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Might it be helpful for students to have two different symbols for subtraction (-) and negation ( _ )? Subtraction, after all is a binary operation (with 2 operands). Negation is a unary operation (with 1 operand). It also usually has a higher order of precedence than subtraction.

The order of operations could be taught using the nmonic BENDMAS with N standing for negation or negative.

Order of Operations (Precedence)

  1. B: Brackets
  2. E: Exponents (right to left ), e.g. $2^{3^4} = 2^{(3^4)}$
  3. N: Negative (right to left), e.g. _ _ $2 =$ _$($ _ $2)=2$
  4. D: Division, M: Multiplication (left to right)
  5. A: Addition, S: Subtraction (left to right)

Note that _ $x^2 =$ _$(x^2)$ as in most algebra textbooks. To be consistent and not have to make exceptions for constants, you would require _ $2^2 =$ _ $4$ and $($_ $2)^2 = 4$.

This would save converting negative signs to multiplication by minus 1, and other embarrassing patches on something as fundamental as the order of operations.

Dan

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    $\begingroup$ Standard notation uses the same symbol for both; readers must get used to that. Negation does not have higher order of precedence than subtraction. $\endgroup$
    – Daniel R. Collins
    Commented Dec 11, 2017 at 23:44
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    $\begingroup$ This is very much a solution in search of a problem. $\endgroup$
    – Adam
    Commented Dec 12, 2017 at 0:00
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    $\begingroup$ @DanielR.Collins Students should know that they are indeed different operations. I guess students are expected them to tell apart by the context. What rule if any do student used to tell them apart? And where do you suppose this other minus operation fits into the order of precedence? $\endgroup$
    – Dan Christensen
    Commented Dec 12, 2017 at 2:45
  • $\begingroup$ By definition, unary negation indicates the additive inverse of a given value. It is a theorem that this is therefore the same as multiplying by $-1$. $\endgroup$
    – Daniel R. Collins
    Commented Dec 12, 2017 at 18:10
  • $\begingroup$ @DanielR.Collins So, everywhere you see a negative sign, you substitute $-1\times$. But not if it's really a subtraction. The student will first have to parse the given expression and determine whether each "$-$" is a negation or subtraction using a simple, unambiguous rule (yet to be determined). That would work nicely for a computer. Not so sure about a distractible 8-year-old human who might appreciate being told which are really subtractions and which are really negative signs. $\endgroup$
    – Dan Christensen
    Commented Dec 12, 2017 at 20:43

4 Answers 4

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I do recall some elementary texts that do this. Subtraction $$6 - 5$$ written in a different way than a negative number $$ {}^-8 $$ so we can do calculations $$ 5 - {}^-2 = 5 + 2 = 7 $$ Presumably at some point, the students are switched to the conventional notation.

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    $\begingroup$ That was how negative numbers were dealt with when they were introduced to us in middle school (7th & 8th grades), but I don't have my texts to see what was said. I do have a copy of my 9th grade beginning algebra text (Dolciani's "Book 1" algebra), and raised signs are used there near the beginning but not later on. In tracking down when the switch occurs, I found a discussion in Section 3-4 about additive inverses (denoted with lower negative signs) with this comment (given in italics): "Throughout the rest of this book, lowered minus signs will be used in the numerals for negative numbers." $\endgroup$
    – Dave L Renfro
    Commented Dec 12, 2017 at 12:16
  • $\begingroup$ It might be best to hold off reverting to conventional notation until after mastering the solving of simple linear equations, maybe even after mastering factoring or the quadratic formula. Those seem to me to be areas where different symbols for subtraction and negative might be most useful as a learning tool. $\endgroup$
    – Dan Christensen
    Commented Dec 12, 2017 at 13:40
  • $\begingroup$ I would really rather students get in the habit that two operational symbols can never be juxtaposed. This just confuses the issue. $\endgroup$
    – Daniel R. Collins
    Commented Dec 12, 2017 at 18:13
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It is perhaps worth noting that on most graphing calculators there are already different symbols for subtraction and negation, instantiated on different keys. For example on the TI-84 Plus (see image below) there is a key labeled (-) at the bottom right corner, for negation, and a separate key along the right-hand side for subtraction.

enter image description here

Is this "better" than having a single key that handles both functions? Personally I doubt it. More often than not the need to distinguish between the two keys leads to syntax errors -- as, for example, when a student wants to set the XMIN, XMAX, YMIN, YMAX variables for a viewing window and accidentally uses the subtraction symbol instead of the negation symbol.

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  • $\begingroup$ If you have a result on a calculator, you can continue the calculation by simply pressing a button. E.g. if you did some stuff and came up with $10$, you can simply press $+5$ and continue the computation with this result, getting 15. I think that is why there are two different symbols: - continues the computation, so e.g. "-5" will give 5, while (-) negates the current number, giving -10. The use of different symbols is thus due to the way the calculator is programmed. If mathematics should generally be adjusted to the hardware used is a big discussion though, too big for a comment... $\endgroup$
    – Dirk
    Commented Dec 12, 2017 at 10:45
  • $\begingroup$ The best way to make sure that students understand whether they are doing subtraction or negation is to have two distinct buttons / symbols for these quite distinct operations, forcing them to decide. $\endgroup$
    – Dan Christensen
    Commented Dec 12, 2017 at 17:27
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    $\begingroup$ @mweiss if you have ever tried to write a mathematical expression parser, you will realize that this issue is an extreme headache, especially with the convention that juxtaposition generally indicates multiplication. So y3 should mean y times 3, but y-3 shouldn't mean y times -3. Truly terrible. $\endgroup$
    – Steven Gubkin
    Commented Dec 13, 2017 at 12:39
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    $\begingroup$ @StevenGubkin It was while I was writing just such a parser that the illogic of having the same symbol for both operations struck me. It can be done, but the internal logic would be a real mess -- VERY inelegant with hideous exceptions as I recall. And why should they be the same? $\endgroup$
    – Dan Christensen
    Commented Dec 13, 2017 at 13:15
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    $\begingroup$ @mweiss If the expressions are ambiguous enough that it is hard (or impossible) to teach a computer how to parse them, then it will also be hard to teach students. The reason being that the rules are actually inconsistent, or at the least extremely convoluted. $\endgroup$
    – Steven Gubkin
    Commented Dec 14, 2017 at 12:47
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It's a common mistake that negative numbers are just the result of an operator on a positive number: they have a right to exist and the - sign in front of them is just for recognition purposes, so next to subtraction and negation, there also is indication.

Let me explain you:

There are two kinds of numbers:

  • positive ones : 0, 1, 2, ...
  • negative ones : 0, -1, -2, ... (indication)

Negation is the function which flips positive numbers into their negative equivalent and vice versa:

  • - (1) = -1 (negation leads to indication)
  • - (-2) = 2 (negation leads to removal of the indication)

There is an operator (the subtraction) between two numbers, which is the addition of the first number with the negation of the second number:

  • 5 - 3 = 5 + (-(3)) = 5 + (-3)
  • 6 - (-4) = 6 + (-(-4)) = 6 + 4

In the example of 5 - 3, we have the following usage of the - sign:

  • 5 - 3 => subtraction operator
  • 5 + (-(3)) => negation operator
  • 5 + (-3) => indication that the second number is negative (no operator)

Whether or not it's a good idea to use different characters for the different meanings, I think you can only succeed if you do it for the three, not just the two you mentioned.

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  • $\begingroup$ Under standard definitions (in English?): "Indication" is not a standard term. $0$ is neither positive nor negative. There is no difference between $-(3)$ and $-3$. $\endgroup$
    – Daniel R. Collins
    Commented Dec 13, 2017 at 13:52
  • $\begingroup$ "Indication" is not the standard term, I agree, but you can understand what I mean. And about the difference between -(3) and -3: "-(3)" means "take the negation of 3" and "-3" is the negative number, on a distance of 3 left to the number 0. And about 0: there is "positive" which is equal to ">=0" and strictly positive which is ">0", as a result 0 is both positive and negative, but not strictly positive or strictly negative. $\endgroup$
    – Dominique
    Commented Dec 13, 2017 at 14:40
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Really bad idea. This is sacrificing the 99 kept sheep to save the 1 lost sheep.

A. It's not that hard.

B. Everything else they read will be different from how you teach.

C. What makes you think you should use a teaching position to drive a difference in usage (versus having teaching follow usage).

D. (And Chris will kill me for saying this but "Ich kann nicht anders") Don't you have something else more useful to improve?

Oh...and 99% of people don't get the subtle differences. Me included.

P.s. Sorry, cable reset my IP again. Hence the low post count.

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    $\begingroup$ -1 neither respectful nor helpful. $\endgroup$
    – MathGuy
    Commented Dec 12, 2017 at 1:57
  • $\begingroup$ The difference is not that great. The symbols even resemble one another. I was thinking that once students have mastered both operations, it shouldn't too difficult to revert to more standard notation. I do recall that many students had difficulties with these concepts and that their textbooks were quite vague on these points. But this was all some time ago. Perhaps this is no longer the case? $\endgroup$
    – Dan Christensen
    Commented Dec 12, 2017 at 3:00
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    $\begingroup$ "P.s. Sorry, cable reset my IP again. Hence the low post count." You know, the good folks at SE planned for that years ago and would've let you register an account ;-) $\endgroup$
    – quid
    Commented Dec 12, 2017 at 19:36
  • $\begingroup$ -1, Hier steht guest, I suppose $\endgroup$
    – Chris Cunningham
    Commented Dec 12, 2017 at 22:26

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