32

I posted this question earlier today on the Mathematics site (https://math.stackexchange.com/q/3988907/96384), but was advised it would be better here.

I had a heated argument with someone online who claimed to be a school mathematics teacher of many years standing. The question which spurred this discussion was something along the lines of:

"A horseman was travelling from (location A) along a path through a forest to (location B) during the American War of Independence. The journey was of 22 miles. How far was it in kilometres?"

To my mind, the answer is trivially obtained by multiplying 22 by 1.6 to get 35.2 km, which can be rounded appropriately to 35 km.

I was roundly scolded by this ancient mathematics teacher for a) not using the official conversion factor of 1.60934 km per mile and b) not reporting the correct value as 35.405598 km.

Now I have serious difficulties with this analysis. My argument is: this is a man riding on horseback through a forest in a pre-industrial age. It would be impractical and impossible to measure such a distance to any greater precision than (at best) to the nearest 20 metres or so, even in this day and age. Yet the answer demanded was accurate to the nearest millimetre.

But when I argued this, I was told that it was not my business to round the numbers. I was to perform the conversion task given the numbers I was quoted, and report the result for the person asking the question to decide how accurately the numbers are to be interpreted.

Is that the way of things in school? As a trained engineer, my attitude is that it is part of the purview of anybody studying mathematics to be able to estimate and report appropriate limits of accuracy, otherwise you get laughably ridiculous results like this one.

I confess I have never had a good relationship with teachers, apart from my A-level physics teacher whom I adored, so I expect I will be given a hard time over my inability to understand the basics of what I have failed to learn during the course of the above.

Torsten Schoeneberg
  • 843
  • 8
  • 13
Prime Mover
  • 925
  • 5
  • 18
  • 17
    This isn't substantial enough to leave as an answer, but basically you're correct for the exact reasons you stated. And although nobody likes to be scolded by a teacher, especially when the teacher is wrong, that rarely (never?) happens here as long as questions are in good faith like yours is. I'm sorry about your previous encounters with teachers; maybe we can make up for them a bit here. – Thierry Jan 17 '21 at 18:25
  • 14
    I've seen numerous instances of this teacher's error, converting to a new unit of measure and producing an absurd level of accuracy. I was taught at a very young age that normal human body temperature is 98.6 degrees Fahrenheit. More recently, 100.4 degrees is the official border for suspicion of covid19. These temperatures happen to result from converting Celsius temperatures of 37 and 38 degrees, respectively. (See also my answer at https://matheducators.stackexchange.com/questions/1572/ for a similar example.) – Andreas Blass Jan 17 '21 at 18:53
  • 8
    ancient mathematics teacher --- Apparently not too ancient, as this would have been ridiculously silly until calculators arrived (in my school this was 1975, when two or three students had one). I looked at several books I have from within 20 years of this, and most didn't even have English-metric conversions. Two that did, Dolciani's Modern Algebra. Structure and Method. Book 1 (1973 edition) and Lankford/Clark's Basic Ideas of Mathematics (1953), only gave the approximations 1 km = 0.6 mile (Dolciani, p. 577) and 1 km = 5/8 mile (L/K, p. 497) – Dave L Renfro Jan 17 '21 at 21:05
  • 1
    @DaveLRenfro He told me how long he had been teaching. Thick end of half a century, if he was actually telling the truth. Outraged that an ignorant child (er, me) should have the insolence to challenge his wisdom. – Prime Mover Jan 17 '21 at 21:47
  • 8
    Related: Brian Kernighan (of K&R C book) gave a guest lecture to Harvard's CS 50 course a decade back that was almost entirely making fun of innumeracy errors, including this over-precision class of misunderstanding: https://youtu.be/kw9KwjJCJH8?t=2170 – Daniel R. Collins Jan 18 '21 at 02:49
  • 11
    Because of the coastline paradox, it's not really clear that that level of precision is even meaningful. https://en.wikipedia.org/wiki/Coastline_paradox – Ryan_L Jan 18 '21 at 03:26
  • 2
    @Ryan_L Yes indeed, that also was in my mind, along the lines "exactly what route along the non-zero-width very-probably-not-perfectly-straight path are we measuring?" but I decided that the question was long enough as it was, which is why I didn't mention it explicitly here. I did discuss that point in the original discussion, but I was still told I was being presumptuous and uppity to think I had the right to make such unmentioned and undocumented assumptions. – Prime Mover Jan 18 '21 at 06:02
  • 1
    @AndreasBlass huh, strange. Here, the normal human body temp is defined as 36.6 (in C) and with 37 you might be having a slight fever. – Dan M. Jan 18 '21 at 15:03
  • @DanM. When I was young, it was $98.4 , ^\circ \mathrm F$ (which works out as about $36.9 , ^\circ \mathrm C$) and over $99$ gave me a day off school, while over $100$ gave me a day in bed and a doctor's visit. – Prime Mover Jan 18 '21 at 15:46
  • @PrimeMover it looks like it generally depends on the most common way to measure the body temperature. For axillary (under the arm) it's about 36.6, while oral is a bit higher (and ear is a bit higher as well). Interesting. Although it's not that important since it's normal for a completely healthy person to exhibit some degree of variance in a body temperature. – Dan M. Jan 18 '21 at 15:57
  • Just for fun, you can also just turn it around, and argue that it isn't 1.60934km per mile. It's actually 1.609344km per mile. So THERE! – Cort Ammon Jan 19 '21 at 00:21
  • 3
    Ask the guy to state the circumference of a 22-inch diameter bike wheel. Any answer with fewer than 31,415,926,535,897 digits is not sufficiently accurate. (Note, the guy who set this record for the number of known digits in pi clearly had a sense of humor when choosing how many digits to calculate...) – alephzero Jan 19 '21 at 00:44
  • It is perfectly reasonable to quote the body temperature example to 0.1C, since modern digital clinical thermometers read to that level of accuracy. Of course the choice of a "normal range" of body temperature is always going to be arbitrary to some extent. – alephzero Jan 19 '21 at 00:47
  • 4
    Actually, this is a trick question. The correct answer relies on understanding the difference between the current metric equivalent of a mile based on 1 inch =25.4 mm exactly, and the USA "survey mile" based on 1 meter = 39.37 inches exactly. The two definitions of "a mile" will give different answers, to the number of digits quoted in the so-called answer. The date when the ride took place is therefore relevant input to the calculation!! – alephzero Jan 19 '21 at 00:55
  • Is there any chance the original question was intended as a trick question, since kilometers had not even been defined at the time of the American war of independence? – Mark Foskey Jan 19 '21 at 04:55
  • 1
    @MarkFoskey Irrelevant. Let's say you were writing a history book. "It was a ride of 22 miles (that's 35 km) through a dense wood with only a hurricane lantern to guide the way ..." Doesn't matter that km had not been invented then. If you want to communicate the information to a modern readership you need to understand the basics of unit conversion -- and reporting to an appropriate level of accuracy is part and parcel of that exercise. – Prime Mover Jan 19 '21 at 06:29
  • "A particularly grumpy, ancient mathematician would take 35.405598 km to transverse the forest. A fun-loving, youthful geographer would say that it took about 35 km... – OnoSendai Jan 19 '21 at 07:51
  • Hmmm "not using the official conversion factor of 1.60934 km per mile" is slightly in error as 5280122.54/100/1000 is 1.609344. If one wants to use a precise conversion factor, why stop 1 digit short of the exact one? – chux - Reinstate Monica Jan 19 '21 at 12:48
  • @chux-ReinstateMonica I may have misquoted slightly. – Prime Mover Jan 19 '21 at 14:18
  • "I may have misquoted slightly." --> Interesting - yet consistent with the larger issue. – chux - Reinstate Monica Jan 19 '21 at 14:31
  • @Ryan_L (and Prime Mover) 2 SF would seem reasonable for a quoted distance in the general case; a surveyor could do better of course even before modern times. Of course there will be some error, but saying 20 Mi includes 15--25 and you should be able to estimate better than that. Even with GPS 3 SF is pushing it, comparing some 67km commutes from my Strava. – Chris H Jan 19 '21 at 15:50
  • 1
    Actually, the correct answer to this question would depend on what class it was for, and what the level of instruction for it had been. – RBarryYoung Jan 19 '21 at 18:51
  • 1
    @DanielR.Collins: Excellent presentation, thanks a lot for the link. I'll feel less alone next time I tell soon-to-be-engineers that they really should check the output of their simulation, and they don't need any computer at all to realize that those results must be wrong by a factor of at least 1000. – Eric Duminil Jan 19 '21 at 22:11
  • Cross-posted: https://math.stackexchange.com/q/3988907/14578, https://matheducators.stackexchange.com/q/19373/862. Please do not post the same question on multiple sites. – D.W. Jan 20 '21 at 00:42
  • Prime Mover - sorry, my comment was a little silly. You are totally correct. – Mark Foskey Jan 20 '21 at 02:31
  • @D.W. User Prime Mover clearly indicated in this question that he had originally asked it on another site, and then followed the suggestion there to ask it here. The only thing they could have done better is to give an explicit link. – Torsten Schoeneberg Jan 20 '21 at 03:30
  • After your sixth paragraph I thought to myself 'this guy is an engineer'. Two paragraphs later my suspicion was confirmed. (I'm an engineer ) – Lamar Latrell Jan 20 '21 at 04:45
  • 1
    @LamarLatrell: as far as I can tell, you have no reason to assume the opposite, though : that "this engineer is a guy". ;) – Eric Duminil Jan 20 '21 at 09:37
  • "not reporting the correct value as 35.405598 km." look's like another "may have misquoted slightly." as that is neither 221.60934 (approximate conversion) nor the 221.609344 (correct conversion). – chux - Reinstate Monica Jan 20 '21 at 14:48
  • @chux-ReinstateMonica Does it matter? – Prime Mover Jan 20 '21 at 16:42
  • Dang, yes, guilty. In the same breath I guess I should have rolled with metaphorical gender and referred to Prime Mover as she? – Lamar Latrell Jan 20 '21 at 18:17
  • Yes. In a post about how precise one should be, best to present the data factual. "not using the official conversion factor of 1.60934" implies the official was sloppy too (@CortAmmon). "not reporting the correct value as 35.405598 km" instead of the expected 35.405568 implies the official again performed sloppy math - or was it reported wrong here? Perhaps this is no worse than spelling/grammar errors? I see it as an indicator of the veracity of the presentation. – chux - Reinstate Monica Jan 20 '21 at 18:44
  • @LamarLatrell: "OP is an engineer" could be an easy solution. – Eric Duminil Jan 20 '21 at 22:40
  • Even trained, let alone experienced engineers are not maths teachers, let alone students at the level the example suggests.

    I think every good teacher would first give the accurate conversion and its explanation, and then point out how every-day practicality often trumps pure accuracy.

    What's clearly unhelpful to the poor student, is using either method without explaining the other.

     – Robbie Goodwin Jan 21 '21 at 00:35
  • @DanielR.Collins For what it's worth, his estimation of the number of golf balls is one order of magnitude too high. His approximations are okay when applied separately. But unfortunately, they all overestimate. The diameter of a golf ball is 1.65in, and not just 1in, so cubed, it's already a factor 4.5. Then 12³=1728 is rounded up to 2000, and 8 is rounded up to 10, twice. Still, that was a really entertaining and interesting lecture. – Eric Duminil Jan 21 '21 at 15:19

11 Answers11

43

You're right. The random, anonymous person you met online is not competent. This is basic mathematical literacy, as taught in every freshman chemistry and physics class.

  • 24
    Your answer may have shown the source of the problem. I was taught the principle "don't produce significant digits out of thin air" in a physics class, but never in a mathematics class. If that experience is widespread, people may have picked up the idea that this principle is limited to applications classes, whereas in mathematics classes we should "be precise" even if the precision is garbage. – Andreas Blass Jan 18 '21 at 16:31
  • 7
    +1 for the abruptness of the answer – David Z Jan 18 '21 at 19:34
  • 1
    @AndreasBlass's point could extend to thinking about spurious precision in the input – Chris H Jan 19 '21 at 15:52
  • 3
    Outside of SE and some other isolated pockets, I'd suggest the first sentence of this answer is near universal. – Lamar Latrell Jan 20 '21 at 04:51
  • 1
    @AndreasBlass: Yes, I'm sure you're right. I think the underlying problem is that to get this kind of thing right, you need strong number sense, which by far the majority of K-6 teachers lack. The following is a question that the majority even of my college-level science majors can't do without extensive help: If a real number is rounded to the nearest integer, what is the maximum rounding error? You'll also see this lack of number sense in the desire to teach and learn heuristics as if they were absolute rules, or the belief that such things are class rules set by each teacher. –  Jan 20 '21 at 14:43
  • 1
    @LamarLatrell And if that was the extent of it it would be ok--just listen to your teachers and ignore most online content. Unfortunately, incompetency extends beyond anonymous internet posts, and unless you understand math it's extremely difficult to tell if someone else doesn't. K-6 teachers get a lot of (mostly valid) criticism here, but let's not forget the higher grades. It's pretty common, in the US at least, to not have a single math teacher who understands math until college, and the fact that this conversation happened online is basically irrelevant. – Thierry Jan 20 '21 at 18:17
  • @BenCrowell: Is it "0.5", or was it a trick question? Or did you mean relative error? – Eric Duminil Jan 20 '21 at 22:41
  • 1
    @EricDuminil: It's 0.5. –  Jan 21 '21 at 15:19
  • 1
    @LamarLatrell I was briefly confused till I realized you were referring to the second sentence of the answer! – Kevin Carlson Jan 21 '21 at 22:29
  • @KevinArlin, Yeah, I saw a comma instead of a period... – Lamar Latrell Jan 21 '21 at 23:45
40

The product of two numbers should be given with as many significant digits as the least precise of the numbers multiplied (see https://www.nku.edu/~intsci/sci110/worksheets/rules_for_significant_figures.html). 1.60934 km/mile has six significant digits (or, if a mile is defined to be an exact number of km, then the conversion factor has an infinite number of significant digits). 22 miles has two significant figures. We take the smaller of these two, which is two significant figures from the 22 miles. This means that rounding to 35 km is correct.

It is a good idea to use, during one's work, at least one significant digit more than the final quantity needed, so it would have been good practice to use the conversion factor of 1.61 if this were a test, but for a casual online conversation, 1.6 is fine.

The importance of getting significant figures correctly pales in comparison of basic decency. Even if this person had been correct, scolding you would not be. If you believe that someone is in error, you should express that view politely. It appears that this person's civility may have atrophied from having a captive audience with such a power differential that they have been able to dispense with basic politeness.

Acccumulation
  • 1,440
  • 7
  • 6
  • 12
    What dismays me most is the thought of those generations of students whose experience of mathematics will have been irretrievably compromised. "Follow these rules! Don't argue! You are not allowed the privilege of even expecting it to make any sense!" – Prime Mover Jan 18 '21 at 07:19
  • 2
    The one struggle I still have with significant digits - convert 20 miles. Use 1.6, and get 32. Will you choose to keep significant digits to 1, or treat the zero of 20 as significant? (I realize, scientific notation takes care of this, indicating whether the zero was significant or not.) – JTP - Apologise to Monica Jan 18 '21 at 12:59
  • 7
    @JTP-ApologisetoMonica This is a problem of all numbers whose SF are fewer than the number of figures on the left of the DP. Are the zeroes actually significant figures or are they merely placeholders? In such a situation it is up to the person communicating the number to specify to how many SF "20" is reported to -- either 1 or 2. This issue crops up over and over again. – Prime Mover Jan 18 '21 at 13:26
  • 1
    @JTP-ApologisetoMonica You would have to take the context into consideration, e.g. if there are other, similar, measurements to 2 S.F. in the same document, use that. Or you could write "about 32 km" to indicate that the amount is not necessarily as accurate as written. – Andrew Morton Jan 18 '21 at 13:26
  • 2
    If they wanted an answer of 35.405598 km they would have had to specify the rider rode 22.000000 miles. – Schwern Jan 20 '21 at 23:10
28

Here's a joke I like to tell when people could use a reminder about precision vs accuracy:

A tour guide at Giza was explaining how the Pyramids were 4507 years old. Someone in the crowd asked: "That's oddly specific. How do we know this?"

"Well. I was told they were 4500 years old when I started working here 7 years ago."

I'm not sure the grumpy teacher you mentioned would be amused, though.

Eric Duminil
  • 411
  • 3
  • 8
  • 3
    Heh! I had that in mind when I was pondering on this while driving around earlier: "This hominid skull is one hundred thousand and seventeen years old. And five months and two days." – Prime Mover Jan 18 '21 at 23:31
  • 3
    What a great joke! Good one! – Fattie Jan 19 '21 at 18:07
17

Just to play the devil's teacher's advocate here: one can make a point that rounding should be generally avoided but measurement uncertainty instead be expressed explicitly. Specifically, rounding errors should always be much smaller than measurement errors. Now, if you have a figure of 22 miles, I'd interpret this as $(22\pm0.5)\mathrm{mi} = (35.4\pm0.8)\mathrm{km}$. I specified one more digit, but not only did I represent the center value better (which in your rounding adds a whopping 50% error), I also captured that the inaccuracy of that result is even bigger than simply $35\:\mathrm{km}$ would suggest. In particular, $36\:\mathrm{km}$ is also within the range!

How many digits to write out is then uncritical; in physics convention is to write two non-significant digits in both the value and uncertainty figure. One is usually enough, but when completely omitting non-significant digits you do introduce excessive extra error. If the numbers are just stored in a computer, you should typically keep all the digits of the binary number representation – with double precision that means you keep a rather absurd 16 decimals! It doesn't really increase the precision, but it also doesn't really cost anything or suggest too high precision (because uncertainty is stored separately), and it makes sure that rounding really will have no contribution to the error of the final result.

leftaroundabout
  • 693
  • 4
  • 9
  • 3
    Where does this convention of two non-significant digits come from? I'm not sure I've ever heard of it, so I would certainly disagree with saying that it's the convention in physics. – David Z Jan 18 '21 at 19:39
  • 2
    @DavidZ it's not a universal convention, but it does seem to be used by most big-scale experimental physics projects nowadays. It is also the form in which NIST lists physical constants, e.g. the vacuum impedance is $376.730 313 668(57): \Omega$. – leftaroundabout Jan 18 '21 at 20:19
  • 2
    Comments are not for extended discussion; I've moved the conversation that was attached to this answer to chat. – Chris Cunningham Jan 19 '21 at 20:16
  • 1
    Or nitpick if "International Mile" or "US Statue Miles" was used? those have different conversion factors and for civil war era the quoted 1.60934 would be wrong if survey miles are used, those would be rounded to 1.60935. So the whole thing is kind of pointless if you have no agreement on measurment precision and time. – schlenk Jan 20 '21 at 00:07
  • @schlenk or much worse, a nautical mile (1852m). Depending on the context, people might simply omit "nautical". – Eric Duminil Jan 20 '21 at 08:20
  • How did you come up with the standard deviation of 0.5 for the distance measurement? I agree with most of your answer, but even this seems to suggest the result contains more information than we actually have. – M. Stern Jan 20 '21 at 12:03
  • 1
    @M.Stern yeah...it's a bit over-pessimistic, and not so much a standard deviation as a worst-case bound. Arguably, $\pm\sqrt{\tfrac1{12}}$ would be a better figure ($\approx 0.29$: standard deviation of the rectangular distribution from 21.5 to 22.5). – leftaroundabout Jan 20 '21 at 12:41
  • 2
    @EricDuminil Actually there is somewhat infamous contract during colonization of Africa, where german negotiators wrote miles in a contract and the local chieftain only knew british miles, but the negotiators insisted they meant german miles (7532,5 meters). Quite a significant difference and led to the Hereo and Nama genocide later. So mind your miles or people might die. – schlenk Jan 20 '21 at 22:17
  • 1
    @schlenk for reference, https://en.wikipedia.org/wiki/Herero_and_Namaqua_genocide — I think claiming that mile-disagreement “led to” the genocide is rather trivialising it. In fact it was mostly the Germans being imperialist pre-nazis, who would have used any excuse to slaughter the locals. – leftaroundabout Jan 21 '21 at 08:25
15

When a tutoring student asks me about rounding, I tell them that absent specific instructions from a teacher, common sense should apply.

For a conversion, 22 miles isn’t 22.0000 miles, there’s the assumption it’s been rounded. You can’t convert and find yourself with 6 digits of accuracy beyond the decimal. As you note, there’s a number of digits that result to be the nearest meter, millimeter, etc. which is absurd. Before GPS, I’d give directions accurate to 1/10 mile, as that’s what a car odometer reflects. Even that was often called a bit obsessive.

My home scale gives me my weight to .1 lbs. Would it really be of value to have an extra digit of accuracy?

A person’s height? The nearest inch will do.

The one thing I warn about - don’t round while doing interim steps. This is a sure way to find that the final result may be off by enough to be graded as wrong. This issue commonly presents itself with trig functions which ask for a triangle side to the nearest 1/100. Rounding should be done as the final step.

JTP - Apologise to Monica
  • 9,770
  • 43
  • 70
  • There is more than an assumption that the figure has been rounded. There is a context into which the question has been placed. I think I understand what the main problem here is now: people seem to assume that the context is there merely to provide a pretty little story to keep the unmotivated students on board. No, the context is there to provide a scenario to be analysed. – Prime Mover Jan 20 '21 at 09:25
  • 1
    Didn't my examples offer that context? We can go off on multiple tangents here, from the observation that not all math problems offer the context required, to the fact that ultimately, perhaps unfortunately, I often ask a (tutoring) student "Do you want to be right, or do you want the credit for your answer?" You are not going to change that person. But you do have 10 answers here that are in your favor. I don't see one that sides with that teacher. – JTP - Apologise to Monica Jan 20 '21 at 11:23
  • I was not arguing against you, but I was specifically elaborating your statement "there's the assumption it's been rounded." Oh, and I'm not sure where I saw it now, but there are some answers here which appear to suggest that what you do in a maths class (i.e. calculate the numbers exactly) is different from what you do in a physics etc. class, because "maths is pure" or some such. And someone did say that the context in a word problem is just there to make it interesting and engage the students. Utter piffle, of course. – Prime Mover Jan 20 '21 at 12:06
  • I agree, 100%. Good talk. – JTP - Apologise to Monica Jan 20 '21 at 12:12
9

When I was in school, I once got an answer marked as error for having too many digits. IIRC it was in trigonometry and I had just written down as many digits as the calculator displayed. (I was able to discuss it away, but was told to avoid unreasonable amounts of digits in the future)

That was in the 1990s in continental Europe, but I think it is still good enough for s counterexample: Not all teachers are like that.

Jan
  • 191
  • 2
2

IF the horse ride were 22.00000000 miles then the other person would be right.

Else if it were 22 miles then you should round the answer to zero decimal places.

Some people are illogically pedantic without any rational reason for what they promulgate.

practical man
  • 21
  • 1
  • 5
    It's not decimal places you should have in mind, but significant figures, surely? – Prime Mover Jan 18 '21 at 17:37
  • I presumed that the numbers given were to sig figs limit – practical man Jan 20 '21 at 03:56
  • @practicalman You would still round to sig digs though, not decimal places. Your number has 10 sig digs total, two left and 8 right of the radix. Your wording implies that your result should always have 8 digits right of the radix. For example: 22.00000000 miles = 35200.00000000 meters, when it should be 35200.00000 meters (assuming 1 mile = 1600 meters with infinite precision for simplicity). – DKNguyen Jan 20 '21 at 14:04
2

You're both right, depending on the domain of discourse and the rules of engagement.

In pure math, the traditional expectation is that the numbers given are exact unless stated otherwise, and answers are also to be exact unless stated otherwise. So when the mathematician read "22 miles," he's using a tradition that means "exactly 22 miles."

But in the physical sciences, all measurements are understood to be inexact and approximations and rounding are either "allowed" or "expected" (depending on the logical rigor applied).

In this case, the correct answer boils down to a question of semantics and assumptions. What about this question:

If a man traveled 22 miles, how far did he travel in kilometers?

How would you answer that? The "If" complicates things. Some would say that it turns the question into a hypothetical that ignores the physical difficulties in measuring exactly 22 miles and turns it into a "given." It's not a stretch to read the original question as a hypothetical, even without an explicit "If" at the beginning.

Some traditions say that integers are always expected to be exact and that the question should have used "approximately 22 miles," "22.0," or a bar on the last significant digit to show it's a real number instead of an integer.

Even in the physical sciences, scenarios used for pedagogical purposes are sometimes idealized in order to remove confounding factors that might distract from the main point being taught. I don't think we know enough about the source of this question to know about what assumptions or simplifications are being made.

You may argue that the use of "a man riding on horseback through a forest in a pre-industrial age" implies a real situation and an actual, inexact measurement. A counterargument is that the use of abstract identifiers "A" and "B" to to designate the starting and stopping point suggest an idealized situation.

I would agree this is a good question for Mathematics Educators Stack Exchange. It emphasizes that in the classroom (as well as life) it's important to be explicit about assumptions and expectations and to lay out the ground rules.

Adding a summary, based on comments, that tries to be more direct:

  1. Use of significant digits only applies to inexact numbers such as measurements.
  2. In a problem like "Convert 22 miles to kilometers," there is no reason to think 22 miles is a measurement. Rather, it is a "given": Something that is to be assumed or taken for granted for the sake of the problem.
  3. I think this question boils down to this: In the original question, is "22 miles" to be taken as a given or a measurement?

I don't think we can tell. (At least not without more context about where the question came from and why it was asked.) The original question could merely be "Convert 22 miles to kilometers," dressed up in a story to make it engaging or interesting.

My reading of some of the comments suggests a point of view of "If the problem resembles a real-world situation, then it must be interpreted as a real-world situation." Or more succinctly: If 22 miles could be interpreted as a measurement, then it must be interpreted as a measurement. Or that by phrasing the question in a historical, real-world context, that somehow forces the measurement interpretation. I don't follow that. It ignores the way real-world people write, talk, and teach.

Syntax Junkie
  • 1,376
  • 9
  • 10
  • 3
    The subtext here, if mathematics and the sciences are taught this way is: "This is mathematics, here within this walled garden. The other side of the wall is physics and chemistry and messy stuff like that, and we don't have anything in common with them. We use numbers precisely, and those smelly hairy apes over there use (shudder) approximations." Oh, and the reason for using "A" and "B" is because I could not remember the actual place names. It is implicit that this is a real-world scenario being modelled mathematically ... – Prime Mover Jan 18 '21 at 23:34
  • 2
    ... and so the expectation is that the student reads all of the question and puts the entire situation into context. You cannot honestly say: "We're mathematicians and so we scoff at the real world because we work with ideals." You are given the real world situation and it is an important part of mathematics to be able to translate accurately and appropriate the full context of a "word problem" into the correct mathematical model. TL;DR: This is not a pure mathematics problem. It is at base an a exercise in mathematical modelling. – Prime Mover Jan 18 '21 at 23:38
  • 3
    It seems like a very odd choice to treat real-world, continuous measurements as integers, since it effectively implies infinite precision of the measurement. Integers seem appropriate for counting problems (i.e. there are exactly 22 apples in the basket), but not at all for this type of measurement problem (it's not even possible to measure something as exactly 22 miles away with infinite precision). – Nuclear Hoagie Jan 19 '21 at 14:31
  • 3
    "It ignores the way real-world people write, talk, and teach." Does it? This lack of explanation and succinctness of this statement implies that you think it is an given. It is very unclear to me that this is actually the case. You have some explaining to do because if anything, the way people write, talk, and teach goes completely in the opposite direction in that the assumption is lack of precision in the absence of explicit information, rather than infinite precision. – DKNguyen Jan 19 '21 at 20:34
  • 2
    @DKNguyen. I regret that I gave that impression. I don't think we have enough information to know what interpretation works best. If my post seems to advocate for the "given" interpretation, I think it's because most of the posts that I read seemed to imply felt it was "obviously" a measurement (my phrasing, not theirs.) So I did want to make a case that were was an alternate interpretation. I appreciate your feedback. – Syntax Junkie Jan 19 '21 at 21:00
  • @PrimeMover I don't think I am the one you are intending to reply to. – DKNguyen Jan 20 '21 at 06:10
  • @DKNguyen You are correct -- I apologise. Just that there is much muddy thinking about exactly what the problem here is. I intend to write a thesis on exactly this topic in the not-too-distant future so as to try to put across all the subtleties and nuances here. Many, many comments and replies only go just so far in analysing this question. (It's more than just a question in a test, designed so as to tick the right boxes, it's a subtle teaching experience.) – Prime Mover Jan 20 '21 at 06:16
  • 2
    @SyntaxJunkie But we do have "enough information to know what interpretation works best". We are told that this is a journey on horseback through a forest in a pre-industrial era. Hence there is no reason to expect "22 miles" to be an exact number, and there is no point in specifying the resulting distance to any greater accuracy than "35 km". The domain of discourse is stated. In a good "word question", the words are not just there to make the page look pretty and to give the airheaded students a fun story to read to stop them getting bored ... – Prime Mover Jan 20 '21 at 09:12
  • @SyntaxJunkie ... the words are there to specify the context. From that context, the student is then (or at least, should then be) expected to interpret that context. At the very least, it is up to the teacher to explain how to interpret that context, and to ask specifically directed questions as to how we may estimate the accuracy and precision of the figure given, and hence to provide a realistic answer based on that. And finally: yes, the 22 miles IS "obviously" a (possibly invented) measurement. What on earth else can it possibly be? – Prime Mover Jan 20 '21 at 09:16
  • @Prime Mover. I can think of at least 3 interpretations of the original question. (1) A real trip made by a real person and a real measurement. (In that case, I agree the only reasonable approach is rounding according to significant digits.) (2) It's a story made up for didactic purposes where 22 is meant to be interpreted as an exact number (such as a "word problem" in a geometry or basic algebra class). (3) It's a didactic story where 22 is meant to be interpreted as a inexact number (such as a surveying or maybe law class). How do you know which of these interpretations to use? – Syntax Junkie Jan 20 '21 at 17:03
  • @SyntaxJunkie I am assuming that the question was written by an intelligent and deep-thinking educator who wished to explore the students' understanding of the importance of uncertainty measurement. Hence 2 is out of the question because as an exercise it is a pointless exercise in pressing buttons on a calculator. 1 is possible, quoting an episode in history, perhaps. But probably 3, the question having been specifically crafted to test understanding. However, note that if it is 1, we have to ponder how the measurement was done -- hence how accurate it is likely to be. – Prime Mover Jan 20 '21 at 17:21
  • @PrimeMover It depends on the experience level that it's targeted at, right? If the students are inexperienced enough, it might be a moderately challenging exercise in pushing buttons on a calculator. – user3067860 Jan 20 '21 at 17:37
  • 1
    @user3067860: Yes. Some students have trouble differentiating relevant from irrelevant information when presented in text instead of a "formula to solve." That's one reason knowing which buttons to push on a calculator is not a pointless exercise. Another reason: I have had college-level students who weren't always sure whether to multiply or divide by a conversion factor. I disagree that interpretation 2 is "out of the question." – Syntax Junkie Jan 20 '21 at 17:52
  • @user3067860 In which case you'll be teaching a class called "calculator awareness" or some such. Or you're on the other side of the pond, I suppose. – Prime Mover Jan 20 '21 at 20:31
  • @PrimeMover No need to be rude. The point stands that it depends on the context of what you're teaching. In a university-level physics class you would get points off for not showing the full calculation of the relative error and using that to choose an appropriate precision for the conversion. In fifth grade, the teacher may be happy that you managed to do the arithmetic correctly. If it's a community/life skills class for adults who don't speak English and only have a 3rd grade education in their home country then "multiply by 1.5 so it's about 33 km (to get to the store)" might be progress. – user3067860 Jan 20 '21 at 21:10
  • @user3067860. You and I may have slightly different interpretations which I only ask about because I enjoy the nuance. I apologize to you and Prime Mover if I'm misinterpreting. I take your last comment to say that sometimes significant digits are ignored for pedagogical purposes. I agree with that. I'm interpreting Prime Moving as saying, "To be technically correct, significant digits must always be considered. And I agree with that. My point is that in some contexts, the source data is, by convention, assumed to be exact, so significant digits don't ultimately change the answer. – Syntax Junkie Jan 20 '21 at 21:57
  • @SyntaxJunkie I think the reason why we have a convention of treating the numbers given in word problems in a math class as being exact is because it's convenient to ignore sigfigs for pedagogical purposes, though. "Lies to children". https://en.wikipedia.org/wiki/Lie-to-children – user3067860 Jan 20 '21 at 22:32
  • -1 and @NuclearHoagie has the central problem with this answer: just because you see an integer doesn't mean anyone should assume it's infinitely precise. E.g., the first web search I did comes up here; it carefully distinguishes between measured numbers and counting numbers ("Counting Numbers are Infinitely Precise"). If the number has a continuous unit of measurement -- like miles -- then, bam, significant digits are an issue. That's an easy call and universal any place I've seen. More importantly, it just makes sense. – Daniel R. Collins Jan 21 '21 at 01:30
2

It depends

I agree with just about everyone that the answer is 35, or perhaps 35.4 (a number I like better, see below). An answer of 35.405598 km is precise to the millimeter. I've ridden horses; they don't work in millimeters.

Update: For what it's worth, after all this discussion, I think that the right number is "about 35 and a half" (not 35.5) kilometers. Thirty five and half has about the same uncertainty as "22 miles" (maybe even more), and is within "horseshoes and hand-grenades" of the exact answer of "just about 35.4 exactly".

As you acknowledge, the intermediate answer you came up with (using an approximate conversion factor) of 35.2 km is wrong; 35 km is a correct answer, but 35.2 km is just plain wrong. It makes sense to consider that a distance of "22 miles" is likely more precise than "something between 21.5 and 22.5" which is what considering 22 as having only two significant figures means. It's more like 22.0 miles (i.e., between 21.95 and 22.05 (which gives you an uncertainty of about 500 feet (about 160 m)).

But, when you multiply 22.0 by 1.6, then your answer should definitely only have 2 significant figures (not because of the 22, but because of the 1.6). You can tell that your 3 significant figure result is off, the "completely precise" number is off by 0.2 km (200 m) from your figure. Horses are more accurate than hundreds of meters.

What you want to do working with numbers is to get an understanding of both the precision and the accuracy of the measurement. Saying something is about 22 miles, give or take 500 feet makes 22.0 about the right number to use.

When doing a conversion, it's always best to use the most precise number you have for all intermediate work, and only round back to the correct number of significant figures at the end of the calculation. When doing distance calculations, I always use the fact that one inch is exactly 2.54 centimeters (i.e. 2.54000000000, as many zeros as you want). If I've got a calculator (or a slide-rule) handy, I'd do this:

22 miles * 5280 ft/mile * 12 in/ft * 2.54 cm/in / 100 cm/m / 1000 m/km
= 35.405568 km

Note that that number is off by 30 millimeters from what you quote. My number is correct. Also note that I carried the units through the calculation. That way, I can do some dimensional analysis and see that I get an answer in km, and that it's what I expect: (miles * (ft/mile) * (in/ft) * (cm/in) / (cm/m) / (m/km) works out to km).

I'd look at that number and say "yes, it's 35.4 km." Also note that all those intermediate conversion constants are exact (the number of inches in a foot is exactly 12 - so you can treat 12 like 2.54, it has as many zeros as you want).

But then again

Way back when I was a student, I had a math prof who'd get upset at us engineers for saying the answer is about 35.4km. He's say that two numbers can be equal, but "about equal" or "approximately equal" have no mathematical meaning. Then he'd point out that it would be pretty easy to figure out that one was about equal to zero - and at the point, everything breaks.

So, if you are in a math class and the teacher says "The relationship between miles and kilometers is 1.609344 km/mile, how many kilometers are there in 22 miles?", then the answer is 35.405568 km, not 35.4 km.

Note the absence of the horse in this phrasing of the question.

Flydog57
  • 595
  • 3
  • 7
  • 1
    I never gave $35.2$ as an answer. It was an interim calculation based on the necessities of the question. $22$ miles is only ever going to be a guessification. You don't need a conversion factor of anything better than $1.6$ km per mile in such circumstances. It's the same when I drive across Europe. I like to know how many miles and how many km left. I only need an approximate number. So I multiply by 8 and divide by 5 (or vice versa) and I can do it in my head. – Prime Mover Jan 20 '21 at 06:00
  • Your final paragraph illustrates exactly that incorrect thinking that my original correspondent showed. That was not the question asked. It was given a context. What is the point of asking such a silly question as the one you asked? It would just be an exercise at pressing buttons on a calculator and writing down the answer, and pupils learn absolutely nothing. – Prime Mover Jan 20 '21 at 06:03
  • And $35.4$ km is wrong. It should be $35$ km. – Prime Mover Jan 20 '21 at 06:04
  • If you're truly a math class, why is the professor using units at all? Or real world examples of things? That would be like a physics or engineering class giving a problem and not giving units for the numbers...or physical context. The only "unit" I know of that comes purely from math and is required in math are things like radians and steradians...and those aren't even really units, being dimensionless. – DKNguyen Jan 20 '21 at 06:18
  • 1
    @DKNguyen Applied mathematics uses units. And unit conversion was part and parcel of the mathematics curriculum throughout the whole length of my high school, and even earlier, although early education did not put "names" on the classes so much, it was all just "lessons". – Prime Mover Jan 20 '21 at 07:28
  • @PrimeMover I'm referring to pure math where units don't have a place. In applied math it is applied so it would necessarily have units but would also be nonsensical to argue infinite precision since it is applied and not physically abstract. – DKNguyen Jan 20 '21 at 13:54
  • @PrimeMover I was also addressing Flydog, not you. – DKNguyen Jan 20 '21 at 14:29
  • @DKNguyen: I agree with Prime Mover. Math and the real world are completely tied together at the hip (particularly math at the high school level) . I remember my prof in an introductory calculus class saying "It's not a big surprise that two Europeans independently came up with the foundations of calculus about 100 years after cannon became common on the battlefields of Europe." – Flydog57 Jan 20 '21 at 15:30
  • @Flydog57 I was commenting on "Way back when I was a student, I had a math prof who'd get upset at us engineers for saying the answer is about 35.4km. He's say that two numbers can be equal, but "about equal" or "approximately equal" have no mathematical meaning. Then he'd point out that it would be pretty easy to figure out that one was about equal to zero - and at the point, everything breaks" – DKNguyen Jan 20 '21 at 15:31
  • @PrimeMover: Sorry, I've smoothed out a sentence. My point is that when someone says 22 miles (and not "about 22 miles"), the number can be more exact than just two significant figures. This example would actually be a good learning exercise about why using sig figs is important (if you just use the 1.6 conversion factor). Getting students to think about the uncertainty in the inputs and how that should be reflected in the certainty of the output can be useful. But, if someone says "22 miles", then it's really about 2.5 significant figures (because they didn't say 22 and a half). – Flydog57 Jan 20 '21 at 15:41
  • @Flydog57 But in this context, when it says "$22$ miles" it is very probably that it isn't even $22 \pm 0.5$ miles in the first place. In fact, in this context, in this particular question, it is more than possible that it may even be $\pm 1$. They wouldn't have had great techniques for surveying a route through a forest in the first place, and more to the point, they would also probably not even have the means to estimate the uncertainty in their measurement, which may well be not much better than an estimate. – Prime Mover Jan 20 '21 at 16:49
  • 4
    @DKNguyen. Q: "If you're truly a math class, why is the professor using units at all?" a: to demonstrate that units need to be treated as algebraic quantities than can be manipulated as any other variable; (2) To demonstrate that having an answer with sensible units can help a student check their work. Q: "Or real world examples of things?" A: To test the student's ability to extract the relevant details from a body of text and convert that into mathematical symbolism. – Syntax Junkie Jan 20 '21 at 17:25
  • 1
    There are contexts (such as land measurements in the United States) where 1 inch is not exactly 2.54 cm.

    https://www.civilgeo.com/blog/when-a-foot-isnt-really-a-foot/

     – Jasper Jan 21 '21 at 01:08
  • @Jasper: Damn, I've lived here 30 years and never heard that before. Of course, being Canadian (and having gone to engineering school there), I've only dealt with Imperial inches and "International" inches before (each of which is 25.4mm). For what it's worth, your trivia is time-stamped, it's going away in less than 2 years: https://en.wikipedia.org/wiki/Inch#US_Survey_inches – Flydog57 Jan 21 '21 at 01:09
  • @DKNguyen: I could guess from your comment that you're primarily some type of engineer. From Stein/Barcellos, Calculus and Analytic Geometry, 5E, p. xxii: "At the Tulane conference... in 1986 we heard the engineers say, 'Teach the concepts. We'll take care of the applications.'... Oddly, mathematicians suggest that we emphasize applications, and the applied people suggest that we emphasize concepts. We have tried to strike a reasonable balance that gives the instructor flexibility to move in either direction." – Daniel R. Collins Jan 21 '21 at 01:42
  • @DanielR.Collins Yes, but in all my university pure math classes I do not once remember units coming up. – DKNguyen Jan 21 '21 at 01:44
  • 1
    ... It's very common, often a requirement, that math courses include applications with real-world units. See any math textbook in a course that engineers might take. Pedagogically most people find that making work concrete helps students get traction. Moreover, how else can students practice manipulating units before entering their applied/engineering courses? My dept.'s liberal-arts math course is mostly about teaching precisely that topic. – Daniel R. Collins Jan 21 '21 at 01:45
  • @DanielR.Collins I can confirm that math honours courses don't care about any of that lol – DKNguyen Jan 21 '21 at 01:46
  • @DKNguyen: Skeptical. I teach math honors courses (e.g., from Rosen, Discrete Mathematics and Its Applications). Perhaps you can cite such an undergraduate-level engineering-appropriate math textbook lacking applications? – Daniel R. Collins Jan 21 '21 at 01:47
  • @DanielR.Collins We used a set of bounded notes written by someone in the department rather than a text. – DKNguyen Jan 21 '21 at 01:48
  • 1
    @DKNguyen: So I'll take that as "no evidence available". – Daniel R. Collins Jan 21 '21 at 01:50
  • @DanielR.Collins Actually looks like the courses have notes posted https://www.ualberta.ca/mathematical-and-statistical-sciences/undergraduate-studies/courses/honors-courses.html – DKNguyen Jan 21 '21 at 01:51
  • 1
    Oh I see examples in the notes where units are used. Never had those problems assigned though. – DKNguyen Jan 21 '21 at 01:52
  • Good answer. Isn't "22 miles * 5280 ft/mile * 12 in/ft * 2.54 cm/in / 100 cm/m / 1000 m/km" ambiguous? "a/b/c/d" could be "a/b/(c/d)" or "((a/b)/c)/d", and AFAICT, your answer could be in km or in 1/km³, for example. Anyway, for units, I'd just use Qalculate (https://qalculate.github.io/) and type "22 miles to km" and get 35,405568 km as an answer. – Eric Duminil Jan 21 '21 at 13:27
  • @EricDuminil: No. Students should be taught how to do dimensional analysis. In this case it is unambiguous. Notice that I put spaces around the multiplication and division operators, but wrote the units without spaces. When I was an engineering student, I did this analysis on every problem I worked on. I had an unambiguous notation I used to make sure I didn't mix up mult/div with units (because it is easy to do). I suspect that "math educators" frown on students copying their problems into Google/Bing/whatever to get their answers – Flydog57 Jan 22 '21 at 17:42
  • Your expression is ambiguous when written in one line on a computer, even if it's clear that it's for dimensional analysis. AFAIK, spaces are not relevant for order of operations, so "a/b / c/d" is still "a/b/c/d", which is either ambiguous or, read from left to right, "((a/b)/c)/d". You could write "22 miles * 5280 ft/mile * 12 in/ft * 2.54 cm/in * m / (100cm) * km / (1000m)" to avoid ambiguity. Obviously, if you're writing it on a piece of paper, you can make it clear by using different levels of division. 48/2(9+3) is also a common problem which leads to heated debates on the Internet. – Eric Duminil Jan 22 '21 at 19:03
  • Finally, I think it's perfectly fine to use Google/Bing/Qalculate for unit conversion. But first, the user should absolutely have a pretty good idea of what should come out. So the user should answer "which physical quantity"? "Which unit?" "Which order of magnitude?" first, without any external help, and only then use the tool. It's easy to know that the result should be ~30km, so if Bing answers "35.4 / km³" or "1234567.89 km", the user would know it's wrong. – Eric Duminil Jan 22 '21 at 19:15
1

It depends on the level of the class.

I would expect someone who has a recent undergraduate degree in mathematics to have experienced significant figures at at least some point in their life, either in high school or in college. I would also expect common sense to kick in and say that the level of accuracy proposed is unreasonable.

But it is reasonable to dodge the topic when teaching arithmetic, algebra, etc., because the students usually have a hard enough time as it is. Sometimes you can arrange for numbers that come out evenly anyway, but if you're stuck trying to teach an awkward conversion (miles to km) or if the task is to teach something about decimals or fractions then you may be unable to avoid it.

For example, "Alex had five cookies and split them evenly with Blake. How many did each of them get?" Two and a half, and we aren't going to quibble over how precisely half of a cookie was achieved.

If your students are advanced enough to be working with more precise numbers (and, presumably, starting to question what level of accuracy is acceptable) then the best way to dodge it is to simply specify what rounding you want in the question: "Answer to x decimal places." That way you can specify the correct precision without the students having to understand how to calculate what the correct precision should be.

That's much simpler for the student to understand than the official way, which is according to NIST:

The precision of your conversion should be based on relative error. If error isn't specified, then you can infer it from the number of digits in the values given. Use a conversion factor with equal or more precision to that to preform the calculation. Then you round the result to produce a relative error that is of the order of the original.

$22$ miles implies an error range of plus or minus $\frac{1}{2}$ mile which is $2.\overline{27}\%$.

Using a conversion factor of $1.61$ kilometers per mile (which has better than $2.\overline{27}\%$ accuracy, note that $1.6$ is not accurate enough)... $22*1.61=35.42$ km. We could also use $1.609$ or any more precise conversion, it does not matter because we will be rounding. (For example, in this case, $22*1.609 = 35.398$ km.)

Now we round... $40$ km would have a relative error $5\div40\approx12.5\%$ which is too much, $35.4$ would have a relative error of $0.05\div35.4\approx0.141%$ which is too little. $35$ km has a relative error of $0.5\div35\approx 1.43\%$ which is just right. Note that we get the same (rounded) answer regardless of how much precision we used in the conversion factor, as long as the conversion factor met a minimum level precision.

Question: Why do we assume plus or minus one-half mile? Wouldn't a distance of 22 miles be measured more accurately than that?

Answer: No. If anything it is likely to be much worse. (Disclaimer, I'm not doing sigfigs in this section, I just can't be bothered.)

In American revolutionary war era from the New York Public Library, Thomas Jefferson measuring exactly 22 miles would have actually gone over 22.3 miles (and he was a bit obsessive about measurements):

Before he left on the trip, Jefferson bought from a Philadelphia watchmaker an odometer that counted the revolution’s of his carriage’s wheel. He had measured distance based “on the belief that the wheel of the Phaeton [his carriage] made exactly 360. revoln’s in a mile.” After the trip, though, he re-measured circumference of the wheel and found that it made only 354.95 revolutions in a mile. So for every seventy-one miles Jefferson thought he traveled, he had actually traveled seventy-two.

But I use my car odometer, not a carriage! It's much more reliable! ...nope. From motus.com, if your car odometer says 22 miles then it could be anywhere between 21.12 to 22.88 miles:

Surprisingly, there is no federal law that regulates odometer accuracy. The Society of Automotive Engineers set guidelines that allow for a margin of error of plus or minus four percent.

Actually I use GPS, that's very accurate! ....nope again. GPS has a margin of error on every position measurement made, plus error from the distance between measurements. Essentially your path is like a coastline and the GPS can suffer from the coastline paradox. From singletracks.com (with pictures and a good explanation: In a very, very bad case (steep trail, lots of switchbacks) your GPS may report 22 miles when you've actually gone 44 miles! Holy guacamole.

[...]GPS reports the full loop is right at 1 mile long. In fact, everyone else who rides this trail gets roughly the same measurement. But the trail always “feels” much longer than that.

Recently I started riding with a wheel-based cyclocomputer, which I calibrated and verified during one of our track tests. Measuring this particular trail with the cyclocomputer reveals the trail is actually closers to 2 miles long, meaning our GPS units are underestimating distance by half!

I'm not going to find sources for inaccuracy of distance calculated by counting steps, the time it took to travel, etc. It's pretty obvious that no one (and no horse) actually moves at that even of a pace for 22 miles.

So accepting 22 miles on a trail as being between 21.5 and 22.5 is actually pretty generous. Better to just call it a day and say it was "some distance".

user3067860
  • 246
  • 1
  • 6
  • Exactly. However, when working with familiar integers in conversation, it's not the same as working from a specification. If someone says "22 miles", I'm assuming that the uncertainty is less than +/- 0.5 mile. That would be "about 22 miles". Perhaps my estimate of +/- 500 feet is too narrow, but I suspect that it's not that far off. The real number may be "35 and a half" (not 35.5), there's a lot of slop in that wording. Question, if someone says "10 miles", how much uncertainty is there in that number - is it 1 sig fig, or 2, or, 1.5? I believe that "22 miles" has less slop than "10 miles" – Flydog57 Jan 20 '21 at 23:56
  • @Flydog57 No. I added into my answer at the bottom, because it was much too long for a comment, but the short answer is that everyday tools for measuring distances traveled are really inaccurate. – user3067860 Jan 21 '21 at 13:12
0

I would say that based on the words of the question, the answer is 35.

It is not a distance or measurement of 22 miles between points A & B. It is a journey between locations A & B.

Next time you see the guy who scolded you, ask him how one should answer if asked, “What is the numerical value of pi minus e?”

Alan
  • 1