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.
First, some entertainment. Here Wildberger helpfully recites every single digit of a 100-or-so digit approximation of √2 that he has written on the whiteboard.
Thanks, Norman — I was wondering what those strange symbols ‘6’, ‘3’, ‘1’, etc, meant.
After that belabored recitation, he argues that the “…” at the end of the 100 digits is meaningless because “the three dots at the end incorporate all kinds of additonal stuff that has not been done. We have not finished the job.” But the three dots aren’t meaningless. They represent all of the digits that haven’t explicitly been written out, and they represent only those digits, in their exact order.
I can write
, and those dots stand for “421356237…” and nothing else.
, and those dots stand for “1356237…” and nothing else.
, and those dots stand for “6237…” and nothing else.
The digits aren’t arbitrary. They are fixed by the expression on the left-hand side of the equals sign: √2 .
The dots represent one and only one sequence of digits.
Wildberger claims that the expansion he has written out — the 100 digits followed by “…” — is an approximation, but that’s incorrect. The dots make all the difference. 1.414 is only approximately equal to √2, but add the three dots and the equality becomes exact:
And so what if we haven’t “finished the job” of writing out the entire expansion? Why should that be a prerequisite for numberhood? Did the number 702,099,471,888 only spring into existence the first time someone wrote out its decimal expansion?
Wildberger tries a reductio ad absurdum, asking the “doubters” whether the following is meaningful:
My answer is yes. While it’s unconventional to write it that way, it is meaningful and correct. As I noted above, the “…” simply represents all the digits that haven’t been explicitly stated, and in this case none of the digits have been stated. The dots therefore represent “1.41421356237…”. They represent that sequence and that sequence only, because any other sequence makes the statement false.
This issue isn’t limited to decimal expansions. The three dots are commonly used to represent finite things. I can define a set
…to represent the even integers from 2 to n. Would Wildberger argue that the “…” is meaningless in that case, simply because no one has written out the numbers it represents?
As with most such discussions, it is, one should be, about terms and definitions.
My opinion is that if you agree on an operational definition, the controversy evaporates.
petrushka:
I’m not sure that’s true in this case. Wildberger knows the standard definitions of ‘irrational number’. He just doesn’t think that such objects exist, because they’re not constructible.
So what notion of ‘exist’ are we using in this dicsussion? Or are we using multiple ones?
Personally, I think that numbers only exist in the sense of being concepts in our minds and nowhere else in the world. Do mental concepts have to be fully expressable in digits? I don’t see why, there are infinite mental concepts that can’t be expressed in that way.
Am I missing something?
Exactly so.
Define ‘exist’, operationally.
If it means carrying out calculations to a halting point, then perhaps it doesn’t exist.
If it means a quantity that can be manipulated in equations to produce an exact result, then perhaps it doesn’t exist exist, at least in operation as that produce exact results.
Is it really difficult to accept that both views can be valid?
faded_Glory:
That’s my position too. Cross-posting from the other Wildberger thread:
Yes, you can take the word ‘exist’ as suggesting a Platonic realm in which all mathematical objects exist, independently of our minds. But non-Platonists don’t avoid the word — they just use it in a looser sense. For example, I’m a fictionalist — I believe that mathematical objects don’t exist in reality — but I don’t hesitate to say things like “a solution exists for the equation x + 3 = 5”. Strictly speaking, what I really mean is something like “if the object we call ‘3’ actually existed, with all the properties we ascribe to it, and the object ‘5’ actually existed, with all the properties we ascribe to it, and if the set of real numbers actually existed, with all the properties we ascribe to it, then that set would contain an object x that when operated on by the rules of arithmetic would make the statement ‘x + 3 = 5’ true.” Accurate, but awkward, so I just say that a solution exists for the equation x + 3 = 5.
An analogy: I think most people would agree that the statement “Superman loves kryptonite” is false, despite the fact that Superman and kryptonite don’t exist. What they really mean is “if Superman existed, with all the characteristics we ascribe to him, and if kryptonite existed, with all the characteristics we ascribe to it, then the statement ‘Superman loves kryptonite’ would be false.”
I don’t actually know whether Wildberger is a Platonist, so I don’t know which sense of ‘exist’ he’s using. All I know (as of now) is that he thinks the use of infinite objects is illegitimate in mathematics. That’s compatible with Platonism, if you argue that the Platonic realm contains no infinities. It’s also compatible with non-Platonism, in which case you could argue (as finitists do) that an object has to be “constructible” in order to be legitimate.
faded_Glory:
No, I think you’re spot on. Numbers are distinct from their representations, but Wildberger seems to lose sight of that, conflating the decimal expansion of a number with the number itself. If you can’t write out the expansion, you don’t “have” the number, in his view. In the video, he keeps asking whether we have “found” √2, when what he is really asking is whether we have found a finite decimal expansion of √2.
For me, something qualifies as a number* if it has an exact numerical value, regardless of whether we know (or can know) that value and whether the decimal expansion of that value is finite. If a representation corresponds to one and only one numerical value, it is a representation of a number.
“Five”, “5”, “5.000”, “V” (in Roman numerals), “the number one less than six”, and “√25” are all representations of a single number. Each of those representations specifies an exact numerical value, and it doesn’t matter that some of them aren’t decimal expansions.
Likewise, “√2” specifies an exact numerical value. It represents a number. The fact that its decimal expansion is infinite doesn’t change that.
* I’m talking about real numbers here — the kind that can be placed on the real number line. Complex numbers are a different story. Not sure how Wildberger feels about those and whether they qualify as numbers.
I think it’s pretty clear from his “Algebraic” treatment of the square root of two, but I am not surprised that you missed this.
Jock:
The parallel is obvious, but it doesn’t follow from that that Wildberger treats complex numbers the same way as the irrationals. Consistency isn’t his strong suit, as you may have noticed. Hence my statement:
More on the “algebraic √2” trainwreck later.
Maybe, depending.
Seriously, math might be a figment of our imagination, but it has proved enormously useful in constructing things that are certainly not imaginary. I have read that there is a philosophical issue concerning how amazingly faithfully our abstract mathematical concepts (including numbers) can describe the universe we observe. Why is this? Would you prefer a prediction based on statistical extrapolation, or would you favor Jeane Dixon?
Robert A. Heinlein made the claim that “if it can’t be expressed in numbers, it doesn’t exist”. Was he wrong? This claim is also attributed to Lord Kelvin.
Flint:
Yes. There is a seminal paper on the topic by the physicist Eugene Wigner titled “The Unreasonable Effectiveness of Mathematics in the Natural Sciences”.
EDIT: A pdf is available here.
Perhaps this is because humans are part of that same universe. I’m not telling you anything new by pointing out that we weren’t catapulted into this world from some separate, alien domain of existence. We evolved as part of this world and our lineage goes back billions of years to the dawn of life on this planet.
Humans are not exceptional in this sense. Have you ever wondered how a dog can catch a ball? Does it use maths to calculate the trajectory of the ball to predict where it will be in a few seconds in the future, and then use maths to program itself to be at that same spot at the same time with its mouth open? Or does it have this innate and unconscious ability because it’s lineage too has evolved to fit in with the world around it, as a precondition to survive?
So perhaps our concepts in mathematics are an evolutionary product to better cope with our environment?
Bonus question: imagine an alternative strongly non-Euclidian universe. Would an alien life form that evolved there under radically different circumstances have different mathematical concepts from ours?
First, this continues to be the “number vs its representation” issue, which I think is a clear distinction. I like keith’s line “Did the number 702,099,471,888 only spring into existence the first time someone wrote out its decimal expansion?” What about17^673? Does that not exist because no one has, or is likely to, write is all down in base 10?
Also, faded_glory Personally, I think that numbers only exist in the sense of being concepts in our minds and nowhere else in the world. “ I agree. In regards to infinity, I think the issue is a form of the “actual vs potential” infinity issue. As a concept actual infinities exist, in the sense faded_glory mentions. In the real physical world, including writing down digits, only potential infinities exist.
And hello faded_glory: I think you are a person whose history in these discussions goes way, way back! True?
You write, in reference to the issue of the “Unreasonable Effectiveness of Mathematics”,
“Perhaps this is because humans are part of that same universe. I’m not telling you anything new by pointing out that we weren’t catapulted into this world from some separate, alien domain of existence. We evolved as part of this world and our lineage goes back billions of years to the dawn of life on this planet.”
This reminds me of a favorite line of Alan Watts concerning ourselves as individuals: “We don’t come into this world, we come out of this world, like a leaf comes out of a tree”.
And I think Wildberger is tilting at windmills.
And I am again reminded of interminable arguments with Kairosfocus about the … ellipse, and his insistence that there were numbers “beyond the ellipse”.
How memory fades! 😂
Hello aleta, nice of you to remember me! I hope you are well yourself, and I remember you too. Yes I have been on these boards before, so your memory is doing fine. Occasionally this penguin flies by (that is just a concept, mind you) to see what is happening, and for some reason I got drawn into some conversations this time. Not sure for how long though!
Thans, FG, but actually I remember your name from way before I was “aleta”. I did a search on my computer and found saved discussion threads with posts from you as far back as 2006!
So I had some fun looking back at my various internet personas today. I downloaded most of Uncommon Descent before it closed down. I also have a smattering of posts from ARN, KCFS, and Dembski’s ID site (I forget its name.)
aleta:
I must have gone through a dozen at UD alone, thanks to the ban-happy DaveScot and Barry Arrington.
It’s online again as an archive at the original URL, if you want to download the rest.
By “Dembski’s site”, do you mean ISCID?
Yes, ISCID – that’s it. I was Evan there, and RBH (Dick Hoppe) and I (mostly Dick) had some fun with proposing multiple designer theory.
Also, I think the UD online is missing some critical parts that maybe I got with my download. I doubt anyone will ever track those down, but I think I remember that some of Dembski’s comments aren’t all available.
aleta:
Yes, and Wildberger’s inability to accept that concept ends up warping his entire mathematical philosophy.
In the other thread, I noted how Wildberger avoids the use of the ∞ symbol even when writing infinite sums, which he needs for his general solution to polynomial equations.
Instead of the standard
…he’ll write:
Likewise, he calls infinite sequences “on-sequences” (short for “ongoing sequences) in order to avoid the word ‘infinite’.
The dodge doesn’t work in either case. If we’re talking about “ongoing” sequences and sums, we’re talking about sequences and sums that are growing. Statements about them that are true today might be false tomorrow.
How fast do they grow, and how does Wildberger determine this? Who is adding the terms? Wildberger? The mathematical community? The National Institute of Standards and Technology? Is there a website somewhere that gives us the real-time values of all the infinite sums currently being used in mathematics?
As of 5/18/2025 @ 23:07:41 UTC, the value of
is 0.9999999999999991118216
Does he take the growth rate into account in his mathematical reasoning? The idea is nonsensical.
What about the natural numbers? Is that an “ongoing” set, too? What is the current largest natural number, and who is keeping track? It brings to mind this legendary comment from JoeG:
You might argue, “Cut Wildberger some slack. He’s really just saying that only finite sums exist, but that any such sum is already complete. No matter how many terms, as long as the number is finite, they can be regarded as having been instantaneously added together.”
The problem is that he isn’t saying that. He actually does reference the time and material resources available in the universe when deciding whether mathematical objects exist. The idea of instantaneously summing an enormous number of terms won’t fly with Wildberger because it isn’t physically possible. And if he did allow for the instantaneous summation of an arbitrarily large (but finite) number of terms, then why not extend that to the instantaneous summation of an infinite number of terms?
” Is there a website somewhere that gives us the current value of all the infinite sums currently being used in mathematics?”
Good line!
Or maybe a list of all the “real” integers that have actually been written down?
And yes, Joe G was an interesting source of numerous misconceptions, to put it mildly.
I went over to JoeG’s blog, looking for more “Largest Known Number” goodness, and found this. Joe had issued a challenge:
So I had given him a proof:
That got me thinking about where Wildberger would claim that my proof goes wrong. As far as I can tell, the only step he might object to is step (c):
In other words, there has to be at least one natural number n in Wildberger World such that n + 1 doesn’t exist. Or if it does exist, it isn’t a natural number. Which would be weird.
Wildberger World gets weirder when you consider the possibility that the natural numbers are a finite but growing set. At any given time, there will be a finite subset of the natural numbers that is not a proper subset. But surely we can add 1 to the largest number in existence. After all, isn’t that the way that the set grows in the first place?
Which means that step (c) in the proof holds. Or does it? Not really. Once we create the new natural number, we have to recompute all the possible subsets, and when we do that, we once again find a subset that is not a proper subset. So the set of natural numbers remains finite, which makes sense since it was finite before and we only added one element to it.
So is Wildberger off the hook? No, because the point of the proof is to show that the set of natural numbers is infinite. By assuming that the set of natural numbers is finite but growing, he would simply be assuming that the proof is wrong rather than demonstrating it.
The concept of thinking of infinite as a finite and growing set of numbers does my head in. At what rate do these numbers grow? When did they start growing? What is the step size of the growth, keeping in mind that between every two steps you care to define there are an infinite number of smaller steps? Why can I instantly beat the growth and jump to a number beyond where the last one ‘currently’ is by simply plugging in a larger N than the one the ‘growing series’ is currently at?
This concept of a growing series of numbers may be fun (especially when we talk about my bank account) but it is thoroughly misleading when referring to infinity. An infinite series exists the very moment we conceive of it, it doesn’t have to ‘grow’ to get there (wherever ‘there’ is).
It is hard to understand a finitist’s (or anti-infinitist’s) crusade against a very definite number like square root of 2 to begin with. This number only looks infinite when expressed in a particular way, such as in decimal. When expressed as √2 it is nicely compact, concrete and does not display any bias towards infinity.
Square root of 2 is no different from, say, a third. If you insist on spelling and pronouncing it as “oh dot three three three three three three …” it seems very scary and best avoided for any practical purposes. However, the spelling as 1/3 and pronunciation as “a third” is a very practical and compact alternative, not to mention older. Moreover, understanding it as a third is perfectly accurate to what it is while implying no infinity whatsoever. What objections could a finitist possibly have?
To Erik: Yes, I think we are agreed that part of Wildberger’s problem is confusing the number from a particular representation.
To keith: Yes, the key idea seems to be that for any finite subset there must be a largest number N, and thus N + 1 is not in that subset. This harkens back to what we mention earlier about the point Kairosfocus couldn’t get. All numbers are finite, but there are an infinite number of them. There are no “infinite numbers”, which is different than saying there are an “infinite number of them.”
I see here that the phrase “infinite number of numbers” is confusing and misleading, because it makes it sound like there is some “infinite number” which counts all the integers. This seems to be the problem Cantor solved by positing the transfinites and aleph-null as the size of the infinite set of integers.
Obviously, it would seem, Wildberger would not accept transfinites, as he seems to reject abstractions that can’t be concretely represented.
Erik:
Right. Wildberger is fixated on a particular representation. I commented on that in the other thread:
√2 has a value, and it’s infinitely precise. It can be placed on the number line, where it occupies a single point. It shares those characteristics with the number 3. Why consider the latter to be a number if the former isn’t? Numbers are not their representations, so the fact that 3 can be finitely represented as a decimal, while √2 cannot, should be irrelevant to the question.
Pocket change is a pain, so whenever I receive some, I dump it into a large jar in my bedroom. I have no idea how many dimes are in that jar right now, but it’s a definite number. I can’t write down a single digit of that number, much less a complete decimal expansion. Should I conclude that the number doesn’t exist? Or that there may or may not be a number that describes the number of dimes in my jar?
To know that something is a number, you only need to know that it has a single, definite value. You don’t need to specify that value.
Here’s a representation of a number: “the first prime number greater than Saxby Chambliss’s Social Security number”. I have no idea what that number is, and I can’t write down its decimal expansion, but it’s definitely a number, with one and only one value.
I found a video where Wildberger talks about his “on-sequences”, aka “ongoing sequences”, which mainstream mathematicians refer to as “infinite sequences”. Here’s how he justifies his rejection of the latter term.
On his whiteboard, he has a table of two columns, one labeled “Reality” and the other labeled “Fiction/poetry”. The table looks like this:
finite sequence <–> infinite sequence
visible person <–> invisible person
mortal cat <–> immortal cat
terrestrial fish <–> extraterrestrial fish
breakable wall <–> unbreakable wall
stoppable mouse <–> unstoppable mouse
Above the table, he has written “The following are not opposites!!”.
The obvious flaw: Visibility is an inherent characteristic of people, but finitude is not an inherent characteristic of sequences, unless we define them that way. If Wildberger defines them that way, he is assuming his conclusion.
EDIT: And one could easily add
terminating sequence <–> non-terminating sequence (aka “ongoing sequence”)
…to his list, which would undercut his point.
I figured out the rate at which an infinite “ever-growing” series of numbers grows: it grows at a rate of infinite numbers per number.
Simply because every time you add a number to the series, this will also add an infinite amount of numbers between the new and the old ‘end’.
Next time you speak with JoeG, tell him that and watch his head explode 😉
faded_Glory,
You’re right, and even though Wildberger denies the existence of the irrational numbers, your objection still applies to him, because there are infinitely many rational numbers between any two rational numbers. If he denies infinities, he can’t even fill all the slots between 0 and 1.
Which numbers are missing? And if you name one of those numbers, is that slot suddenly filled?
This reminds me of an argument Jock made during our epic 8-month discussion. He asserted that no two abstract numbers can be approximately equal to each other because there are always infinitely many numbers between them.
Meaning, for instance, that
654,876,993.0000000000000000000000000000000000000000000000000001
and
654,876,993.0000000000000000000000000000000000000000000000000002
are not approximately equal.
I found that argument bizarre.
And I found your incomprehension sad.
Do you believe that you have addressed Wildberger’s argument?
I have seen a lot of examples that you believe show that he must be wrong, and assertions that he is wrong. For example, I have not seen you support your claim that root2 can be placed on the number line. He argues otherwise — did I miss your counter-argument?
“That’s just silly!”, repeated endlessly, is not the amazing rebuttal that you believe it to be.
My intuition is that root2 cannot be placed on the number line, because the number line is composed of actual values of infinite precision. It strikes me that root2 does not have an actual value even at infinite precision.
Flint,
The idea being: it is not so much a line, more a series of dots. You can define an infinitely small interval on that “line” that contains root2, but within that interval, there are an infinite number of (rational) numbers, and none of them is root2.
Jock:
Do you still believe that no two abstract numbers can be approximately equal? Including the two I mentioned in my comment above?
Jock:
Are you talking about the arguments (plural) that he makes in his √2 video above? I have a lot more to say about those.
Do you actually think he’s right? If you do, I’m happy to discuss. Or if you don’t think he’s right, but just want to play the devil’s advocate by arguing for his position, that’s cool too. I think this stuff is fascinating.
Flint:
Could you explain why? The exact number 2 can be squared to produce the exact number 4, so why shouldn’t there be an exact number √2 that can be squared to produce the exact number 2?
Jock:
Better to say “arbitrarily small” rather than “infinitely small”.
Why would you expect one of those rational numbers to be √2, when √2 is irrational? A rational number can’t be irrational. If you limit yourself to rational numbers, you’ll never find √2.
You said it yourself:
The rationals are the dots; the irrationals (including √2) lie between the dots.
I should add that for Wildberger, if he’s being consistent, not even all the dots are there. There are infinitely many rationals in any finite interval, and since Wildberger doesn’t believe in infinite sets, some of those rationals must be missing. The dots aren’t there.
In fact, infinitely many of the dots are missing, because the ones that are there must form a merely finite set in order to pass muster with Wildberger.
Well, you couldn’t say less </C.L.D.>
I wouldn’t expect that, silly: that’s the point. I just think that you are assuming your conclusion, and (for whatever reason) are incapable of understanding this. In the analytic sense, there is nothing between the dots; you just keep asserting otherwise.
I’d suggest a little more caution in attributing beliefs to Wildberger.
As far as I know, he does intuitionist math. He is on the faculty of a respectable university. Unlike JoeG or KF, is not way out of his depth.
Jock:
It’s the reverse. Your argument depends on the assumption that all numbers are rational, when that is really what you’re trying to demonstrate. It’s circular. You’re assuming your conclusion.
Your reasoning amounts to something like this:
1) all numbers are rational;
2) the numbers in the interval enclosing √2 are therefore all rational;
3) √2 is irrational;
4) therefore √2 cannot be one of the numbers enclosed in said interval;
5) therefore √2 is not a number;
6) therefore it cannot be placed on the number line.
If you eliminate the assumption in step 1, the argument fails.
Can you justify your assertion that there is nothing between the dots?
I think there is something between the dots, and you have inadvertently confirmed that yourself. If there were truly nothing between the dots, they would be contiguous. They would form a continuum, a line. Yet you have affirmed that they don’t:
Points on the number line are arranged in increasing order. That is, the values increase as you move to the right. There are gaps between the dots, as you have affirmed. Why are those gaps there? Because they correspond to values that cannot be represented as integer ratios. Numbers are just values, so the gaps contain numbers. Irrational numbers, including √2.
Above: “I have not seen you support your claim that root2 can be placed on the number line.”
How about this. Take the hypotenuse of an isosceles right triangle with sides of 1. Put one end on the origin, lying in the positive direction. The other end will lie at root 2.
QED?
keiths:
Neil:
I’m going by what he has said and written — what else is available? He’s been quite explicit about not believing in infinite sets. The set of rational numbers in the interval [0,1] must therefore be finite in his view. I think it’s infinite, and I would guess that you do too. He thinks it’s finite. If we are right, he is missing an infinite number of rational numbers — an infinite number of “dots”. Do you see any way around it?
He’s a finitist, but not an intuitionist. He accepts proofs by contradiction, including the Pythagorean proof that √2 is irrational, which he describes in the video.
He’s far more competent than JoeG or KF, no question, and he does have credentials, but that doesn’t make him infallible. I have to judge him by what he says. Can you see any way in which he can deny the existence of infinite sets without denying the existence of the infinitely many rational numbers in the interval [0,1]?
aleta:
I say yes, but I think Jock would say no. For Jock, the other end of the hypotenuse will fall in one of the gaps between the rational “dots”, and that point is therefore not a number.
It’s pretty wild. For Jock (and Wildberger), the hypotenuse does not have a numerical length.
Earlier I wrote, regarding Wildberger:
I’ve since learned that the Pythagorean proof is technically a “refutation by contradiction”, not a “proof by contradiction”. Intuitionists are apparently OK with the former but not with the latter, so Wildberger’s acceptance of the Pythagorean proof doesn’t by itself mean that he isn’t an intuitionist.
However, intuitionists accept the existence of real numbers, and Wildberger doesn’t, so on that basis we can say that he’s not an intuitionist. He’s definitely a finitist, though.
Maths is one of my weaker points so I really shouldn’t keep posting in this thread, but another thought occurred to me. I think that somehow our conventions on units are tied up in this.
Take the isosceles triangle with sides of 1, where ‘1’ is in some sort of unit, say ‘foot’. The hypothenuse is root 2, which is a value that cannot be expressed exacly in feet.
But now take that same triangle, and give the same hypothenuse the value of 1, of course in a different unit system, call it whatever you like, ‘tibia’? The other two sides will now suddenly become irrational with lengths of 1/root 2 tibias.
Does this prove that there is no exact conversion possible between feet and tibia? This seems odd, if instead of worrying about triangles we would simply take the segments on their own (forget that they once were part of a triangle) and measure them within a third, independent unit system, say centimeters. Would it be impossible to measure the segment that was once a hypothenusa accurately in cm’s? Same for the other two segments that were defined in feet, can’t we now express their length in cm’s just because in a different unit system they are irrational since they were once defined as part of a specific triangle? Seems bizarre to me.
I also wonder about the concept of a number line. It doesn’t seem to work very well when we move away from the rational numbers, or even when we consider repeating decimals like 0.333… Do we perhaps just use it because we humans like to count stuff? Is there another, better, way to mentally imagine numbers and their interrelations? Or should we not even bother to imagine numers a all?
ETA I suppose I’m falling into the trap of confusing abstract mathematical concepts with real life objects.
faded_Glory:
It depends on what you mean by ‘exact’. In one sense, an exact conversion from feet to tibia is possible, and you did it yourself: the conversion factor is exactly
Converting the sides:
Converting the hypotenuse:
But if by ‘exact’ you meant the ability to write down exact finite decimal representations of both the sides and the hypotenuse simultaneously, the answer is no. No matter what units you choose, either the side lengths will be irrational or the hypotenuse length will be. You’ll have to truncate or round the irrational lengths, and that will introduce an error.
It’s all about the side-to-hypotenuse ratio, and the choice of units won’t affect that ratio:
That’s because the conversion factor between feet and fargles will be a constant, since length is length.
Conceptually, the number line works fine with both rational and irrational numbers. That’s because every real number (whether rational or irrational) has its place on the number line and occupies one and only one point. The points are strictly ordered, so for a given point, the points to the right correspond to larger numbers and the points to the left correspond to smaller numbers, and that remains true whether the point in question corresponds to a rational number or an irrational number.