2966: Exam Numbers

|< < Prev Comic #2966 (July 31, 2024) Next > >|

Exam Numbers

Title text: Calligraphy exam: Write down the number 37, spelled out, nicely.

Explanation

The comic depicts various similarly formatted examination questions that might appear on test papers at various points in a student's potential academic career. While they all share a similar feel, they are asking for different things, some of which might be considered more serious and examinable proofs of study than others. The joke is that the first and last questions are the same, but have very different interpretations based on the context.

Exam numbers

Exam level	Question	Answer
Kindergarten math	Write down the biggest number you can think of	At a kindergarten-level education, it is assumed that an individual might write down a relatively small number like 300, depending upon whether they have encountered the concept of hundreds. It might also be interpreted as "what's the highest number that you (think you) can count up to". Given the nature of a child's exuberant glee at learning about really big (but otherwise normal) numbers, they may even try an answer something like "a million billion squillion gazillion... and seven!". It is not certain what criteria would be used to mark this question correct or otherwise, it may actually be a stealth question in child psychology or a question that everybody "gets right" so long as they answer it. Although an advanced concept taught years later, infinity is not out of the question here, as employing it is often the only way to explain a substantial number of very simple math questions.
Pre-algebra	Write down the value of x if x=3x-8	3x refers to the multiplication of 3 and the originally unknown number x, as a convenient shorthand. By subtracting 3x from both sides, -2x = -8. Divide both sides by -2 to find x=4. Alternately, subtract x from both sides to give 0 = 2x - 8, and as taking 8 from two xs makes it zero, one x is half that (Verify by plugging x=4 into the original equation. 4 = (3*4) - 8 -> 4 = 12 - 8 -> 4 = 4.) 4 is also the subject of 221: Random Number.
Calculus	Write down the value of ⌠π │ x sin²x dx ⌡0	The integral can be solved using a trigonometric identity and integration by parts: ⌠π │ x sin²x dx ⌡0 ⌠π = │ x (1−cos 2x)÷2 dx ⌡0 ⌠π ⌠π = ½│ x dx − ½│ x cos 2x dx ⌡0 ⌡0 │π ⌠π = ¼ x² │ -¼│ x (sin 2x)′ dx │0 ⌡0 │π │π ⌠π = ¼ x² │ -¼ x sin 2x │ +¼│ (x)′ sin 2x dx │0 │0 ⌡0 │π │π ⌠π = ¼ x² │ -¼ x sin 2x │ +¼│ sin 2x dx │0 │0 ⌡0 │π │π │π = ¼ x² │ -¼ x sin 2x │ +⅛ cos 2x │ │0 │0 │0 = ¼ π² = 2.4674…
PhD Cosmology	Write down the Hubble constant to within 1%	The Hubble constant is a component of Hubble's law, which describes the relationship between the distance between galaxies and their speed of separation. Its exact value is not known to this level of precision; it is about 70 (km/s)/Mpc. Different methods of measuring it have come up with significantly different values, with their own error bars, generally of 2% to 5% (both plus and minus, or asymmetrically, but with a similar overall range) and resolving this difference (the Hubble tension) is one of the great challenges of modern cosmology. Getting a PhD involves advancing the field, so it seems this particular PhD student has completed a thesis solving this problem. This question might have a different acceptable answer if asked again of the student, depending upon further developments in cosmology, and the 'constant' itself is supposed to change over cosmological time (the nature of this change being yet another subject requiring further study, calculation and observation). What would a correct answer look like? Because of how significant digits are interpreted, 69 (km/s)/Mpc could be correct to within 1% while 70 is not.
Game Theory	Write down 10 more than the average of the class's answers	Game Theory studies "games" (and 'game-like' situations) in which two or more participants take actions that will succeed or fail based on other participants' decisions. In this case, all students' answers will be averaged (likely a simple arithmetical mean), and the best answer would be one that is 10 more than that average. For instance, if the answers end up being 30, 40, 50, 60, and 70, the mean would be 50, making 60 the best possible answer. As an individual test taker who doesn't know what your classmates are going to write down, the objective is to answer 10 more than your estimate of the average including your own guess, knowing that everyone else will be doing the same. Attempts at numerical solutions, including Monte-Carlo methods, will not converge because the problem as stated is unstable. The best answer will always be ten more than what the other students think the best answer is, and they have the same information as everyone else, which results in a contradiction. However, the average of any set of numbers that includes infinity is infinity, and ∞ + 10 is still ∞. Something similar to this question is found in the title text of 2385: Final Exam.
Postgraduate Math	Write down the biggest number you can think of	This question echoes the very first example, but would be expected to be answered very differently (unlike a revisiting of most of the others). Postgraduate math students can probably think of very large numbers. While a tempting answer could be "infinity", most mathematicians do not consider infinity to be a number,[actual citation needed] but rather a class of numbers. (Writing down "Infinity" in this context would be as wrong as writing down "Primes" or "Positive integers"). Even if infinity is an acceptable answer, some infinities are bigger than others. Students familiar with the field of Googology or Transcology may give an answer such as Rayo's number, which was the winning entry in the Big Number Duel. This might heavily depend upon the branch of mathematics you are studying. Named (finite) numbers, or ones with specific and useful notations, might satisfy some questioning contexts, whilst the existence of a whole further set of trans-finite numbers (i.e. increasingly large types of "infinity") would be important considerations in others. For those associated with more computational mathematics, any infinity would be Not a Number, and their answer might instead be the ceiling of some binary representation (typically 28n-1 for some value of n), the largest unsigned value reliably storable in a given byte form for an integer (e.g. a double quadword). On the other end of the spectrum, many abstract algebraists might answer with some variation of "What ring are we working in, and is it even well ordered?" It also might be a trick question: if you can envision a real number greater than one, are you even doing real math (in a given field)? As with the kindergarten question, there may be no previously anticipated "correct" answer. It could be another "correct just so long as you answer it" (or perhaps "sensibly" so) or the mark goes only to those giving the greatest valid number across all submissions.
Calligraphy (title text)	Write down the number 37, spelled out, nicely	Calligraphy is the art of artistic writing. The title text expands the joke outside the realm of math and points out that since calligraphy does not require any math skills, the only way a calligraphy exam would even mention numbers is if one had to write them out in such a way as to showcase their calligraphic skill and aesthetic judgement (choosing a form and adornment of script that is "nice", which may be a highly subjective choice). In this case, it could be rendered as "thirty-seven" or "thirty seven", or possibly, "one score and seventeen" in old-fashioned writing. The subject may choose to render it in a language other than English — for example "dau ar bymtheg ar hugain" would provide significant scope to show off calligraphic skill. 37 is a number that some people believe mysteriously appears more often than it should; this was a subject of a recent Veritasium video.

Note that "PhD Cosmology" and "Postgraduate Math", unlike the other items, aren't specific courses but instead refer to areas of study, and as such wouldn't have a exam (unless a PhD thesis defense is considered an "exam").

This style of final exam question, un-numbered and therefore possibly the only question upon the whole of each final paper, in some ways (for some instances) echoes the question "What is your name?" that Randall will be aware was the sole question given to Discworld's Victor Tugelbend in an attempt to ensure he comprehensively passed (or utterly failed) his final student-wizard's exam, after many prior times of deliberately not-quite-passing.

Transcript

This transcript is incomplete. Please help editing it! Thanks.

[6 different math test questions.]

[The first panel:]

Kindergarten math final exam

Q. Write down the biggest number you can think of

A. [empty box]

[The second panel:]

Pre-algebra final exam

Q. Write down the value of x if x=3x-8

A. [empty box]

[The third panel:]

Calculus final exam

Q. Write down the value of [integral sign, from 0 to pi] x sin^2 x dx

A. [empty box]

[The fourth panel:]

PhD cosmology final exam

Q. Write down the Hubble constant to within 1%

A. [empty box]

[The fifth panel:]

Game theory final exam

Q. Write down 10 more than the average of the class's answers

A. [empty box]

[The sixth panel:]

Postgraduate math final exam

Q. Write down the biggest number you can think of

A. [empty box]

 add a comment! ⋅  add a topic (use sparingly)! ⋅  refresh comments!

Discussion

pre-algebra: 4, calculus: pi^2 / 4 (about 2.467), physics: cosmological constant: depends on how you measure it 162.158.167.48 18:11, 31 July 2024 (UTC)

Game theory: -5x10⁶ (maybe helpful, maybe not... just be thankful I didn't include an i factor in there somewhere...) 172.70.162.185 18:20, 31 July 2024 (UTC)

Interesting; I went with ∞+10. So, between our answers, that makes the average... ProphetZarquon (talk) 05:21, 1 August 2024 (UTC)

Could somebody reformat all the math here in whatever LaTeX plugin this wiki uses? --162.158.222.102 18:35, 31 July 2024 (UTC)

Probably not, because the MathML here is broken. But, also, nothing I see requires anything particularly complicated, it can all stay in fairly straightforward (standardly formatted) text. 141.101.98.224 18:44, 31 July 2024 (UTC)

I had to look up "TREE(3)." Seriousness aside, I think the largest number would be the astrological sign 1 that has its end_points_ as galaxy clusters. 172.68.245.184 19:26, 31 July 2024 (UTC)

Which astrological sign? Search engines aren't helping. Onestay (talk) 20:41, 31 July 2024 (UTC)

The nonexistent one I just made up that looks like a "1." 😃 172.71.222.6 21:06, 31 July 2024 (UTC)

'OAK'? 'ELM'? 'ASH?' 'BOX'? 'YEW'? 141.101.98.165 08:52, 1 August 2024 (UTC)

If infinity _is_ a number, it might be a possible solution to the game theory question. The average of any set of numbers that includes infinity is infinity, and infinity + 10 is still infinity. I probably wouldn't try that in most classes, but a game theory professor might approve "gaming" the system, as it were. 172.70.39.44 (talk) (please sign your comments with ~~~~)

If I would prefer no-one (else) to win, I might submit -∞ as my answer. 172.70.90.74 20:13, 31 July 2024 (UTC)

If I really wanted to mess with them, I would submit i. 172.70.160.248 08:54, 1 August 2024 (UTC)

I did a bit of a deep dive into wikipedia and the googology wiki and the answer to the last question depends on a few things (along with assuming ZFC). If transfinite ordinals count as numbers, then those at the end of List of large cardinal properties take the cake (if i'm reading it right). Otherwise, something based off Rayo's number is the best googologists have come up with so far. 172.69.246.149 20:18, 31 July 2024 (UTC)Bumpf

How about "On, in the context of MK set theory"? MK is a standard way to extend ZFC by allowing classes as mathematical objects, so On (the class of all set-size ordinals) is a class-sized "ordinal". But MK doesn't allow proper classes to be contained in any object, so "On+1" doesn't exist except as a definable hyperclass. Thus, On is the biggest "number" in a model of MK set theory. 172.68.205.151 (talk) (please sign your comments with ~~~~)

I personally would not call a class a number, but that is very very subjective! And why would we take a standard extension of ZFC in the first place and not just keep ZFC? yadda yadda yadda you get the idea :) 172.69.246.142 21:43, 2 August 2024 (UTC)Bumpf

Isn’t the joke in the pre-algebra that it would require algebra in order ro calculate? 172.68.70.135 20:36, 31 July 2024 (UTC)

Yes. I agree that it would be worth adding wording along the lines that “the joke here is that you need algebra to solve the equation”. Dúthomhas (talk) 20:56, 31 July 2024 (UTC)

I interpreted the 'pre-' bit as being more like 'proto-' - i.e. it's not fully proper algebra, but it's the kind of work you would do in preparation for tackling proper algebra.172.68.186.156 08:58, 1 August 2024 (UTC)

That is actually exactly correct, at least in the US. Pre-algebra teaches the basics of algebra, and any seventh-grade student _should_ actually be able to solve the given problem. IDK if Randall gave this thought when formulating the joke, though... Dúthomhas (talk) 05:33, 2 August 2024 (UTC)

You know, formatting math on this wiki would be a lot easier if the Math extension were correctly installed, but evidently it's not: Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \int_0^\pi x \sin^2 x \;dx Zmatt (talk) 22:22, 31 July 2024 (UTC)

Is that integral really correct? I asked Wolfram Alpha and it gave me

integral x sin^2(x) dx = 1/8 (2 x (x - sin(2 x)) - cos(2 x)) + constant

which does not seem to be the same as

−2x sin(2x)+cos(2x)−2x)/28 + C.

But maybe there's something with half-angle formulas that makes them the same? … but I don't think so, they don't evaluate the same for x=0. JohnHawkinson (talk) 02:56, 1 August 2024 (UTC)

Yup, looks like it was supposed to be

-(2x sin(2x)+cos(2x)-2x^2)/8

but they messed up the places of the negation and square.

Though the important part here isn't what it is at any f(x), but what it is for any f(x)-f(y). In this particular case, f(pi)-f(0). 162.158.41.121 04:49, 1 August 2024 (UTC)

The answer

(-2x sin(2x)+cos(2x)-2x)/28 + C

is indeed incorrect. However, your reasoning for thinking this is wrong. Two functions can have the same derivative while also having different outputs for the same input. For example, sin^2(x) and -cos^2(x) have the same derivative, but sin^2(0) = 0, whereas -cos^2(0) = -1. ISaveXKCDpapers (talk) 13:59, 3 August 2024 (UTC)

As to biggest numbers: I thought most people would say the answers revolved around "nine-stuffing." For a kindergartener, stuff in as many bare 9s as possible. For a postgrad, mix in exponentiation and write your numbers even smaller than a kindergartener can. 9^9^9^9... or perhaps 99^99^99... or…I'm not sure what's optimal. Of course, I'm not math postdoc ;) Or maybe some integrals or big-∏ notation. JohnHawkinson (talk) 12:41, 1 August 2024 (UTC)

Mastered division in preschool, learned exponents and logarithms in kindergarten. When I got asked this very question, my answer was (10^(10^9-1)-1), which is 999999999 "9"s. When told off for a "wrong answer format", I asked the teacher exactly how long she expected me to spend writing out literally nearly a billion digits to answer "properly" and whether she could afford that much stationery. She have me an A+. I knew the lower numbers don't actually matter so much, but it still took me until first grade to properly get into the programmatic mindset, and now the biggest finite integer I can properly consider is (2^(2^48)), which...

10^9-1 = 999999999

2^48 = 281474976710656

And those are the higher numbers, so even though the lower ones are 10 versus 2, it's pretty clear which number is bigger, no? 172.71.150.197 23:22, 1 August 2024 (UTC)

If you answer "the biggest number you can think of" with "<some number> minus one", then I think you would rightfully have done yourself out of a mark... 141.101.98.128 23:59, 1 August 2024 (UTC)

What even is thinking? Does math necessarily count? What are numbers? Whole numbers only? No living human can yet prove whether or not pi^(pi^(pi^pi)) is an integer, even though it's less than 2^(2^61). To me, "nine nines of nines" is entirely reasonable for a kindergartener who knows about exponents to answer, and while I may be slightly surprised at knowing proper expression, b^p-1 (seen both as 10^9-1 for "nine nines" and 10^(...)-1 for "of nines") appears in a lot of math things, e.g. the biggest signed 32 bit integer, 2147483647, is 2^31-1. Just because a normal internet connected computer can only count octets from 0 to 255, does that mean 255 being 2^8-1 is a more wrong answer than 2^8, which it "thinks" is 0? Does that make 0 automatically the largest number anybody can think of? 172.68.23.200 00:29, 2 August 2024 (UTC)

I learned about tetration in college, and always thought it was the coolest thing. It is to exponentiation that exponentiation is to multiplication. The playground rules of infinity + 1 basically applies here. Whatever largest number you can write down, Just say 999^^999 for example, which being 999 to the 999th power 999 times. a googolplex is 10^^4, so this is an extremely fastly growing number, but not being 100% math nerd, don't know if this counts in any way. Rayo^^rayo, does it even make sense, who knows.--Youj ying (talk) 04:36, 2 August 2024 (UTC)

googolplex = 10^googol = 10^10^100 = 10^10^10^2. Whereas 10^^4 = 10^10^10^10 = googolplex^(10^10^10-10^10^2), a number so much larger that the number of times you could write out googolplex using the same number of digits as 10^^4 has is itself a number so large as to have about 9999999900 digits. Even 4^^4 is bigger: ln(a^b)=ln(a)*b so ln(ln(10^10^100))=ln(ln(10)*10^100)=231.09254... and ln(ln(4^^4))=ln(ln(4)*4^4^4)=355.21799... which can be double-checked on many calculators. 162.158.41.180 09:32, 2 August 2024 (UTC)

The PhD Cosmology question is easy, actually. Just write down H0 (imagine that 0 is subscript, I don't know how (if) I can format this comment). It doesn't ask you to write down the value of the constant, just the constant itself.172.71.103.118 14:56, 1 August 2024 (UTC)

Well if you're going to get smartass about it, the last question is easy too - you just write out "THE BIGGEST NUMBER YOU CAN THINK OF"172.68.186.56 15:17, 1 August 2024 (UTC)

If you asked a psychologist what is the biggest number you can think of, they'd probably say "about 5". Anything more than that, and you're not really thinking of the number - you're just thinking of the name of the number.172.68.186.128 15:44, 1 August 2024 (UTC)

I can absolutely visualize a grid of 9, 16, or 25. 162.158.41.22 04:23, 2 August 2024 (UTC)

Exactly, though - a grid. You're really thinking about 5 and 5 and an arrangement pattern.172.70.85.19 08:13, 2 August 2024 (UTC)

Given that postgraduate math should be real math, according to 899: Number Line the largest number should be 8. My first thought for that question was card(R), as it is not really possible to prove that the number you thought about is larger than that. (Granted, the powerset of the reals is larger, if you fix an interpretation of R). --172.68.253.131 17:25, 1 August 2024 (UTC)

Prime notation in integrand?: Here we go again, I guess. In this edit, 172.68.2.126 changed the working of the integral to include expressions with a prime inside the integrand, like ¼ ∫0π x (sin 2x)’ dx. I…have no idea what the prime is supposed to be indicating here…differentiation? Even if this notation is meaningful (as it surely must be?), I don't think we should be using it here. Sure, many lay readers won't understand calculus at all, but for those that do, keeping it at a level understandable to a high-school calculus student seems wise. But since I don't understand it, I wanted to post before changing it back. What does it mean? JohnHawkinson (talk) 19:58, 1 August 2024 (UTC)

What is a number?

Infinity is _not_ a number. Dúthomhas (talk) 19:39, 31 July 2024 (UTC)

Infinity is absolutely not a number, and is the one answer I would mark as unambiguously wrong for the last one. Just say TREE(G_64) or something. 162.158.154.31 20:15, 31 July 2024 (UTC)

This is correct. No one in post-grad math would write “infinity” and expect that answer to work. Infinity is NOT a number except for seven-year-olds. Yet the explanation above continues to posit it as a possible correct answer. Dúthomhas (talk) 20:49, 31 July 2024 (UTC)

I qualify as a "post-grad math", and yet, I think infinity would have been a perfectly valid answer. Let me explain. The term "number" without further context is a bit vague, because there are several possible generalizations of natural numbers (something that presumably everyone agrees to call a "number"), and they are not compatible, ie. there is not a single generalization that generalizes them all. So we have to choose which generalization makes sense in the current context. Since the question is about thinking how big a number is, I naturally thought that the adequate generalization would be one that focuses on the order on natural numbers, ie. ordinals. In that case, my answer to this question would be "the class of numbers I can think of is not bounded, therefore there is no such thing such as a 'biggest number I can think of'". But if I had to write down a big number, I would write ε_{ε_{ε_{...}}} up until I filled the page, because that's the most efficient way I know to write a big, *big* infinity. Which is a number. (and I'm not seven, just to be clear) Jthulhu (talk) 08:35, 1 August 2024 (UTC)

In IEEE floating point math, Infinity is not Not A Number. The latter is an indication of error (in a context where errors can't be signalled immediately) and an entirely separate concept to infinity. But both are not Normal Numbers. Or even Denormalized Numbers. Floating point math is a whole lot trickier than it appears to be at first glance, and only extremely tangentially related to mathematical reals. --172.68.205.54 00:48, 1 August 2024 (UTC)

I would have written this, but I saw that your comment already explained the two points I would have made, so, well, well done! Jthulhu (talk) 08:35, 1 August 2024 (UTC)

If I write a song titled "Infinity" that was part of an opera, then it would be a number. 162.158.175.141 13:26, 1 August 2024 (UTC)

A number, by definition, is a construct used to classify and/or compare values. How rigorous this needs be for one limits the extent to which they accept things as being a number. Even things like "apple" could be interpreted as (dimensioned) numbers, with a possible value being "1 fruit"; In that regard, one may consider things like apple=orange<grapes.

Just "infinity" is nearly useless in this regard, as it's "no end thing". Usually interpreted (when necessary) as the countable infinite cardinal x=aleph_null, this prevents most useful comparisons, including dimensional analysis since x^n=x for all counting (aka. finite positive integer) n. Spacetime may or may not be boundless, but we can't tell how many edges may or may not loop. Is it infinity? Yes. Is it infinite? God only knows. Can you *count to it*? God can. Does that make it a number? Depends. Is "infinity plus one" a sane concept? No, it can't be finite, ordinal, and/or real in a way addition is defined; It's without end, and if you could add to it, that would indicate an end.

In contrast, classification has its roots in trade, and barter, and tipping. How much of a thing is enough, but not too much. Somebody may accept between 1/2 and 2/3 of a pie you're splitting, because less wouldn't be fair and more may give them a stomach ache; Is 3<=6x<=4 a number? It's similar in uselessness to "infinity", but whether something is less or more can at least still be established within its range. In the limit, Surreal numbers are the principal example of classification, taking the arithmetic mean of the maximum and minimum of their lower and upper bounds, or the predecessor or successor, or zero. For example, y={y|1} is the biggest number less than one, with z<=y<1 for all z<1. It's less than one, but not any "smaller" than one, with an immeasurably infinitesimal difference 0<1-y.

Choice of axioms is very important for all this, since its full extent can render everything except finite non-negative integers "not a number" (by Presburger Arithmetic), or allow everything up to and including unique antichain cardinalities (by Martin's Maximum).

The sixth power of the smallest ordinal with the cardinality of the continuum in the constructed universe (w_1^6 where beth_n=C(w_n)) is the biggest number I can personally conceptualize, although I can consistently work with w_2 in this system as well. Does the fact that this is infinite make it any less useful as a number than 2.5? No. It says I can think accurately about all the standard ways of comparing things in up to 6 infinitely divisible dimensions. Just because one cannot necessarily picture something others can't doesn't mean it doesn't exist. If a one-eyed person can only see a 2 spatial + 1 temporal dimensional image, that doesn't mean depth doesn't exist, it just means it's "hidden" from that perspective. 3+1+2 has two "hidden" dimensions compared to normal 3+1 spacetime, and beth_1 is infinitely divisible unlike the quantum (at most beth_0) nature of our known universe, but I can still work with 3+1+1, and 3+1+2 in the same way people can think about a (possibly looping) universe where everything can be bigger or smaller, and spatial geometry itself may be some degree of spherical, and people have been working with fractions since antiquity, so why should I limit myself to what other people can grasp?

In summary: "number" is too vague for claiming most things "aren't" to be reasonable. Infinite values (that aren't just "infinity", that's vague enough by itself to be almost as unreasonable) are just one one example of a valid answer most people seem to be up in arms about. 162.158.41.181 01:06, 1 August 2024 (UTC)

All right, all right. I yield. That’s some... _impressive_ reasoning. If we are going to redefine words to meaninglessness then there is no hope of engaging in useful discussion. I’m sure Randall will at least get a good laugh out of the idea that post-grad math students would submit “infinity” as the largest number they could think of. I still think it a disservice to readers to posit infinity as a _valid_ answer, though. Dúthomhas (talk) 05:05, 1 August 2024 (UTC)

This isn't redefining words to meaninglessness. Do you know how many branches of math there are that generalize the concept of "number"? 2500 years ago the Greeks didn't believe irrational numbers were numbers (and Hippasus supposedly drowned for discovering them), and it was a long time before 0 was accepted as a number, but now we have entire hierarchies of numbers. 100 years ago Georg Cantor created another upheaval in mathematics when he invented Set theory along with the infinite ordinals and cardinals, which are usually referred to as "numbers". Complex analysis defines the "extended complex numbers", modeled by the Riemann sphere, which includes a "point at infinity" (and of course you can extend the real numbers] similarly). Non-standard analysis defines not only infinite but infinitesimal numbers as Hyperreal numbers. Granted, I'll agree that using "infitity" without any qualification or context is not really precise enough to do anything useful with. 172.69.23.49 03:12, 3 August 2024 (UTC)

Um, hi... Nothing really to do with what you wrote, but how you wrote it. You use URL-links for wikipedia. You have [https://en.wikipedia.org/wiki/Riemann_sphere Riemann sphere] ("Riemann sphere") instead of the more nice templated {{w|Riemann sphere}} ("Riemann sphere").

But you also seem aware of the 'plural modifier' suffix without doing it either 'normal' way. You have given [https://en.wikipedia.org/wiki/Hyperreal_number Hyperreal number]s ("Hyperreal numbers") instead of either [https://en.wikipedia.org/wiki/Hyperreal_number Hyperreal numbers] ("Hyperreal numbers"), with no point in adding the pluralisation outside the link at all, or {{w|Hyperreal number}}s ("Hyperreal numbers") which does most of the work for you...

Wouldn't have commented on this at all (would have rationalised any case in an Explanation, but wouldn't change a person's Discussion contribution, even purely as a background aesthetic), and there have been a lot of 'lazy' []-linking by IPs recently where the {{w}}-template could have been used, but that last one seems to show an intent to do it right/nice. Just crucially missing the probable intent. So... FYI. General informative meta-comment that I hope helps you/others a bit, if you care to pass by this way again and appreciate it. 172.69.195.175 08:48, 3 August 2024 (UTC)

Y'all, the answer is clearly 1. Sincerely, someone who has studied probability.162.158.137.155 14:04, 1 August 2024 (UTC)

I came to the same conclusion by different means: A number that fills 100% of the answer area is the biggest valid answer, and is very clearly a one turned on its side. 108.162.238.68 18:59, 2 August 2024 (UTC)

No actual cosmologist denotes the Hubble constant in Hz. It's about 70 (km/s)/Mpc. 162.158.41.22 04:23, 2 August 2024 (UTC)

The true Cosmic cosmologist surely knows that H₀ is 1 (exactly and always), redefining all subsequently derived units in terms of that and the other 'true constants' of the universe... ;) 141.101.99.172 10:43, 2 August 2024 (UTC)

Infinity comes up in kindergarten more often than most people think. Kids learn to keep asking "why?" in sequence around four, and they don't stop until they get an answer which makes them face some terrifying reality, usually concerning mortality. Infinity is small potatoes and involved in the answer to a suprising number of simple math questions. 172.70.214.124 05:50, 3 August 2024 (UTC)