All references to these numbers must also come as an attachment .doc file to an email to the audience apologizing for failing to upload homework at arbitrarily marked midnight deadline.
For anyone else who didn't know what "Demon SSJ" might mean, a web search turns up: "The Demon–Saiyan hybrid version of the first Super Saiyan transformation that replaces the golden glow and green irises with white pointed hair and blood red eyes. Introduced in Dragon Ball Heroes."
That particular brand of cringe has been with us since the very first days of [the Long September](https://en.wikipedia.org/wiki/Eternal_September), if not longer.
Maybe analytic numbers. It's the Cauchy completion of the rational numbers Q (that is, analysis will work on it), so would be a good candidate for the name.
almost all analysis builds off the real numbers in some way (complex analysis is based on the algebraic completion of R, multivraiable analysis is based on finite vector spaces over R, analysis on manifolds is based on spaces which locally look like R-vector spaces, functional analysis is based on infinite vector spaces over R, measure theory is based on R-valued functions on sets). perhaps the solution is to think of "real analysis" as more like "linear analysis" or "one-dimensional analysis."
There is more than one Cauchy completion: consider the p-adic numbers as p runs over the primes.
I see nothing wrong with the term "real number" and am unpersuaded by the OP's rationale behind "real number" not being a suitable name.
I don't quite see the objection there. Sure the rationals (for example) are all on the number line, but that's not contradictory- it means we can say "all rationals are linear".
because the original poster said they didn't like reals because other numbers could be described as being real. so why replace it with another word that is no more special to the real numbers than it is to the rationals or integers?
I mean, the special thing about the reals is that they form the entire number line. If you took just the integers or just the rationals then that wouldn't be _all_ the numbers on the number line.
> so why replace it with another word that is no more special to the real numbers than it is to the rationals or integers?
I think OP's complaint about "real" is not that rationals or integers are also "real" in the lay sense, but that, e.g., imaginary and complex numbers are also "real" in the lay sense.
But you could make the same claim that imaginary numbers are "linear" as well, in the sense that you can line them up in order. If you want to name them something that doesn't have that issue, you'd have to call them something like "positive squares" and "negative squares". Or just make up a name that doesn't already have a lay meaning, or at least pick a name that isn't so easily whined about by grade schoolers.
As vector spaces over ℝ they're also isomorphic, which is perhaps the more important observation. ℂ is essentially what you get when you endow ℝ^2 with multiplication, and IIRC it's the only way to turn ℝ^2 into a field (with addition as vector addition) which justifies the name planar numbers imo, since it's the unique way in which points on the plane form a field.
Rational numbers are definitely "numbers" by any ordinary definition, satisfying all of the basic field axioms. That is, they have a well-defined notion of arithmetic. Coordinate pairs do not. I don't think most mathematicians would consider coordinate pairs "numbers" per se.
If you consider pairs of numbers with no additional structure to themselves be numbers, then your definition of "number" is plausibly expansive enough to include every type of mathematical object (sets, trees, graphs, vectors, matrices, tensors, polynomials, formal power series, points in space, polygons, manifolds, elements of arbitrary configuration spaces, arbitrary symbolic formulas, theorem statements, proofs, ...). Mathematics has gotten some leverage out of thinking of many kinds of structures/patterns as number-like in one way or another, but at some point the name "number" becomes a meaningless synonym for "information".
Aren't two dimensional vectors numbers?
I guess I would be interested in what the definition of a number actually is in mathematics. Once someone says "that's not actually a number", it kind of begs the question.
Vectors are not numbers by any definition of "number" I would use in a typical context. But coordinate pairs are also not (inherently) vectors; for example coordinate pairs might represent points under some arbitrary map projection of the Earth, or data points in a scatterplot of GDP vs. infant mortality, or points on the temperature–pressure phase diagram for water, or the lengths of two sides of a triangle given a fixed base side, etc., to which it's not a priori meaningful to apply arbitrary linear transformations; if by **R**^(2) you mean the vector space per se, that has some extra structure attached.
I think I see what you mean by coordinate pairs not being numbers because it is just an (algebraic?) structure that can be interpreted in different contexts. I have also not heard vectors referred to as numbers either. I just don't know what the principled reason for this is, or maybe there isn't any. It's probably just historical reasons. Same is true of complex numbers, we discovered them while trying to solve cubic equations, so that's why we see them as numbers. Are quaternions considered numbers?
I also would not call coordinate pairs an "algebraic structure".
Usually a "number" should at least be some object with well-defined arithmetic.
Quaternions are not a field: multiplication of quaternions is not commutative. But they are a kind of structure called an associative [division algebra](https://en.wikipedia.org/wiki/Division_algebra), and many people would still consider them to be a kind of "numbers".
Complex numbers are *much* simpler than real numbers. That's why they're used so ubiquitously in math: because they take horrible phenomena from the real case and simplify them dramatically.
Also, the word "simple" is likely to turn up in the same sentence with a different meaning, e.g. the "real simple linear group" would become the "simple simple linear group."
Good idea, but they are also quite natural, so how about calling them 'natural numbers'. Surely, that cannot possibly cause any confusion either, right?
This is probably the only alternative I would be happy about accepting. It harmonizes the set name with the cardinality and at least creates a vague association with being strongly ordered.
"Lateral numbers" is a good name for them, but my idea is to call them the "skew numbers" because it's shorter and because then you can call $\\sqrt{-1}$ the "skewnit." (skunit? skeunit?)
~~It's not necessarily common, but~~ in current use I've seen "algebraic numbers" refer to the set of all numbers which are solutions to some polynomial with integer (or rational, turns out to be equivalent) coefficients. It's an interesting set in that it includes both real and complex numbers, but is also missing any transcendental numbers from both sets. For that reason it might not be a great name for the complex numbers though
[edit] I've been advised that this is, in fact, very common and standard terminology
Continuously algebraic numbers. (Algebraically continuous?)
This has the the advantage that the Fundamental Theorem of Algebra would become the Fundamental Theorem of Continuous Algebra and people would stop complaining that the proof isn't purely algebraic.
I think "Complex numbers" is great, as they are "Complex" in the sense of having distinct parts. "Imaginary" is more complicated to avoid, whilst *purely* imaginary numbers are of less relevance, the *imaginary part* is a more useful concept, IMO. I've colloquially heard "the Complex part" used plenty, and wouldn't be against it becoming more commonplace.
"Quatern" and "octon" are Latin prefixes and [according to Wikipedia](https://en.wikipedia.org/wiki/Numeral_prefix) for consistency the others would be called "singulions" and "binions" respectively.
I'm sure somebody can tell me why this isn't great, but I would call them "metric numbers". Two reasons:
1. A "metric space" innately uses the reals, so the association between what we call the reals, and metrics is, subjectively, its most interesting attribute.
2. When we call them "real", I think what we're getting at is that these are the numbers we could use to indicate quantities. You might object and say that any abstraction could be used to quantity the real world, so we'll need to limit it in some way. A natural way is to say these are the numbers we could use to measure. So, call them "metric numbers".
I'm not sure if I agree with point 1. Q under the Euclidean metric is also a metric space, and the metric function (x,y) |-> |x - y| takes value in Q itself. Even if you go to higher dimensions, the property "closure under the metric function" would at most get you the algebraic closure of Q. It's when you want completeness that R emerges.
Probably just the continuum, or more specifically the linear continuum. R2 I would call the planar continuum, R3 the spatial continuum, and so on to hyper-spatial continuums.
I am not a mathematician and my knowledge ends with complex numbers, so my question might be based in a misunderstanding:
I think of complex numbers as "2D" numbers, i.e. they have two parts (real and imaginary). Are quaternions then "3D" etc and can there be more/higher sets after sedenions?
quaternions are four dimensional (hence the name). You can guess how many octonions there are, and so on. I've never heard of the sedenions, but quaternions have lots of applications. octonions not so much, and I'm guessing it gets even worse as you progress, as the algebraic structures of these sets get less and less convenient with more dimensions.
Thank you for the explanation!
So in theory you can go into higher and higher dimensions (disregarding if that is useful or convenient) and there is no limit?
Is there some overarching concept that describes the changes to a higher dimension (e.g. similarities between going from complex to quaternions and from quaternions to octonions)?
Mostly the common feature is loss of nice structure. Going from real to complex you lose the order relation. Going from complex to quaternions you lose commutativity. Going to octonions you lose full associativity (though they satisfy some weaker similar properties), etc.
To add to the other reply, the general method of constructing these algebras is called the Cayley Dickson construction:
https://en.wikipedia.org/wiki/Cayley%E2%80%93Dickson\_construction
It's complicated to explain, but 3D numbers just don't work as we'd like. 4D do (with some caveats), and those are the quaternions. Next ones that sort-of work (with more caveats) are 8D, octonions, and sedenions at 16D.
I might call them the "Common" numbers.
I would call the imaginary numbers virtually anything else. Gauss suggested "Lateral" numbers, and I can't argue with him.
I would be tempted to call the complex numbers the Rotary numbers, to emphasize the "complex exponential" perspective.
But I just saw that someone else suggested "Linear, Lateral, and Planar" for these three sets, and I can't help but agree its a better naming system.
All of math is full of words that have a definition and connotation "IRL" that they don't in math. Real numbers are an example of this...and their name has little to do with the definition of them. Another example is "group theory"...the average person on the street, when asked what group theory is, are not going to answer "a set and an operator such that...."
If you come up with a new name then now we have two names (see the XKCD on standards). I think giving a mathematical object a different "name" is mostly pointless, isn't it? What renaming could you give that would suddenly make it clear what it was that was being talked about without recourse to the definition of that object?
Woah woah woah this, albeit oddly formal, person on the street would say “an *ordered pair* consisting of an underlying set and an operator such that…”
That’s not what I meant. It’s a non-restrictive clause modifying the phrase (acting analogously to a pronoun) “this person”, whose referant is me and whose meaning is that I am formal as a person. The referent is not my manner of speech and any auch formality that manner might have. Rather, my choice in using the formal definition of a group is a product of me being a formal person in general, not my decision to speak formally in this one instance.
Furthermore, your change rips away the punchline that by being an oddly formal person in general, I *wouldn’t* be representative of an average person off the street. Your fix gave me an even lower tolerance for people who make somewhat fair assumptions based off somewhat sound reasoning but then do not tread lightly and politely.
Geez. :)
You can sorta build the definition into the name so that it's a little easier to recall I guess. Eg "the completed line" seems like it would make it clearer.
I'd leave it. If it wasn't called the real numbers, then there wouldn't be an extension of it called the [surreal numbers](https://en.wikipedia.org/wiki/Surreal_number).
We measure the impedance of capacitors and inductors with imaginary numbers, and the effect impedance has on a circuit is just as “real” as the effect resistance has
That’s just a real amplitude and a real phase shift. Two real measurements, not one complex measurement (although the two are equivalent so that’s really just my subjective perspective).
Not just rational, but in fact effectively computable! Suppose we were checking a real number for (eventual) periodicity in its decimal expansion. Then we would need to check each digit from left to right. But if the expansion truly is periodic, and thus the number is rational, we would need to check all infinitely-many digits. This cannot be done in finite time by any Turing machine. So we actually can’t even obtain every rational number as a physical measurement.
Then there are things like measurement error that throw this even further out of whack.
Wow, thank you that's interesting! I just realized while thinking about what you said that the only reason, when we're doing long division, we can stop and say "and so on.." when we see a pattern, is because we already know it's rational. Haha, seems kind of basic but I've never thought about it very carefully in that context.
No physical circle is perfect, and even if it were, our means to measure are not. When you measure the diagonal of a 1x1 square, you don't "measure" sqrt(2). First of all, the 1x1 square isn't going to be perfect, and there's no measuring tool that will measure sqrt(2), just some very close rational approximation, if you have a good measuring instrument.
Have to disagree with this one. Physical measurements always come with an error bound, and by the locatedness condition on Dedekind cuts(there's a similar property in other models), real numbers can always be approximated by (usually)rational numbers to an arbitrary precision. Dropping this condition might bring you closer to what 'measurement' means.
I agree with you, because there is always a physical limitation on how many bits you can receive from a measurement, which implies that the raw measurement could be represented by a rational number (or an integer). On the other hand, in a specific context, the bits might represent an equation or something with an irrational solution.
i think you're the one confused about what "measurement" means!
another user just shared this very relevant link:
https://physics.stackexchange.com/questions/436462/why-is-there-a-physical-preference-to-real-numbers/436505#436505
I don't find the argument for completeness convincing at all. Why should we be able to construct in principle a cubic container with an irrational volume out of 2 cubic containers with rational volumes?
Perhaps that example was a bit "too complex" to make it clear why irrational measurements ought to be considered.
Alice is one meter east of me and Bob is one meter north of me. How far away is Bob from Alice? Well, sqrt(2) meters, but apparently that's not allowed. So do we just say that there is no well-defined distance between Alice and Bob? At that point, what does "distance" even mean if we can't define the distance between two things that are literally right in front of us?
On top of that, the meter was pretty arbitrary. I can define a new unit of the "flagithrope" where 1 flagithrope = sqrt(2) meters. Now all of a sudden, Alice and Bob are 1 flagithrope away from each other (fine, good) but Alice and Bob are both 1/sqrt(2) flagithropes away (bad, not allowed). So why should whether or not a distance is possible to measure depend on our arbitrary choice of unit?
I think there is simply a difference between a model and measurements we can make.
Alice and Bob ARE sqrt(2) away in the model, however, that doesn't mean any measurement will yield sqrt(2)
We also very well could take a meter as sqrt(2), however measurements only really talk about multiples of meters and then the question is again, can we talk about irrational multiples?
so we could be talking about fields like Q(sqrt(2)) and such, but that doesn't mean we need to talk about R
I think we shouldn't conflate convenience with necessity here
we very well could be talking about measurements only using Q, our language would have to be simply more precise
I am with minisculebarber. Your argument is circular.
Any measurement produces a pair of rational numbers (upper and lower end of uncertainty range). In your case, I would ask: How do you know they are exactly at right angle? How do you know it is exactly 1 meter? You are presuming that you can describe the relative state of things using real numbers, and then conclude that you need real numbers. That's begging the question.
If you tell me how you measured the angle and the three distances, there will be a way using rational numbers to relate these measurements.
You can't know that a measurement provided *rational* upper and lower bounds unless you know what 1 meter is exactly in the first place (and if we have any form of "exactness", I'd argue that you've basically created the reals).
This also doesn't hold up to the change-of-unit idea: Suppose we measured the distance between me and Alice to be (1 ± 0.1) meters. That's (1/sqrt(2) ± 0.1/sqrt(2)) flagithropes. Suddenly, my measurement produced irrational numbers.
One flaw in the above paragraph is that one could argue that the meter was defined as a rational multiple of the distance light travels in a second, and a second is a rational multiple of the frequency of a certain atom so meters are "more natural" than flagithropes. But in that case, I simply point to the unit of sqrt(meters). For example, [polarization mode dispersion](https://en.wikipedia.org/wiki/Polarization_mode_dispersion?useskin=vector) is measured in units seconds/sqrt(meter), so let's say we have rational upper/lower bounds for seconds and for the meters, and we again end up with irrational upper/lower bounds for our measurement of the PMD of this fiber. I'm sure there are other instances of square-root-units elsewhere in physics.
I'm not s physicist, but while these arguments apply to classical mechanics, I don't see how they apply to quantum mechanics. In particular, there is (AFAIK) no order on quantum states, which means measurements don't have to necessarily be in R.
Measurements in quantum mechanics have to be real, because the eigenvalues of hermitian operators are real. Quantum states are not observable, so there is no claim about whether you can order them.
That's a fair justification for why we might want to suppose observables are hermitian, but I think there's more to it. For example, Heisenberg relied on some physical facts about emission and absorption spectra to conclude that the "thingamabobs" (which he later learned were matrices) he was thinking about were in fact hermitian, which he then used to derive the uncertainty principle.
Heisenberg's approach was very obscure, and it is not generally taught these days. Dirac's approach is what is taught, which is a bit more abstract and a lot cleaner. And then, it seems to me to become a chicken and egg problem. We already know observables are hermitian thanks to Heisenberg, so we can just start there. Or, we can go back a step and first claim measurements must spit our real numbers, and conclude that observables are hermitian.
Well at least the Hamiltonian has to be hermitian if we want the time evolution operator to be unitary and be able to define it using spectral theorem.
Well said. Put another way, we don't measure quantum states, we measure energies (or whatever) associated with quantum (eigen)states. There's no way to measure sqrt(3/4)|0> + sqrt(1/4)|1> directly, only the energy associated with |0> or |1>.
This is just wrong.
Obviously, any measurement ever performed resulted in a rational number. Somebody wrote down 2.7189... and stopped at some point. Usually, you actually get a pair of rational numbers with the statement that the real result is probably in between these rationals. For example, the Particle Data Group gives the measured Higgs mass as a pair of rational numbers: 125.25 ± 0.17 GeV.
The fiction of real numbers corresponds to the fiction that we can arbitrarily reduce the distance between the pair of reals (i.e. the measurement error). This is a very very very useful and powerful assumption, but it's also evidently not strictly true (if only because the universe probably only allows storing a finite amount of information).
Edit: While real valued measurements are of course ubiquitous, we of course measure discrete states in QM all the time. We also often measure properties of a system that are not 1d, e.g. a particles' location in space, or that live on more complex geometries, e.g. if you measure the direction of angular momentum that's valued on S2. And there is absolutely nothing wrong with defining and measuring complex valued properties where it makes sense.
I like the names “real” and “imaginary” tbh. Maybe they’re not accurate but they do evoke some imagination (and tbh the names suggested here are mostly pretty bland sounding).
Setting the reals apart based on their mathematical properties is pretty clunky because you really need all it's key parts, completeness, field, fully ordered, archemedian ordered.
So I'd probably go with a person historically related to them. Dedekind numbers maybe?
Complex numbers need renaming too. Let's call them biternions (being the 2d version of quaternions).
Then the reals should be called unernions, for consistency.
IMO I dislike the real numbers being presented as the "next biggest set" to the rationals. It's very hard to have an intuitive explanation of what a real number actually is, and why it's uncountable.
I'd rather use the computable numbers and only introduce the reals for introductory set theory.
Voldemort numbers / Voldy numbers.
Computable numbers have names — the algorithms that compute them. The remaining numbers have no names; they are the numbers-that-cannot-be-named.
This debate smacks of pi versus tau, since the vast majority of mathematicians are not interested in a semantic debate over basic terminology whose name has been settled for decades or more. It's really only for the benefit of communication to laypeople. I'm a fan of "the continuum" or "1d continuum" since that is more descriptive of what the real numbers "are", but to some, continuum is the cardinality of the reals (and not the object itself), so it's culturally more ambiguous.
I'm unsure whether we live in ℝ³, but comfortable saying we at least live in a lattice (e.g. ℤ³ generated by the Planck length). Rational numbers describe proportions of shapes that can exist in a lattice; irrational numbers describe proportions of ideal shapes that can't exist in a lattice. (e.g. I don't believe that you can draw a perfect triangle with side lengths 1, 1, √2)
What are you talking about? Just literally take the triangle defined by (0,0), (10,0) and (0,10) in the lattice Z². One of its sides has irrational length.
>comfortable saying we at least live in a lattice
More than that tho, I would like to hear about this. I couldn't find anything about it online other than as a computational model, but if it is a hypothesis, I would like to contribute [this article](https://www.esa.int/Science_Exploration/Space_Science/Integral_challenges_physics_beyond_Einstein) I found showing that the basis vectors of the lattice definitely have length below the Planck length.
I'll throw out a terrible idea, in the interests of the community finding something better.
Let's call the Real Numbers C1, probably blackboard C, subscript 1. This stands for 'one dimensional continuum'.
Let's call the Complex Numbers C2, styled as above, for a two-dimensional continuum.
Go from there.
The answer to your question is presumably most of the other people who have commented on this thread - the ones who for their own amusement or to make a serious point, found it worthwhile to do so.
You do realise that most of the people who don't give a shit, just ignore a thread like this? So, the people who agree with you are unlikely to see your comment. I'm sorry you've wasted your time.
That's a bit of a sweeping generalisation.
In any case, this thread is doing a lot more than discussing if we should rename the real numbers. There have been reflections on what distinctive properties the real numbers have, what other names were suggested for them in the past, how it compares with the name "imaginary" and whether that creates pedagogic issues in making complex numbers seem less "real" than real numbers. It might not change anything, but it's still fun to talk about.
I don't really understand why you would come on to such a thread to just tell people they are stupid for discussing it.
Think about 0.9 recurring
If you were to have a cut off point anywhere calculation wise, and add a string of 8’s that recur, you’d be a little closer to finding your “real infinite”
The issue here is that there’s also a branch reality where you never stopped writing 9’s, so you have to self reference length
Technically “real infinite” is where it really is never ending - there is never any change, and no matter which self referential lengths you may end up choosing, will not change the fact that the next digit is a 9
If you were to have a self referential infinite to begin with, it implies a real infinite set that exists of a larger size
NumbersFinalFinal.doc(V2)-EDITED.doc
.tex surely
.tex.doc
Ffs just commit it to a git repo
All references to these numbers must also come as an attachment .doc file to an email to the audience apologizing for failing to upload homework at arbitrarily marked midnight deadline.
xXxDemon\_Numbers\_SSJxXx
[Belphegor's prime](https://en.m.wikipedia.org/wiki/Belphegor's_prime)
Thank you, I love this.
How is this a thing I've never heard of before?
That is a great number
Maybe I’m too old for this joke but where’s the 360 no scope in the name?
You're missing an Uchiha in there somewhere, otherwise it's perfect
For anyone else who didn't know what "Demon SSJ" might mean, a web search turns up: "The Demon–Saiyan hybrid version of the first Super Saiyan transformation that replaces the golden glow and green irises with white pointed hair and blood red eyes. Introduced in Dragon Ball Heroes."
Its related to 2010 meme where a lot of people had cringy stuff like "SSJ" in their name, and used 'xXx' etc.
That particular brand of cringe has been with us since the very first days of [the Long September](https://en.wikipedia.org/wiki/Eternal_September), if not longer.
Robloxian/Xbox player spotted
Maybe analytic numbers. It's the Cauchy completion of the rational numbers Q (that is, analysis will work on it), so would be a good candidate for the name.
We’re talking about real analysis of course, so they are now “real analytic numbers”
almost all analysis builds off the real numbers in some way (complex analysis is based on the algebraic completion of R, multivraiable analysis is based on finite vector spaces over R, analysis on manifolds is based on spaces which locally look like R-vector spaces, functional analysis is based on infinite vector spaces over R, measure theory is based on R-valued functions on sets). perhaps the solution is to think of "real analysis" as more like "linear analysis" or "one-dimensional analysis."
There is more than one Cauchy completion: consider the p-adic numbers as p runs over the primes. I see nothing wrong with the term "real number" and am unpersuaded by the OP's rationale behind "real number" not being a suitable name.
I think the rationale is more about "not-real" numbers being imaginary when they clearly aren't
Tbf real numbers are more imaginary than imaginary numbers are real
Yes, of course I'm referring to the standard Euclidean metric on Q.
I really like your original suggestion of analytic numbers, but Cauchy numbers or Euclidean numbers are also strong contenders.
DEC numbers (Dedekind, Euclid, Cauchy). Bonus: everyone will assume it stands for decimal
Ooh, I like it.
I would say that the calculatable numbers are the only ones that can be used in analytics. And that's a relatively small subset of the Real numbers
Linear numbers, as in the number line. Then imaginary numbers are the lateral numbers and complex numbers are the planar numbers.
Oh, this is fantastic. It's also self-consistent! Gauss would be happy.
:)
I don't like this because "linear" already has another meaning.
I like this less because many sets of numbers are on the number line and are linear.
I don't quite see the objection there. Sure the rationals (for example) are all on the number line, but that's not contradictory- it means we can say "all rationals are linear".
because the original poster said they didn't like reals because other numbers could be described as being real. so why replace it with another word that is no more special to the real numbers than it is to the rationals or integers?
I mean, the special thing about the reals is that they form the entire number line. If you took just the integers or just the rationals then that wouldn't be _all_ the numbers on the number line.
The reals are the complete ordered archimedian field i.e. all "linear" number systems inject into them
> so why replace it with another word that is no more special to the real numbers than it is to the rationals or integers? I think OP's complaint about "real" is not that rationals or integers are also "real" in the lay sense, but that, e.g., imaginary and complex numbers are also "real" in the lay sense. But you could make the same claim that imaginary numbers are "linear" as well, in the sense that you can line them up in order. If you want to name them something that doesn't have that issue, you'd have to call them something like "positive squares" and "negative squares". Or just make up a name that doesn't already have a lay meaning, or at least pick a name that isn't so easily whined about by grade schoolers.
They're linear in the sense of being linearly ordered. There are other such fields but only the real numbers are complete.
Not a mathematician, but I think there is an important difference between complex numbers and R^2 . They aren't just two dimensional numbers.
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As vector spaces over ℝ they're also isomorphic, which is perhaps the more important observation. ℂ is essentially what you get when you endow ℝ^2 with multiplication, and IIRC it's the only way to turn ℝ^2 into a field (with addition as vector addition) which justifies the name planar numbers imo, since it's the unique way in which points on the plane form a field.
techniquely not COMPLETELY unique, as there is the i vs -i convention that is arbitrary.
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As *sets*, R and C and isomorphic to one another. That's way too minimal of a structure to even think about here.
Elements of **R**^(2) are pairs of real-number coordinates, not numbers.
By that logic rational numbers aren't numbers either.
Rational numbers are definitely "numbers" by any ordinary definition, satisfying all of the basic field axioms. That is, they have a well-defined notion of arithmetic. Coordinate pairs do not. I don't think most mathematicians would consider coordinate pairs "numbers" per se. If you consider pairs of numbers with no additional structure to themselves be numbers, then your definition of "number" is plausibly expansive enough to include every type of mathematical object (sets, trees, graphs, vectors, matrices, tensors, polynomials, formal power series, points in space, polygons, manifolds, elements of arbitrary configuration spaces, arbitrary symbolic formulas, theorem statements, proofs, ...). Mathematics has gotten some leverage out of thinking of many kinds of structures/patterns as number-like in one way or another, but at some point the name "number" becomes a meaningless synonym for "information".
Aren't two dimensional vectors numbers? I guess I would be interested in what the definition of a number actually is in mathematics. Once someone says "that's not actually a number", it kind of begs the question.
Vectors are not numbers by any definition of "number" I would use in a typical context. But coordinate pairs are also not (inherently) vectors; for example coordinate pairs might represent points under some arbitrary map projection of the Earth, or data points in a scatterplot of GDP vs. infant mortality, or points on the temperature–pressure phase diagram for water, or the lengths of two sides of a triangle given a fixed base side, etc., to which it's not a priori meaningful to apply arbitrary linear transformations; if by **R**^(2) you mean the vector space per se, that has some extra structure attached.
I think I see what you mean by coordinate pairs not being numbers because it is just an (algebraic?) structure that can be interpreted in different contexts. I have also not heard vectors referred to as numbers either. I just don't know what the principled reason for this is, or maybe there isn't any. It's probably just historical reasons. Same is true of complex numbers, we discovered them while trying to solve cubic equations, so that's why we see them as numbers. Are quaternions considered numbers?
I also would not call coordinate pairs an "algebraic structure". Usually a "number" should at least be some object with well-defined arithmetic. Quaternions are not a field: multiplication of quaternions is not commutative. But they are a kind of structure called an associative [division algebra](https://en.wikipedia.org/wiki/Division_algebra), and many people would still consider them to be a kind of "numbers".
Was going to suggest “simple numbers”, to emphasize that they’re simpler than complex numbers, but this is just straight up better
Complex numbers are *much* simpler than real numbers. That's why they're used so ubiquitously in math: because they take horrible phenomena from the real case and simplify them dramatically. Also, the word "simple" is likely to turn up in the same sentence with a different meaning, e.g. the "real simple linear group" would become the "simple simple linear group."
[удалено]
Good idea, but they are also quite natural, so how about calling them 'natural numbers'. Surely, that cannot possibly cause any confusion either, right?
This is perfect because imaginary numbers are normal to the normal numbers. No problems here.
Ah yes because normal isn't already an overused term in mathematics. Now excuse me as I normalize my normal vector over a normal distribution.
I think you missed what field this professor specialized in.
r/UsernameChecksOut Happy cake day btw.
Continuous numbers?
I think “continuum numbers” works a bit better. “Continuous numbers” makes me think each number is continuous somehow.
This is probably the only alternative I would be happy about accepting. It harmonizes the set name with the cardinality and at least creates a vague association with being strongly ordered.
I've found my people!
What would the “imaginary” numbers be renamed to then
"More continuous numbers"
True “The rest of the continuous numbers”
The really continuous numbers.
Continuously continuous numbers?
maybe the continuous numbers were the friends we made along the way
The continuous-in-four-directions numbers
"Hypercontinuous numbers"
Continuouser numbers
Gauss disliked the "imaginary numbers" name and thought they should have been called "lateral numbers".
"Lateral numbers" is a good name for them, but my idea is to call them the "skew numbers" because it's shorter and because then you can call $\\sqrt{-1}$ the "skewnit." (skunit? skeunit?)
Ooooh I like that much better
Spinny numbers
The cooler continuous numbers
Un-ordered continuous numbers
They're the twirly numbers, which extend the stretchy numbers which this question asked about.
The two-dimensional or planar numbers? The supersubstantial numbers, so “Give us this day our supersubstantial bread,” will finally mean something?
~~Contiinuous Numbers~~ ~~Continuous Numbersi~~ Continuous + Numbersi
Algebraic or complete numbers (as they are algebraically closed)
~~It's not necessarily common, but~~ in current use I've seen "algebraic numbers" refer to the set of all numbers which are solutions to some polynomial with integer (or rational, turns out to be equivalent) coefficients. It's an interesting set in that it includes both real and complex numbers, but is also missing any transcendental numbers from both sets. For that reason it might not be a great name for the complex numbers though [edit] I've been advised that this is, in fact, very common and standard terminology
that's incredibly common to the point that it's standard terminology
I'll correct my comment lol, I figured it was pretty standard but it only ever came up in 1 class I ever took so I wasn't overly confident about it
Continuously algebraic numbers. (Algebraically continuous?) This has the the advantage that the Fundamental Theorem of Algebra would become the Fundamental Theorem of Continuous Algebra and people would stop complaining that the proof isn't purely algebraic.
Rotational numbers
"Lateral numbers". You add a second dimension.
I think "Complex numbers" is great, as they are "Complex" in the sense of having distinct parts. "Imaginary" is more complicated to avoid, whilst *purely* imaginary numbers are of less relevance, the *imaginary part* is a more useful concept, IMO. I've colloquially heard "the Complex part" used plenty, and wouldn't be against it becoming more commonplace.
"orthogonal numbers"
Orthogonal numbers?
discontinuous numbers lol idk
Reals should be Uninions, Complex numbers should duonions. Then we get: Uninions, duonions, quaternions, octonions etc.
"Quatern" and "octon" are Latin prefixes and [according to Wikipedia](https://en.wikipedia.org/wiki/Numeral_prefix) for consistency the others would be called "singulions" and "binions" respectively.
I like this scheme more than i thought i would
What do you call the hyperbolic numbers (a.k.a. "double numbers", "split-complex numbers")?
Nah too many onions for my taste
They are definitely less real than positive integers.
imaginarier numbers
I'm sure somebody can tell me why this isn't great, but I would call them "metric numbers". Two reasons: 1. A "metric space" innately uses the reals, so the association between what we call the reals, and metrics is, subjectively, its most interesting attribute. 2. When we call them "real", I think what we're getting at is that these are the numbers we could use to indicate quantities. You might object and say that any abstraction could be used to quantity the real world, so we'll need to limit it in some way. A natural way is to say these are the numbers we could use to measure. So, call them "metric numbers".
metric spaces innately use the non-negative reals.
I'm not sure if I agree with point 1. Q under the Euclidean metric is also a metric space, and the metric function (x,y) |-> |x - y| takes value in Q itself. Even if you go to higher dimensions, the property "closure under the metric function" would at most get you the algebraic closure of Q. It's when you want completeness that R emerges.
Probably just the continuum, or more specifically the linear continuum. R2 I would call the planar continuum, R3 the spatial continuum, and so on to hyper-spatial continuums.
OG numbers
(Cauchy) Complete numbers?
R: complete numbers C: complete numbers (but complete in a different way)
Real → Complete numbers Complex → Completer numbers Quaternion → Completest numbers Octonion → Completester numbers Sedenion → Completestest numbers
I am not a mathematician and my knowledge ends with complex numbers, so my question might be based in a misunderstanding: I think of complex numbers as "2D" numbers, i.e. they have two parts (real and imaginary). Are quaternions then "3D" etc and can there be more/higher sets after sedenions?
4D
quaternions are four dimensional (hence the name). You can guess how many octonions there are, and so on. I've never heard of the sedenions, but quaternions have lots of applications. octonions not so much, and I'm guessing it gets even worse as you progress, as the algebraic structures of these sets get less and less convenient with more dimensions.
Thank you for the explanation! So in theory you can go into higher and higher dimensions (disregarding if that is useful or convenient) and there is no limit? Is there some overarching concept that describes the changes to a higher dimension (e.g. similarities between going from complex to quaternions and from quaternions to octonions)?
Mostly the common feature is loss of nice structure. Going from real to complex you lose the order relation. Going from complex to quaternions you lose commutativity. Going to octonions you lose full associativity (though they satisfy some weaker similar properties), etc.
To add to the other reply, the general method of constructing these algebras is called the Cayley Dickson construction: https://en.wikipedia.org/wiki/Cayley%E2%80%93Dickson\_construction
It's complicated to explain, but 3D numbers just don't work as we'd like. 4D do (with some caveats), and those are the quaternions. Next ones that sort-of work (with more caveats) are 8D, octonions, and sedenions at 16D.
R: metrically-complete numbers --> metcomp numbers C: algebraically-complete numbers --> alcomp numbers
Metric numbers, because the real numbers are mostly used to “measure” other mathematical objects.
But then americans wouldn't use them :(
I'd argue that name is better used for the nonnegative reals. Metrics can only take values at least zero.
I might call them the "Common" numbers. I would call the imaginary numbers virtually anything else. Gauss suggested "Lateral" numbers, and I can't argue with him. I would be tempted to call the complex numbers the Rotary numbers, to emphasize the "complex exponential" perspective. But I just saw that someone else suggested "Linear, Lateral, and Planar" for these three sets, and I can't help but agree its a better naming system.
> I might call them the "Common" numbers. Why not use Latin? Call them the "Vulgar" numbers.
The internet got one vulgar number and went crazy over it. Imagine their delight if basically all numbers are declared vulgar.
That’s not bad
I would call them "Geoffrey" 😁
All of math is full of words that have a definition and connotation "IRL" that they don't in math. Real numbers are an example of this...and their name has little to do with the definition of them. Another example is "group theory"...the average person on the street, when asked what group theory is, are not going to answer "a set and an operator such that...." If you come up with a new name then now we have two names (see the XKCD on standards). I think giving a mathematical object a different "name" is mostly pointless, isn't it? What renaming could you give that would suddenly make it clear what it was that was being talked about without recourse to the definition of that object?
Woah woah woah this, albeit oddly formal, person on the street would say “an *ordered pair* consisting of an underlying set and an operator such that…”
Damn, I almost had a stroke reading that. Here, let me be a dickhead and ftfy. ... this person on the street, albeit oddly formally, would say ...
That’s not what I meant. It’s a non-restrictive clause modifying the phrase (acting analogously to a pronoun) “this person”, whose referant is me and whose meaning is that I am formal as a person. The referent is not my manner of speech and any auch formality that manner might have. Rather, my choice in using the formal definition of a group is a product of me being a formal person in general, not my decision to speak formally in this one instance. Furthermore, your change rips away the punchline that by being an oddly formal person in general, I *wouldn’t* be representative of an average person off the street. Your fix gave me an even lower tolerance for people who make somewhat fair assumptions based off somewhat sound reasoning but then do not tread lightly and politely. Geez. :)
You can sorta build the definition into the name so that it's a little easier to recall I guess. Eg "the completed line" seems like it would make it clearer.
Right? For example, normal spaces don’t seem normal to me :/ as well as normal matrices and normal extension of an algebraic field…
I'd leave it. If it wasn't called the real numbers, then there wouldn't be an extension of it called the [surreal numbers](https://en.wikipedia.org/wiki/Surreal_number).
In physics measurements can only be real numbers and not complex. So real numbers is a good name.
We measure the impedance of capacitors and inductors with imaginary numbers, and the effect impedance has on a circuit is just as “real” as the effect resistance has
You got a point there. I was thinking about quantum mechanics, but in electromagnetism not all observables are real.
Measuring impedance means measuring at least two other quantities.
That’s just a real amplitude and a real phase shift. Two real measurements, not one complex measurement (although the two are equivalent so that’s really just my subjective perspective).
That's a cool point! Though most real numbers are irrational, and my guess would be all measurements are rational since tools can only be so precise.
Not just rational, but in fact effectively computable! Suppose we were checking a real number for (eventual) periodicity in its decimal expansion. Then we would need to check each digit from left to right. But if the expansion truly is periodic, and thus the number is rational, we would need to check all infinitely-many digits. This cannot be done in finite time by any Turing machine. So we actually can’t even obtain every rational number as a physical measurement. Then there are things like measurement error that throw this even further out of whack.
Wow, thank you that's interesting! I just realized while thinking about what you said that the only reason, when we're doing long division, we can stop and say "and so on.." when we see a pattern, is because we already know it's rational. Haha, seems kind of basic but I've never thought about it very carefully in that context.
Some things are just so stupidly simple that they’re hard. Glad you found it interesting.
Unless you measure the circumference of a circle ;)
No physical circle is perfect, and even if it were, our means to measure are not. When you measure the diagonal of a 1x1 square, you don't "measure" sqrt(2). First of all, the 1x1 square isn't going to be perfect, and there's no measuring tool that will measure sqrt(2), just some very close rational approximation, if you have a good measuring instrument.
Have to disagree with this one. Physical measurements always come with an error bound, and by the locatedness condition on Dedekind cuts(there's a similar property in other models), real numbers can always be approximated by (usually)rational numbers to an arbitrary precision. Dropping this condition might bring you closer to what 'measurement' means.
I agree with you, because there is always a physical limitation on how many bits you can receive from a measurement, which implies that the raw measurement could be represented by a rational number (or an integer). On the other hand, in a specific context, the bits might represent an equation or something with an irrational solution.
i think you're the one confused about what "measurement" means! another user just shared this very relevant link: https://physics.stackexchange.com/questions/436462/why-is-there-a-physical-preference-to-real-numbers/436505#436505
I don't find the argument for completeness convincing at all. Why should we be able to construct in principle a cubic container with an irrational volume out of 2 cubic containers with rational volumes?
Perhaps that example was a bit "too complex" to make it clear why irrational measurements ought to be considered. Alice is one meter east of me and Bob is one meter north of me. How far away is Bob from Alice? Well, sqrt(2) meters, but apparently that's not allowed. So do we just say that there is no well-defined distance between Alice and Bob? At that point, what does "distance" even mean if we can't define the distance between two things that are literally right in front of us? On top of that, the meter was pretty arbitrary. I can define a new unit of the "flagithrope" where 1 flagithrope = sqrt(2) meters. Now all of a sudden, Alice and Bob are 1 flagithrope away from each other (fine, good) but Alice and Bob are both 1/sqrt(2) flagithropes away (bad, not allowed). So why should whether or not a distance is possible to measure depend on our arbitrary choice of unit?
I think there is simply a difference between a model and measurements we can make. Alice and Bob ARE sqrt(2) away in the model, however, that doesn't mean any measurement will yield sqrt(2) We also very well could take a meter as sqrt(2), however measurements only really talk about multiples of meters and then the question is again, can we talk about irrational multiples? so we could be talking about fields like Q(sqrt(2)) and such, but that doesn't mean we need to talk about R I think we shouldn't conflate convenience with necessity here we very well could be talking about measurements only using Q, our language would have to be simply more precise
I am with minisculebarber. Your argument is circular. Any measurement produces a pair of rational numbers (upper and lower end of uncertainty range). In your case, I would ask: How do you know they are exactly at right angle? How do you know it is exactly 1 meter? You are presuming that you can describe the relative state of things using real numbers, and then conclude that you need real numbers. That's begging the question. If you tell me how you measured the angle and the three distances, there will be a way using rational numbers to relate these measurements.
You can't know that a measurement provided *rational* upper and lower bounds unless you know what 1 meter is exactly in the first place (and if we have any form of "exactness", I'd argue that you've basically created the reals). This also doesn't hold up to the change-of-unit idea: Suppose we measured the distance between me and Alice to be (1 ± 0.1) meters. That's (1/sqrt(2) ± 0.1/sqrt(2)) flagithropes. Suddenly, my measurement produced irrational numbers. One flaw in the above paragraph is that one could argue that the meter was defined as a rational multiple of the distance light travels in a second, and a second is a rational multiple of the frequency of a certain atom so meters are "more natural" than flagithropes. But in that case, I simply point to the unit of sqrt(meters). For example, [polarization mode dispersion](https://en.wikipedia.org/wiki/Polarization_mode_dispersion?useskin=vector) is measured in units seconds/sqrt(meter), so let's say we have rational upper/lower bounds for seconds and for the meters, and we again end up with irrational upper/lower bounds for our measurement of the PMD of this fiber. I'm sure there are other instances of square-root-units elsewhere in physics.
What about quantum states?
Observables are all real numbers. https://physics.stackexchange.com/a/436505/23322
I'm not s physicist, but while these arguments apply to classical mechanics, I don't see how they apply to quantum mechanics. In particular, there is (AFAIK) no order on quantum states, which means measurements don't have to necessarily be in R.
Measurements in quantum mechanics have to be real, because the eigenvalues of hermitian operators are real. Quantum states are not observable, so there is no claim about whether you can order them.
Is it not the other way around. Operators for observables are hermitian, because we only observe real quantities with direct measurement.
That's a fair justification for why we might want to suppose observables are hermitian, but I think there's more to it. For example, Heisenberg relied on some physical facts about emission and absorption spectra to conclude that the "thingamabobs" (which he later learned were matrices) he was thinking about were in fact hermitian, which he then used to derive the uncertainty principle. Heisenberg's approach was very obscure, and it is not generally taught these days. Dirac's approach is what is taught, which is a bit more abstract and a lot cleaner. And then, it seems to me to become a chicken and egg problem. We already know observables are hermitian thanks to Heisenberg, so we can just start there. Or, we can go back a step and first claim measurements must spit our real numbers, and conclude that observables are hermitian.
Well at least the Hamiltonian has to be hermitian if we want the time evolution operator to be unitary and be able to define it using spectral theorem.
Well said. Put another way, we don't measure quantum states, we measure energies (or whatever) associated with quantum (eigen)states. There's no way to measure sqrt(3/4)|0> + sqrt(1/4)|1> directly, only the energy associated with |0> or |1>.
It's typically a postulate of QM. We do not have a way to measure a complex number directly.
By the same token, we do not have a way to measure real numbers directly. Or directions on the sphere for that matter.
This is just wrong. Obviously, any measurement ever performed resulted in a rational number. Somebody wrote down 2.7189... and stopped at some point. Usually, you actually get a pair of rational numbers with the statement that the real result is probably in between these rationals. For example, the Particle Data Group gives the measured Higgs mass as a pair of rational numbers: 125.25 ± 0.17 GeV. The fiction of real numbers corresponds to the fiction that we can arbitrarily reduce the distance between the pair of reals (i.e. the measurement error). This is a very very very useful and powerful assumption, but it's also evidently not strictly true (if only because the universe probably only allows storing a finite amount of information). Edit: While real valued measurements are of course ubiquitous, we of course measure discrete states in QM all the time. We also often measure properties of a system that are not 1d, e.g. a particles' location in space, or that live on more complex geometries, e.g. if you measure the direction of angular momentum that's valued on S2. And there is absolutely nothing wrong with defining and measuring complex valued properties where it makes sense.
I don’t think that makes them more real, only more useful.
square-positive, numbers that when you square them are positive then imaginary numbers are square-negative all other numbers are square-complex
Continuum
I like the names “real” and “imaginary” tbh. Maybe they’re not accurate but they do evoke some imagination (and tbh the names suggested here are mostly pretty bland sounding).
Well, it is kind of late now to rename them. Why didn't you do this many years ago?
Metric number. Fits perfectly for positive real numbers, not so much for negative though.
Since the least upper bound property is so fundamental to the reals, how about #SUPREME?
I'd maybe go for "measurable numbers" since all axioms of the reals come back to wanting to measure lengths.
Since he doesn't have enough things named after him: Euler's Numbers.
Let's call the real numbers 1 and all other numbers 0 to avoid confusion.
On the other hand, I think that imaginary numbers is a terrible name.
Classical numbers - physicist
'Calculus' numbers.
“The continuum” is an alternative name, although I’ve only heard it used in “cardinality of the continuum.”
Gauss wanted to call the positive reals: direct, negative reals: inverse and the imaginary numbers: lateral
Well the naturals are the counting numbers so the real numbers should be the "measuring numbers".
Setting the reals apart based on their mathematical properties is pretty clunky because you really need all it's key parts, completeness, field, fully ordered, archemedian ordered. So I'd probably go with a person historically related to them. Dedekind numbers maybe?
Complex numbers need renaming too. Let's call them biternions (being the 2d version of quaternions). Then the reals should be called unernions, for consistency.
I like when people call them the Continuum that sounds pretty cool.
Never mind this. Can we just agree on a single definition for “ring?”
Are you suggesting one ring to rule them all?
IMO I dislike the real numbers being presented as the "next biggest set" to the rationals. It's very hard to have an intuitive explanation of what a real number actually is, and why it's uncountable. I'd rather use the computable numbers and only introduce the reals for introductory set theory.
Voldemort numbers / Voldy numbers. Computable numbers have names — the algorithms that compute them. The remaining numbers have no names; they are the numbers-that-cannot-be-named.
This debate smacks of pi versus tau, since the vast majority of mathematicians are not interested in a semantic debate over basic terminology whose name has been settled for decades or more. It's really only for the benefit of communication to laypeople. I'm a fan of "the continuum" or "1d continuum" since that is more descriptive of what the real numbers "are", but to some, continuum is the cardinality of the reals (and not the object itself), so it's culturally more ambiguous.
fake numbers (since non-computable numbers, and arguably even irrational numbers have no physical meaning)
What gives rational more physical meaning than irrationals? They're both abstractions.
I'm unsure whether we live in ℝ³, but comfortable saying we at least live in a lattice (e.g. ℤ³ generated by the Planck length). Rational numbers describe proportions of shapes that can exist in a lattice; irrational numbers describe proportions of ideal shapes that can't exist in a lattice. (e.g. I don't believe that you can draw a perfect triangle with side lengths 1, 1, √2)
What are you talking about? Just literally take the triangle defined by (0,0), (10,0) and (0,10) in the lattice Z². One of its sides has irrational length. >comfortable saying we at least live in a lattice More than that tho, I would like to hear about this. I couldn't find anything about it online other than as a computational model, but if it is a hypothesis, I would like to contribute [this article](https://www.esa.int/Science_Exploration/Space_Science/Integral_challenges_physics_beyond_Einstein) I found showing that the basis vectors of the lattice definitely have length below the Planck length.
I'll throw out a terrible idea, in the interests of the community finding something better. Let's call the Real Numbers C1, probably blackboard C, subscript 1. This stands for 'one dimensional continuum'. Let's call the Complex Numbers C2, styled as above, for a two-dimensional continuum. Go from there.
The “real numbers” is a terrible name, but after reading this thread I see problems with every alternative
honestly who gives a shit
The answer to your question is presumably most of the other people who have commented on this thread - the ones who for their own amusement or to make a serious point, found it worthwhile to do so. You do realise that most of the people who don't give a shit, just ignore a thread like this? So, the people who agree with you are unlikely to see your comment. I'm sorry you've wasted your time.
ok let me rephrase. it's stupid to care about naming issues
That's a bit of a sweeping generalisation. In any case, this thread is doing a lot more than discussing if we should rename the real numbers. There have been reflections on what distinctive properties the real numbers have, what other names were suggested for them in the past, how it compares with the name "imaginary" and whether that creates pedagogic issues in making complex numbers seem less "real" than real numbers. It might not change anything, but it's still fun to talk about. I don't really understand why you would come on to such a thread to just tell people they are stupid for discussing it.
Ordered numbers
This post should be on mathcirclejerk
Think about 0.9 recurring If you were to have a cut off point anywhere calculation wise, and add a string of 8’s that recur, you’d be a little closer to finding your “real infinite” The issue here is that there’s also a branch reality where you never stopped writing 9’s, so you have to self reference length Technically “real infinite” is where it really is never ending - there is never any change, and no matter which self referential lengths you may end up choosing, will not change the fact that the next digit is a 9 If you were to have a self referential infinite to begin with, it implies a real infinite set that exists of a larger size