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Using the same symbol for two different concepts may cause confusion. So if one decides to do so, they should justify this choice by showing its advantages over other choices. What about the minus sign and negative sign? What are the reasons which convinces us to use the same sign for both of them?

What would be lost if the negative sign were different, e.g. $-3$ were shown by ${}^{-}3$ (as in some elementary textbooks)?

Of course, this question may be answered form a mathematical or education point of view.

P.S. It seems necessary to explain what I mean by "confusion" in lines above. We expect the students to convert subtraction to addition (so that they be able to use associativity, commutativity, etc.). How do they justify the identity $a-b=a+(-b)$? Unfortunately, by saying that "positive times negative is negative"; obviously a bad justification.

Behzad
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  • They are the same since they represent the same number: $-3=0-3=0+(-3)$. – Sigur Oct 28 '13 at 15:38
  • Why do we use a fraction bar to denote fractions? It's just a useful convention until somebody invents a new, more useful, notation. One of the ingenious inventor of notation was surely Leibniz. He invented “$\cdot”, $\colon$, and the famous $d/dx$. The latter is far away from being a fraction, but one may calculate in a wide area as if it was. Ingenuously simple, simply ingenious. A notation that survived over more than 300 years. – Michael Hoppe Oct 28 '13 at 15:40
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    @Sigur The $-$ sign in $-3$, is a unary operator, but it's a binary operator in $0-3$. They're not the same. – Behzad Oct 28 '13 at 15:44
  • @Behzad, and how this unary operator is defined? It is the opposite of $3$ with respect to the addition. There is no minus, there is only plus. – Sigur Oct 28 '13 at 15:46
  • @MichaelHoppe Why it has more advantages then disadvantages? I - as a middle school teacher - DO notice the confusion students encounter caused by the sameness of the minus and negative sign. – Behzad Oct 28 '13 at 15:47
  • @Sigur Well, you can't simply enter the classroom and say "hey kids! There is no minus, just plus. Everything you learned about minus in elementary school was a big lie!", I mean, we should extend the meaning of minus so that we get to the identity $a-b=a+(-b)$. – Behzad Oct 28 '13 at 15:52
  • @Behzad, I agree. We can not. So this is the reason to use the same symbol. – Sigur Oct 28 '13 at 15:55
  • @Sigur I added a post script to my question, so that the "confusion" become clear. – Behzad Oct 28 '13 at 16:00
  • There is also a related discussion here. – Old John Oct 28 '13 at 16:00
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    @Behzad Dear colleague, it just spares one more sign. Carefully introduced, our pupils will know in which context it's binary or unary as they'll detect -- depending on the context -- in which cases “lives” is pronounced “lee-ves” or “lie-ves”. So lives is not lives and $-$ is not $-$. At last not necessarily. – Michael Hoppe Oct 28 '13 at 16:02

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