though, the question isn’t whether Wildberger believes he has accounted for all the rationals in [0,1]. The question is whether he actually has accounted for all of them, and you and I can both see that he hasn’t. Not if he denies that there are infinitely many.
Jock to Flint, earlier:
Though unintentional, this is a tacit admission by Jock that √2 actually can be placed on the number line. If √2 weren’t on the line, it would be impossible to define an interval that contains it.
In effect, here’s what’s actually happening:
1) Jock knows where √2 falls on the continuous number line.
2) Because he knows where it is, he is able to define an arbitrarily small interval that contains it.
3) Like Wildberger, he doesn’t like irrational numbers, so he systematically removes them from the line.
4) What’s left is the “series of dots” that he describes.
5) The gaps are still there, though, and √2 falls into one of them.
6) The only reason it isn’t on the line anymore is that Jock has poofed the irrationals out of existence, leaving the gaps.
The truth is that √2 can be placed on the number line. It’s just that Jock doesn’t want it there.
He seems to be an intuitionist. However, intuitionists don’t all agree on everything.
He really doesn’t seem to talk about [0,1]. How he conceptualizes numbers is very different from how I see them.
I went back and watched the video again. He describes the analytic method as treating numbers as infinite decimals. But that’s certainly not my view of numbers. I see a number as an ideal object. The idea of an infinite decimal expansion can be used for motivation, but it isn’t what I take a number to be.
He may not even have a concept of “all the rationals in [0,1]”. It seems clear that his concept of number is very different from mine.
I just read the Wikipedia article on Intuitionism, as I’m not very familiar with these formal schools of thought. Concerning infinity, the article distinguishes potential vs actual infinities, as we have, and then says,
“Brouwer rejected the concept of actual infinity, but admitted the idea of potential infinity, and then quotes Kleene:
“According to Weyl 1946, ‘Brouwer made it clear, as I think beyond any doubt, that there is no evidence supporting the belief in the existential character of the totality of all natural numbers … the sequence of numbers which grows beyond any stage already reached by passing to the next number, is a manifold of possibilities open towards infinity; it remains forever in the status of creation, but is not a closed realm of things existing in themselves. That we blindly converted one into the other is the true source of our difficulties, including the antinomies – a source of more fundamental nature than Russell’s vicious circle principle indicated. Brouwer opened our eyes and made us see how far classical mathematics, nourished by a belief in the ‘absolute’ that transcends all human possibilities of realization, goes beyond such statements as can claim real meaning and truth founded on evidence.”
So the issue here is the “existential” nature of abstractions such as N, the set of all natural numbers. If we can’t “realize” them, they do not have “real meaning and truth”.
The question of in what way numbers exist is a subset of the broader question, to me, of in what way do abstractions in our minds exist. I am not a Platonist: I don’t think they exist “out there” someplace separate and independent from the minds that understand them. I also am strongly agnostic about the nature of mind and consciousness, so I don’t know how I know that understanding the nature of the completed infinity N is not a problem, nor how the logic that leads to it and follows from it gives it all the meaning and truth I would want an abstract concept to have.
The “realization” of N in the sense of bringing it alive in the physical world outside my mind comes from our ability to create shared symbol systems, so that I can find that lots of us are able to work together to develop the concepts further. But ultimately, the final abstraction is in my mind, and yours if we understand the same things in the same way. The way the understanding goes in and out of the shared symbolic systems we have created is a wonderful mystery.
Neil:
I just chose [0,1] for discussion purposes, not because Wildberger has mentioned it. He has a concept of the rationals, he believes that they exist, and he believes that infinities don’t exist. Therefore the rationals aren’t infinite. If they’re not infinite overall, then they’re not infinite in the interval [0,1]. But we know better. They are infinite, and that’s how we know that Wildberger hasn’t accounted for all of the rationals in [0,1] either. Some of the “dots” are missing.
My own view is that a number just is a value, occupying a single point on the number line, expressible as an infinite decimal. Most have just one expansion, but some have two (eg 1.000… and 0.999… ). Two distinct numbers a and b never have the same value, and they can always be ordered: either a < b or a > b. All points to the right of a number x on the number line are greater than x, and all the points to the left are less than x. The scheme is simple, useful, and beautiful, and it holds for all numbers, rational or irrational.
Infiniphobia is just a hangup. It does nothing to improve mathematics or set it on a more solid footing.
Weyl, as quoted by Kleene in the Wikipedia article:
That kind of thinking annoys me. If the natural numbers “remain forever in the status of creation”, it implies that only a finite number of them have been created so far. That in turn implies a finite speed of creation. How fast are numbers being created? If the natural numbers are being constructed as we speak, who is doing the constructing? Can’t we just let God do it, since he could presumably finish the job all at once? Does each of us have to do the construction ourselves, or is it sufficient if one person does it, with the rest of us piggybacking on their effort? What is the current largest natural number? If an alien civilization has been creating numbers for far longer than we have, can we borrow their very large numbers? Do we literally have to create every single natural number from its predecessor? Does 760,827,436,777,193,655 not exist until someone applies the successor operation to 760,827,436,777,193,654?
If the numbers aren’t literally being created, one by one, but instead their existence is being inferred because the successor operation will inevitably lead to them, then why not infer the existence of all of them at once? Why should the creation of numbers be assumed to happen in time, rather than outside of it?
I wonder if any intuitionists have addressed questions like these.
I’m with keiths in his post above about the sillyness of infinite numbers ‘being created’. The rate at which infinite numbers are being created must be “infinite numbers / number”, because for each number that is being ‘created’ there now is another infinite series of new numbers between it and the previous one created. This is just a mess that blows up in your face as you’re watching it.
I have a question about Wildberger: how can he accept that 1/3 exists? Isn’t 1/3 the same as 0.3333… i.e. an infinite number of decimal 3’s?
Something else that occurred to me is that the issue with root 2 perhaps isn’t with the number, but with whatever unit system we use? The very same line segment (length, distance, whatever you like to call it) can be rational in one unit system yet irrational in another. An isosceles triange with side ‘1’ has an irrational hypothenuse in a system where the unit is the length of one of the other sides, yet that very same triangle (if you had two of them you could overlay them without gaps or overlaps) has a rational hypothenuse when we use a unit system where that side’s length is defined as ‘1’. In that case it is the other two sides that are irrational (viz. 1/root 2)!
Does he really believe that they all exist?
In the next post you quote an intuitionist as saying “it remains forever in the status of creation, but is not a closed realm of things existing in themselves.”
That may be typical intuitionism. It’s my impression that intuitionists take numbers to exist only in the person’s intuition. So they are creating them as they need them.
You are doing what you so often do. You are attributing to people beliefs that they likely do not hold, and then criticizing them for those alleged beliefs.
You could try asking Wildberger what he really believes about this. I don’t know whether he would reply. But asking would be more honest than what you are doing.
I probably won’t post any more to this discussion. I’ve wasted enough time on it.
That does seem to be his thing.
If you have followed this discussion properly, then you know that keiths has done that. How meaningfully Wildberger has answered is another matter.
Having watched some Wildberger’s videos and reading you trying to bail him out, I hold no hope. There is hardly any realisation on his part that with his finitism or intuitionism or whatever it be he is no longer in the realm of mathematics, but in the realm of philosophy of mathematics, where he is clearly inadequate and failing hard at any attempt of coherence and consistency.
When the interaction between you and keiths is like this, why would you think it would be any better between Wildberger and keiths?
I’m not “trying to bail him out”. I thoroughly disagree with Wildberger’s ideas about mathematics. But any criticism of them should be done honestly.
keiths:
Neil:
No, and that’s exactly my point. There are infinitely many of them, so if he did believe that they all exist, he would be affirming the existence of an infinity. Which would violate his anti-infinitist stance.
From his point of view, then, some of what you and I consider to be rational numbers don’t actually exist. It follows from his own statements. Some “dots” (to borrow Jock’s term) are missing from the number line.
That shouldn’t be surprising. In the OP video, he denies the existence of √2 on the grounds that its decimal expansion can’t be fully written out. Well, the same is true of many of the rational numbers. Some rational numbers simply cannot be written out using the resources available in our universe.
Right. So for Brouwer and Weyl, as for Wildberger, completed infinite sets do not exist. If they are “forever in the status of creation”, it means that new elements are perpetually being created. That leads to the weirdnesses I pointed out in an earlier comment.
That’s one possibility among several that I listed in this comment, and it’s problematic. What does it mean for a number to exist in someone’s intuition? Does it have to be explicitly represented? Did the number 760,827,436,777,193,654 exist in my intuition before I wrote it out? If it exists in my intuition, but not in my next door neighbor’s, does it exist for me and not for her? I would say that every finite number exists in my intuition, because every such number can be reached by repeatedly adding 1. But that’s an infinite set, which intuitionists would deny.
I have attributed to Wildberger the belief that infinities don’t exist. Why? Because he has stated that belief. I have also attributed to him the belief that rational numbers exist. Why? Because he talks about them all the time, they form a core part of his mathematics, and he believes that math should only be done with legitimate mathematical objects, including rational numbers.
Which of those is a belief that he “likely does not hold”, and why do you think so?
faded_Glory:
I had that question too. At first I thought he might be OK with it, despite the fact that it has an infinite decimal expansion, because it does have a nice, finite representation as “1/3”. But by that logic, √2 should be OK too because the representation “√2” is nice and finite.
I later saw a hand-wavy video in which he argued that since we can express 1/3 using overline notation…
…it counts as a finite decimal expansion. That’s a dodge, though. Both of those are equivalent to the power series
…which has infinitely many terms and therefore cannot be considered “completed” by Wildberger. I think he painted himself into a corner with his insistence on finite decimal expansions.
The way I think of it is that length is just length, and we only assign a number to it once we’ve decided on our units. It’s analogous to the number vs representation issue. “0.333…” and “1/3” are different representations of the same number, and “12 inches” and “1 foot” are different representations of the same length.
A length is just a length. By itself it isn’t rational or irrational, but the number you assign to it will be rational or irrational depending on the units you choose to express it with.
Wildberger could clean up his position significantly (though it would still be flawed) if he would just leave material resources out of it and stick to the finite vs infinite distinction.
1.Why should the size of the universe factor into the existence or nonexistence of a particular number?
2. If the universe is actually infinite, does that mean that all of the natural numbers exist, regardless of how huge their decimal expansions are?
3. Or does it depend only on the amount of matter in the observable universe? If so, why?
4. Is it because it’s impossible to manipulate matter outside one’s light cone in order to represent numbers? If the speed of light were greater, would more numbers exist?
5. Given that the amount of matter in the observable universe is increasing as more of it comes into view, does that mean new numbers are popping into existence?
6. Why should the total amount of matter be the determining factor, rather than the amount that could feasibly be employed to represent a number?
7. if feasibility is an issue, which depends on the state of our current technology, do new numbers pop into existence as our technology advances?
8. If we conceptually “use up” resources to represent a particular number, are we allowed to reuse those resources to represent different numbers? Or are we only allowed to use the leftover matter?
9. Do different sets of numbers exist depending on how we decide to allocate the universe’s matter among them? Do we have to “kill’ some numbers in order to represent others?
It’s all so goofy. Leave the universe out of it and just concentrate on finite vs infinite.
This, at least, is correct.
Jock,
I’m glad we agree, but this undermines your claim that √2 can’t be placed on the number line.
Consider aleta’s method for placing it on the number line:
You’ve argued that the number line is actually just a series of dots, each representing a rational number. It’s a perforated line. The irrational number √2 falls into a gap between your dots, and so that doesn’t count in your view as being placed on the line. The endpoint of the hypotenuse looks like it’s on the line, but it really isn’t, because it falls into a minute gap.
Now take that same line but apply different units. Let’s say your original units were inches, and you now want to use feet. The point that was labeled ’12’ gets relabeled ‘1’, the point that was labeled ‘6’ gets relabeled ‘0.5’, and so on. So far, so good — those are all rational numbers, so the rule still holds: rational numbers correspond to dots and are therefore considered to be on the line, while irrational numbers correspond to gaps and are not considered to be on the line.
But you could also choose units such as faded_Glory’s tibias. In that case “1 foot” corresponds to “1/√2 tibias”. The point that was labeled with 1, a rational number, is now labeled with 1/√2, an irrational number. Yet that point is a dot, meaning it’s on your number line. In fact, every point that was labeled with a rational number is now labeled with an irrational one. You suddenly have infinitely many irrationals on your perforated number line!
At the same time, infinitely many points that were labeled with irrational numbers (but not all of them) are now labeled with rational numbers. Yet they still fall into the gaps between your dots. You’ve excluded some rational numbers from your number line.
You now have irrationals on your number line that you didn’t want, and you’ve lost rationals that you did want.
Your perforated number line doesn’t work. Fill in the gaps, though, and you’re good. Every number, rational or irrational, is on the line. Unit changes make no difference. Vive le continuum!
I know we’ve pretty much agreed on this, but I want revisit it in a slightly different way.
Above: “the issue with root 2 perhaps isn’t with the number, but with whatever unit system we use? The very same line segment (length, distance, whatever you like to call it) can be rational in one unit system yet irrational in another.”
Yes. Above, I wrote, “Take the side of 1 on the triangle and divide it into as many equal parts as you wish. No matter what size of part you create, a whole number of those parts will not completely cover the hypotenuse. There is no unit that is compatible with both the side and the hypotenuse.
For what it’s worth, this is pretty much what the ancient proof that sqrt 2 is not a rational number a/b shows.”
First, as we have discussed since then, if works both ways: If you decide to divide the hypotenuse into equal parts, no whole number of them will cover the sides. The two lengths are incommensurate: they have no common unit. Which is rational and which is not just depends on which view one wants to take.
Notice that this conclusion, reached by the Greeks 2500 years ago, says absolutely nothing about decimal representations. It’s based on the meaning of a fraction a/b, where the denominator b is the number of parts into which something has been divided and the numerator a is the number of those parts you have. The Greeks and others did work to try to find good rational approximations of sqrt 2 (99/70 is not bad), which is a practical problem, and later approximations based on partial sums of infinite series allowed us to get approximations arbitrarily close to sqrt 2. But the existence and value of sqrt 2 does not depend on those approximations. The value of sqrt 2 is sqrt 2 just as surely as the value of 2 is 2, and both exist in the same way (whatever one’s view on that is.)
“Vive le continuum!”
Yes. An important intellectual achievement!
I think that sums it up very nicely 🙂
aleta:
Yes. Equal rights for irrational numbers! They’ve been oppressed and discriminated against ever since they were given their derogatory name.
Yes, and let’s not even start talking about imaginary numbers! 🙂
aleta:
Yeah, those guys have had to fight for their existence. I learned recently that Gauss wanted to call them “lateral numbers”, which is a perfect name for them since they’re off to the sides of the real number line rather than sitting on it.
Learning about the imaginary numbers was a WTH moment for me in elementary school. (At that age, the H stood for ‘heck’). “If they’re imaginary, why are they teaching us about them??”
All numbers are imaginary 🙂
Fictions would be a better name, or imaginative mental creations.
Make rule then break a rule. A whole new (well, it was once) vista opens up. Why just a number line? Why not an infinity of them?
You could argue that the imaginary numbers are aready off the line. A number cloud, perhaps?
Alan:
Stay tuned. Wildberger sort of does that, but clumsily, unnecessarily and with poor results.
Wildberger aside, infinite-dimensional space is actually a thing in mainstream mathematics. The location of a point in 3D space can be specified by three coordinates (x,y z) which refer to positions along the x, y, and z axes. Each of those axes is a number line. Scaling up, the location of a point in infinite-dimensional space is specified by an infinite number of coordinates, each of which corresponds to a location on the associated axis. Infinite dimensional space, infinitely many number lines.
Because it upsets some people. But it’s a consistent way of dealing with the irrationals: the algebraic approach, a trainwreck that keiths will be lampooning shortly.
Good point.
I’d like to believe that, but there is the weirdness that sqrt 2 has a value, but you can never get there. 2 is so much better behaved. Hence the problem with the analytical approach.
All I have seen here is people asserting this conclusion. Compelling, it ain’t.
faded_Glory:
They’re considered to lie along their own axis, perpendicular to the real number line. Hence Gauss’s desire to call them ‘lateral numbers’.
A complex number is the sum of a real number and an imaginary number, so you can think of it as a distinct point in the plane formed by the real and imaginary number lines.
EDIT: Just to be clear, the real part of the complex number acts like an x-coordinate and the imaginary part acts like a y-coordinate, so that a complex number specifies a point in the complex plane.
Complex numbers have some surprising and profound properties that come into view when you manipulate them on the complex plane.
When you write stuff like this, it’s not a good look.
This one is up there too, keiths. Its called the number line. The things on it are numbers, not lengths. We all agree that rational unit conversions are infinitely precise (e.g. Smoots/foot), but irrational unit conversions run into problems. They are not infinitely precise.
Jock writes,
“I’d like to believe that, but there is the weirdness that sqrt 2 has a value, but you can never get there. 2 is so much better behaved. Hence the problem with the analytical approach.
All I have seen here is people asserting this conclusion. Compelling, it ain’t.”
What you really mean is that you can never write the value in decimal format, which is very different from “not getting there”. Walk down the hypotenuse on an isosceles right triangle with side =1 and you can “get there”.
We have been doing much more than “asserting a conclusion”. We have been explaining the difference between a number and a particular way of writing the number down, and explaining how the difference between being able to “get there” and not just depends on the features of the number system you use to represent the number.
You agree that “Which is rational and which is not just depends on which view one wants to take.”
So, if you let the hypotenuse of that aforementioned triangle be your unit, then sqrt 2 is quite well behaved, but you can never “get to 2” if you try to write it down in a number system based on the hypotenuse.
To FG: Imaginary numbers are well-defined (or at least well-illustrated geometrically) as being on their own number line which is perpendicular to the real number line. As keith points out, when we extend this to more dimensions we just have more numbers lines, and don’t distinguish which is “real”
aleta,
I just square it.
ETA:
Root 2 in base 2
1.01101010000010011110011001100111111100111011110011001001000010001011001011111011000100110110011011 etc
2 in base root2
100.00000000000000000000000000
aleta:
Jock:
What is the ‘it’ that you square in order to get 2, if √2 is not a number?
You wrote that “The parallel is obvious” between Wildberger’s algebraic approach to irrationals and imaginary numbers, so I had assumed that you could figure that out for yourself.
Perhaps not. </Claire Foy>
Jock,
Except that you can’t use Wildberger’s “algebraic √2” to refer to a length. It’s not a number, remember?