Editing 2966: Exam Numbers

{{comic
| number    = 2966
| date      = July 31, 2024
| title     = Exam Numbers
| image     = exam_numbers_2x.png
| imagesize = 553x400px
| noexpand  = true
| titletext = 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.
{| class="wikitable"
|+ Exam numbers
! Exam level !! Question !! Answer
|-
| {{w|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!".  However, a kindergarten student is not likely to be able to write small enough to fit that number in the space provided; the largest numeral that will fit in a box of arbitrary size is consists of repeated ones, which are narrower than other digits, thereby allowing more digits to fit.  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-{{w|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 ''x''s 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]].

|-
| {{w|Calculus}} || Write down the value of <pre style="line-height: initial;">⌠π
│ x sin²x dx
⌡0</pre>|| 
The integral can be solved using {{w|List_of_trigonometric_identities#Power-reduction_formulae|a trigonometric identity}} and [https://math.libretexts.org/Bookshelves/Calculus/Calculus_(OpenStax)/07%3A_Techniques_of_Integration/7.01%3A_Integration_by_Parts integration by parts]:

<pre style="line-height: initial;">⌠π
│  x sin²x dx
⌡0

  ⌠π   1 − cos 2x
= │  x ―――――――――― dx
  ⌡0       2

  1 ⌠π        1 ⌠π
= ― │  x dx − ― │  x cos 2x dx
  2 ⌡0        2 ⌡0

  1    │π   1 ⌠π
= ― x² │  - ― │  x (sin 2x)′ dx
  4    │0   4 ⌡0

  1    │π   1          │π   1 ⌠π
= ― x² │  - ― x sin 2x │  + ― │  (x)′ sin 2x dx
  4    │0   4          │0   4 ⌡0

  1    │π   1          │π   1 ⌠π
= ― x² │  - ― x sin 2x │  + ― │  sin 2x dx
  4    │0   4          │0   4 ⌡0

  1    │π   1          │π   1        │π
= ― x² │  - ― x sin 2x │  - ― cos 2x │
  4    │0   4          │0   8        │0

  1
= ― π² = 2.4674…
  4
</pre>


|-
| PhD {{w|Cosmology}} || Write down the Hubble constant to within 1% || The Hubble constant is a component of {{w|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 {{w|Hubble's law#Hubble tension|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 {{w|significant digits}} are interpreted, 69 (km/s)/Mpc could be correct to within 1% while 70 is not.
|-
| {{w|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 {{w|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 {{w|Monte-Carlo method}}s, 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 "{{w|infinity}}", most mathematicians do not consider infinity to be a number,{{acn}} 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 [https://googology.fandom.com/wiki/Googology#History Googology] or [https://sites.google.com/view/transcology/lists/numbers Transcology] may give an answer such as {{w|Rayo's number}}, which was the winning entry in the [https://googology.fandom.com/wiki/Big_Number_Duel 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 {{w|NaN|Not a Number}}, and their answer might instead be the ceiling of some binary representation (typically ''2<sup>8n</sup>-1'' for some value of ''n''), the largest unsigned value reliably storable in a given byte form for an integer (e.g. a {{w|Integer (computer science)|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.
|-
| {{w|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. {{w|37_(number)|37}} is a number that some people believe [http://thirty-seven.org mysteriously appears more often than it should]; this was a subject of a [https://www.youtube.com/watch?v=d6iQrh2TK98 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 {{w|List of Discworld characters#Victor Tugelbend|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==

:[6 different math test questions are arranged in a three by two grid. All subject headers include the words "Final exam" underlined, or given a similarly wide horizontal rule before the question itself.]

:[The first panel:]
:Kindergarten math
:<u>Final exam</u>
:Q. Write down the biggest number you can think of
:A. [Empty box.]

:[The second panel:]
:Pre-algebra
:<u>Final exam</u>
:Q. Write down the value of x if x=3x-8
:A. [Empty box.]

:[The third panel:]
:Calculus
:<u>Final exam</u>
: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
:<u>Final exam</u>
:Q. Write down the Hubble constant to within 1%
:A. [Empty box.]

:[The fifth panel:]
:Game theory
:<u>Final exam</u>
:Q. Write down 10 more than the average of the class's answers
:A. [Empty box.]

:[The sixth panel:]
:Postgraduate math
:<u>Final exam</u>
:Q. Write down the biggest number you can think of
:A. [Empty box.]

{{comic discussion}}

[[Category:Math]]
[[Category:Cosmology]]
[[Category:Comics with lowercase text]]