OK, here’s a stupid math question for the brainiacs of SDope Boards to take under consideration: We’ve all been taught the rules of basic math, but how does this really work with negative numbers? Is -2 really an entity unto itself? Consider this…
Is there a difference between {a} -2^2 vs. {b} (-2)^2? One math teacher is claiming that {a} equals -4 and {b} equals 4. Is this correct? And, what about -(2)^2? Surely, this last one is undoubtably equal to -4. But, is the math teacher correct???
The math teacher is correct. By convention, -2[sup]2[/sup] is declared equal to -4. Think of it this way: -x is equivalent to -1 * x, and exponentiation comes before multiplication.
Thus, we are saying that only positive numbers are truly numbers unto themselves??? Or, it is simply coincidence that it appears this way since 12 = 2, so 2 is really short-hand for 21? In short, every number always has a coefficient in front of it, but with positive numbers, the coefficient is inert… So, in fact, no number is an entity unto itself, other than perhaps zero? - Jinx
No, just that mathematics defines precedence of arithmetic operators, and raising a number to an exponent takes higher precedence than unary minus.
There really is such a number as negative 2.
I know what you’re saying: If -2 means apply a unary minus operation to 2, then how the heck do I write the result of that operation? The answer is that it doesn’t matter, these are just ways to write a representation of the same concept. Unless you’re concerned about precedence, in which case you would write
BTW, regarding the subject line of the OP, my 8th grade algebra teacher once stood in front of class and said, “My dear Aunt Sally is dead.” It got very quiet. “My dear Aunt Sally is dead and she’s buried in the gray box on page 67.” Within which was a description of precedence of operations that put multiplication on equal footing with division, and paired addition and subtraction. None of us had even heard of My Dear Aunt Sally before, but he explained it after we read the gray box. Apparently My Dear Aunt Sally was causing hapless students to apply multiplication before division, and addition before subtraction. (This was in 1971; apparently the reports of My Dear Aunt Sally’s death were exaggerated.)
And I didn’t learn about the proper precedence of exponentation until I taught a programming language with built-in exponentation that used the proper artihmetic precedence.
Of course, in programming you NEVER want to rely on order of operations to make your calculations. It is just good practice to stick in parenthesis so that everyone can see what you are trying to accomplish. And also, sometimes people are mistaken in order of operations and calculations they didn’t expect get carried out.
-2 is very much it’s own number. I’m guessing that it’s the addition of the ‘-’ prefix that is causing you to think that negative numbers are somehow “different” than positive numbers. In fact, it’s just a matter of convention that numbers are assumed to be positive unless otherwise stated. You could very well write 2 as +2, but it’s rare to do so.
Though when the integers are defined, I think you often define the naturals first (using peano or something) and then define the negatives (and -2 is the one which, when added to 2, gives 0)
Every number is derived, ultimately, from the axioms of the natural numbers.
The natural numbers are 1, 2, 3 and so on. The jury is still out on whether 0 is a natural number (as a pure mathematician I say no…but I’m not going to start any fights over it). Basically though, each number is defined as the successor (one more than) the previous number - it’s fairly intuitive counting.
The next set of numbers to be derived is the set of integers. This is the set of all numbers which can be formed by subtracting one natural number from another - hence negative numbers where the subtracted number is larger than the original.
From there we go on to rational numbers (numbers form by a division of one integer by another), to real numbers (numbers which can be expressed as the limit of an infinite series) to complex numbers (a number containing a real and imaginary component…and that’s where I give up explaining!)
Anyhow, the up-shot of all this is that you’re right and wrong in one neat bundle. Negative numbers are, as part of the set of integers, numbers in their own right. However, they are derived from the subtraction of natural numbers so they aren’t entirely independent.
That’s the number theory behind things, of course. What your question gets down to is the notation. Is -2 distinct from -(+2)? The result is the same at the end of the day but the strict answer is that, yes, when dealing with anything but the natural numbers -2 is its own entity.
Actually, there are constructions that allow for either possibility. The normal set-theoretic construction does include 0, but I’ve played around with a functional construction that doesn’t. Since Godel’s theorem requires that 0 is a natural number, I prefer to say that it is. YMMV.
I’d like to pick a nit with the statements: “… -2 is its own entity” (Hern) and " -2 is very much its own number" (Joe Random)
Although I think I agree with what you guys are saying, in the purely notational sense, I believe that these statements serve to mislead or confuse people like Jinx. If you think that -2 “is its own entity” then I don’t see how you can escape the conclusion that -2^2 = (-2)*(-2) = 4, which, I hope we agree, is not the convention.
Let me re-phrase Jinx’s question another way. I think he’s asking “Is -2 an explicit representation for negative two, or is it something else? If something else, what is it?”
I think the answer is that “-2” is not an explicit representation of a number, but is an operation on the number we all know as two. The priority of this operation is below exponentiation, on the same level as multiplication. The result is a negative number.
This doesn’t take anything away from the uniqueness of -2 or challenge its legitimacy in any way, its just a fact of our convention for writing numbers that you get to them by operating on the “more familiar” positive numbers.
In order to claim that -2^2 = -4, I think you have to admit that the “-2” is, at least, an implied operation, rather than an explicit numeral.
I probably should have been more clear and said “the number represented by ‘-2’ is its own number.”
The problem is that we represent numbers as being negative by way of a unary operator, rather than by some symbol that is a part of the number. If we used something like an underscore (or, to make this example clearer, color), then it would be much clearer. Imagine that red numbers are negative.
It seems to me that you’re skipping a few steps. If you start off not including zero as a natural number, then it looks to me like you need to put zero in before defining negative numbers, or even subtraction. Something like this:
Start with one number (I presume you’re using 1) and the “successor” operation
Define the “addition” operation, in terms of “successor”
Define 0 to be the identity of “addition”
Define subtraction to be the inverse operation of addition (don’t you need an identity before you can define inverses?)
Define negative numbers in terms of subtraction