What is going on with the Minus Sign?

You think you are familiar with the minus sign. OK so take the following challenge.

Express -(-3-5)^-2  as an integer.

Let’s break this down.

The symbol before the 3 represents the integer “negative three”.

The symbol before the 5 is the subtraction symbol and results in 5 being subtracted from ^–3 to give    -8, the value of the number in the brackets.

The symbol before the 2 represents the reciprocal exponent which is the equivalent of 1 divided by the result of the last step to the power of 2.  i.e. 1/(^–8*^–8) = 1/64. What is negative about a fraction? The minus sign is being used to say this operation is based on division not multiplication.

The symbol before the expression in the brackets calls for taking the opposite of that expression giving -1/64.

So far, the use of the minus sign is as clear as mud.

The sine of 30^o or sin30⁰ = ½. The purpose of the sine inverse function is to reverse what the original function does. The sine inverse function is denoted with a minus sign as sin^-1½ = 30^o. What else will the minus sign be used for?

Another example is the limit function.  ‘The limit of a function (if it exists) for some x-value, a, is the height the function gets closer and closer to as x gets closer and closer to a from the left and the right.’⁴ A one sided limit is the height in the graph of the function that the function gets closer and closer to from the left. This is indicated by a superscript minus sign such as  .

Next, we find that multiplying two minuses results in a plus. The proof of this uses that tricky number 0. We may accept that 0 is a number, but relying on it to support fundamental proofs seems suspect. The proof that -1 x -1 is shown in the Zero Usefulness blog and depends on multiplication by 0.

I think I would compare the logic used by a student in a study in 2009 with the logic that uses multiplication by 0. The student says ‘I would show them that two minus can also be a plus if you take one minus and put it crosswise over the other one.’

So, what’s going on with the minus sign? Why is it being used for so many different operations?