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I want to stress to my students that we should be careful with how we treat the equals sign and that we should always make sure that the units match. However, sometimes I write

$ 11:40 - 15 \text{ min} = 11:25 $

and I am wondering whether I should stop doing it. One problem is that I would have trouble justifying the notation to a student.

Question: Should I avoid writing it in this manner, and what are good alternatives?

Remark: From an abstract mathematical standpoint it seems like the notation really means that we have an action of the group $\mathbb{Z}\cdot \min$ acting on the set of military hours $hh:mm$.

Clarification:

  1. By $11:40$ I mean that it has passed $11$ hours and $40$ minutes since midnight, i.e. $11:40 \text{ a.m.}$
  2. One of my issues is that the operation $20 \min - 9 \text{ a.m.}$ does not seem meaningful, but $9 \text{ a.m.}-20 \min $ does.
  3. There might be some additional issues with the cases $00:15 - 45 \min = 23:30$.
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    $\begingroup$ Where I came from, colon has the same meaning as obelus, hence 11:40 means $11 \div 40 = \frac {11}{40}$. Second, you are subtracting a value with dimension from a dimensionless one. Adding or subtracting works only for values having the same dimension (same units). If "11:40" means time, then you should unambiguously provide all the necessary units, like "11 hr 40 min - 15 min", but to me this is not clean enough because the units are mixed, so I would not write an arithmetic expression like that, and instead either phrase it in words, or convert everything into minutes. $\endgroup$
    – Rusty Core
    Commented Apr 10, 2021 at 20:33
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    $\begingroup$ I think I would say that this was an exception to the rule (about same units). $\endgroup$
    – Sue VanHattum ♦
    Commented Apr 10, 2021 at 21:19

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Absolute time is an example of what is called an affine space; a mathematical structure where you can consider the difference of two points (which in this case would be duration), and that second space is linear. The affine space doesn't have a linear structure itself in the most abstract case, but it gets an interesting structure of its own when you think about all the different facets of the linearity of the translation space. Another common example of an affine space is the Euclidean plane, where you can't add two points but you can add a vector to a point to yield the resultant point of that translation.

What you're doing is perfectly cromulent from a notational standpoint. You don't mention the age of the students you are teaching. But if they call you out on it and you don't want to explain all of this, you could demur and pretend that "11:40" is a different notation for the duration since midnight and you can combine them just like you might say "5'8" + 3 inches".

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    $\begingroup$ Thank you for embiggening my vocabulary today! $\endgroup$
    – user52817
    Commented Apr 11, 2021 at 4:14
  • $\begingroup$ While reading through the answers, I was preparing to write something precisely like this. Thank you for saving me the trouble. (+1) (I would give another +1 for the use of "cromulent", a word which embiggens us all, but, alas, I am only allowed to give one upvote). $\endgroup$
    – Xander Henderson ♦
    Commented Apr 11, 2021 at 15:17
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I read the left-hand side as 11 minutes and 40 seconds minus 15 minutes. (Analogous to 11.4 - 15). So I expected the right-hand side to be negative.

I would not write this, especially as a teacher.

I would prefer $11:40-00:15=11:25$ hours, with the understanding that the units refer to the leftmost grouping

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  • $\begingroup$ Please see the updated clarifications. $\endgroup$
    – Improve
    Commented Apr 11, 2021 at 9:04
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I taught elementary school. I think the clearest way to write this for elementary school children is:

11 hours 40 min - 15 min = 11 hours 25 min

You could then conclude from the above that 11:25 is 15 minutes earlier than 11:40. You could also regroup as necessary.

11 hours 15 min - 40 min = ?

This should be regrouped as 10 hours 75 min - 40 min = 10 hours 35 min. We can now conclude that 40 minutes earlier than 11:15 is 10:35.

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    $\begingroup$ (+1) I would probably write "11:40 $-$ 15 min is 11:25" and not try to make a formal mathematical equation out of this. However, your elementary school approach seems fine (and probably better for elementary school purposes), except for hours and minutes I would either abbreviate both or abbreviate neither, rather than abbreviate one and not abbreviate the other. Note, by the way, that putting 11:40 in math-mode (as the OP has done) results in non-standard spacing (for time designations) around the colon. $\endgroup$
    – Dave L Renfro
    Commented Apr 11, 2021 at 16:20
  • $\begingroup$ +1. Always opt for clarity over brevity, as it actually saves teaching (and student learning) time in the long run. $\endgroup$
    – Adam Rubinson
    Commented Apr 13, 2021 at 12:51
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Consider the following statement:

$\frac{8}{5} - 1 = \frac{3}{5}$

You could argue that the data types don't match: $\frac{8}{5}$ is an ordered pair; why would you be able to subtract an integer from it?

The reason is that you can define the way subtraction works for these objects even though they are not the same data type. You need to prove that your operation is well-defined, but once you do so, you are fine.

You can also cohesively define how subtraction works with a time and a time interval, so... do it. It's fine.

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  • $\begingroup$ I see your example as infix with three numerical objects 8,5,1. The postfix representation is 8 5 / 1 - $\endgroup$
    – user52817
    Commented Apr 11, 2021 at 2:47
  • $\begingroup$ Please see the clarification in the post, especially point 2. Would you still argue that this is a case of data types? $\endgroup$
    – Improve
    Commented Apr 11, 2021 at 9:36
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    $\begingroup$ Why does it bother you that A - B is sensible when B - A is not? What axiom in your head is being violated here? What is wrong with 23:45 + 30 minutes being well-defined as 00:15, while 23:45 + 00:15 is undefined? @Improve $\endgroup$
    – Chris Cunningham
    Commented Apr 12, 2021 at 1:15
  • $\begingroup$ What? @user52817 $\endgroup$
    – Chris Cunningham
    Commented Apr 12, 2021 at 2:06
  • $\begingroup$ @ChrisCunningham Quite often when you can do A-B, you can also do B-A. I am mostly concerned that students will think this is true in this case too, although I admit I am being quite pessimistic here. I also misunderstood your point about "data types" first and confused it with another answer. $\endgroup$
    – Improve
    Commented Apr 12, 2021 at 7:53
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I think the concern about careful use of units here is a bit of a red herring because the representation of time is not numeric.

A good framework to resolve this conundrum might be that of datatypes. Perhaps there are some future computer scientists in your class.

Time represented in the format HH:MM can be thought of as a special datatype, similar to number, text, float, integer, percent. Subtraction would be done by converting first to numeric, and then converting back to HH:MM

In this case, 11:40 would convert to 0.486111111111111, in the sense that noon which is halfway through the day, would convert to 0.50

Fifteen minutes has numeric value 0.0104166666666667. We (the computer) does the subtraction to get 0.4756944444444. Upon converting back to HH:MM datatype we get 11:25.

I would not worry about what feels like sloppy use of units because HH:MM is not numeric. If a student asks, give them a developmentally appropriate "datatype" answer.

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  • $\begingroup$ Please see the clarification in the post, especially point 2. Would you still argue that this is a case of data types? The part about being careful about units is indeed a bit of a bit red herring, as I think my overall problem might be how to denote actions. $\endgroup$
    – Improve
    Commented Apr 11, 2021 at 9:37
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In statistics this type of data is called interval data - data where differences ("intervals") make sense but not necessarily other operations. This is the data-science way to say what Matthew Daly is saying in his answer.

Dates and times are a standard example used in statistics classes. For example, you can find the difference between two years:

  • 2021 - 1999 = 22 years.

This means that if you like, you can rearrange the equation to say things like

  • 1999 + 22 years = 2021.

  • 2021 - 22 years = 1999.

but similar expressions make no sense, for example

  • 22 years - 2021 ??

  • 1999 + 2021 ??

Other common examples of interval data, like temperatures, have the same issue but the issues are hidden by a subtle trick in the way we speak. For example, you can find the difference in temperature between an object that is 75° and another that is 60°:

  • 75° - 60° = 15°

But the "15°" is a different kind of thing; it is an interval, not the temperature of an object.

If I tell you to add this 15° (interval) to the 75° (a temperature), that works just fine.

But if I instead tell you I have two objects, where one is 75° (a temperature) and the other is 60° (a temperature), and then ask you to find the sum of their temperatures, you should feel some cognitive issue here -- can you really add two temperatures? It's never a useful or meaningful thing to do. You can't really add those two temperatures together in any meaningful way, just like you can't add "noon" plus "2 o'clock." This is interval data.

As far as I know, times and dates are the best example of interval data where this issue manifests clearly and does not have any linguistic tricks to obscure the issue.

So -- for interval data, you have two types of quantities: "measurements" and "intervals" (in Matthew Daly's answer he refers to these as two spaces). You can do the following things:

  • Measurement - Measurement = Interval
  • Measurement $\pm$ Interval = Measurement

But other operations don't make sense.

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  • $\begingroup$ (1) Thanks for reminding how the number line is introduced, and how numbers are added or subtracted using the number line: using a pair of compasses you open it to the size of the unit segment, and then make as many equal segments on the number line as the first addend. Then you do the same with the second addend. The tick you marked is the answer. Is it a "measurement" or an interval? I think, both. Absolute value of an integer is the distance from zero. And you can do it in any order, illustrating commutativity of addition. And you can have more than two addends, illustrating associativity. $\endgroup$
    – Rusty Core
    Commented Apr 12, 2021 at 17:18
  • $\begingroup$ (2) So, I have no problems adding 1999 to 2021: both "measurements" are the intervals from the 0 year, so the result will be the interval from the 0 year. Likewise, I have no problem with 22 years - 2021 year, it will be a B.C. date, so what? Temperature is conceptually different, it is a physical property, and temperatures of the objects cannot be added. This cannot be explained with math alone. $\endgroup$
    – Rusty Core
    Commented Apr 12, 2021 at 17:24
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    $\begingroup$ (3) But, this is exactly what authors of the textbooks that Richard Feynman once reviewed, did: "Red stars have a temperature of four thousand degrees, yellow stars have a temperature of five thousand degrees, etc ... John and his father go out to look at the stars. John sees two blue stars and a red star. His father sees a green star, a violet star, and two yellow stars. What is the total temperature of the stars seen by John and his father?" ­­ and I would explode in horror. It was awful! All it was was a game to get you to add, and they didn't understand what they were talking about. $\endgroup$
    – Rusty Core
    Commented Apr 12, 2021 at 17:25
  • $\begingroup$ I was able to find the essay referenced in the above comments here archive.org/stream/Surely-youre-joking-mr.-feynman/… $\endgroup$
    – Chris Cunningham
    Commented Apr 12, 2021 at 17:42
  • $\begingroup$ Another example in this class: Pointer arithmetic in C/C++ (i.e., arithmetic on addresses in memory). In this case, the operational limitations you describe are strictly enforced by the language standard and compilers. $\endgroup$
    – Daniel R. Collins
    Commented Apr 13, 2021 at 2:58
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  • It is as cromulent (thanks, MatthewDaly!) to write $$11.40\text{ a.m.} \color{#00F}- 15\text{ min} = 11.25\text{ a.m.}$$ as it is to write $$(4,3) \color{#00F}-\begin{pmatrix}1 \cr1\end{pmatrix}=(3,2),$$ because $“\color{#00F}-”$ here can be thought of (is implicitly defined) as a variant subtraction operation: one whose first argument is a time/Cartesian -point, second argument is a duration/vector, and output is a time/Cartesian -point.

    When using the 24-hour clock, I write $$11\;40 \color{#00F}- 15\text{ min} = 11\;25.$$

  • I avoid writing $$11:40 \color{#00F}- 15\text{ min} = 11:25$$ though, as it is ambiguous whether it is nearing noon or midnight.

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  • $\begingroup$ I would never even consider writing $(4,3) \color{#00F}-\begin{pmatrix}1 \cr1\end{pmatrix}=(3,2)$. Is this something you actually do? $\endgroup$
    – Improve
    Commented Apr 11, 2021 at 17:45
  • $\begingroup$ @Improve Since a Cartesian point has a direct translation to a position vector (whereas translating 290 mins to 04 50 or 4.50 is odd/confusing/non-straightforward), I prefer to reframe the displacement procedure to stick with writing $“\begin{pmatrix}4 \cr3\end{pmatrix} - \begin{pmatrix}1 \cr1\end{pmatrix}=\begin{pmatrix}3 \cr2\end{pmatrix}”.$ $\endgroup$
    – ryang
    Commented Apr 11, 2021 at 19:00

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