[pez]

Quote:

Originally Posted by goldenratiophi /img/forum/go_quote.gif The - is processed first because it's in the exponent. And when all else fails: ps: I don't argue that -x doesn't always equal -1*x in some algebraic structures, but I think it's safe to assume that we're talking about a field here

Quote:

Originally Posted by manofmathematics /img/forum/go_quote.gif You have to be careful. -5^2 is actually -25. This is because, in this case, -5 is actually -1*5, and the order of operations tell us that (-1)5^2 should have the exponent evaluated first, followed by the multiplicative negation. It is wise to group the "-" appropriately. say, (-5)^2, which would indeed equate to -25 since the multiplication is completed first, once again due to the order of operations. aaron313, has a very good point above. Indeed it holds true for all numbers. "1" is considered one of those "special numbers" and can throw a few curveballs, especially when diving into a little Number Theory. Also remember that 1 is not a prime. I've seen many people argue over this one, but 1 is indeed not prime due to the Fundamental Theorem of Arithmetic, which I really don't feel like explaining, but look it up, it's not a difficult matter really.

-5^2 and 5^2 both equal 25, what you do one way, must come out the same when doing the opposite (i.e. 2x5=10, and 10/5 = 2 and 10/2=5) If you were to say -5^2 is equal to 25, you would have to that would be implying that the square root of -25 is -5, and it's not. The square root of a (-)25 is 5i.

 

[goldenratiophi]

Order of operations - Wikipedia, the free encyclopedia

Quote:

The order in which the unary operator − (usually read "minus") acts is often problematical. In written or printed mathematics, − 3^2 = − (3^2) = − 9, but in some applications and programming languages, notably the application Microsoft Office Excel and the programming language bc, unary operators have a higher priority than binary operators, that is, the unary minus (negation) has higher precedence than exponentiation, so in those languages − 3^2 = ( − 3)^2 = 9, . [3]

Darn you, Excel and bc!

 

[manofmathematics]

5^2 is indeed equal to (-5)^2, but not equal to -5^2. As already shown, this is due to the order in which we complete operations.

What we need to do is understand what exactly -5 means.

Both of the number, 5 and -5, are real numbers, R, which means they are members of what is known as the R Field.

Let's focus on the number 5.

The R Field has particular properties and rules that govern these properties, so, since 5 is a member, these properties and rules must also apply. One such property pertains to the Additive Identity, which basically is saying, for a given X, there is a Y so that X+Y=0. Of course, we know this to be true because adding a numbers opposite, 5+(-5), indeed equals 0. This is actually how we can define subtraction.

But what this property doesn't tell us is, how in the world do we come up with the -5 to add to 5 in the first place. The answer is in a particular rule present in all Fields, and across the Reals. This is simply put that -X = -1X. This rule guides all negatives across the Reals.

Thus when presented with the problem -5^2, we break it down into its parts and have -1*5^2, which due to the order of operations, yields -25.

The "-" may be a unary operator, but throughout Arithmetic, Algebra, Calculus, and other areas of Mathematics, when we see a "-" not being used in its binary form, it is known to represent -1.

 

[rsaavedra]

Quote:

Originally Posted by manofmathematics /img/forum/go_quote.gif This is simply put that -X = -1X.

Yes, that expression is valid. But so is -X = 0 - X. All you said is really mostly BS.

The complement of an element with respect to the sum operator doesn't require the existence of another operator (in this case, multiplication or substraction) to exist, at least not in an abstract algebra structure called a group.

[R, +] is an algebraic "group." It has an identity element with respect to + (namely zero: for every X, X+0 = 0+X = X), and there is an "inverse element" with respect to + for every element X in R (namely, that number Y such that X+Y = Y+X = 0). You can denote that Y (that inverse element of X in that group) anyway you'd like; it could be -X, or X'. In any case, the inverse exists in the group, no need for anything from outside of the group. You don't require the existence of the multiplication operator (*), or of any other operator (not even " - "), by any means, to make [R, +] a valid group so that every X in R has its inverse, namely, what we by convention denote as -X.

 

[manofmathematics]

Now, I haven't studied algebraic structures in almost five years, so I do appologize for any mistakes, but I fail to see how what I have said is bs, except for referring to the described property as the additive identity when I meant inverse.

You bring up a group and the two properties that separate a group from a semi-group. I am looking at the Reals from the higher perspective of a Field where all properties of a group, plus others, hold true.

Let's go back to the Inverse property and more clearly define like this.

-X + X = X + (-X) = 0

For a group, you are correct. The inverse does exist and can exist without "*". But, when examing this while looking at the Real Field, (R, +, .), of course the . should be raised, we find a rule within a Field that describes -X as -1X.

Like you said and I agree, in a Group, every X has its inverse, but the above rule described for a Field helps to better understand what exactly the inverse is.

The example of -X = 0 - X, when properly represented with respect to the properties of a Field should read -X = 0 + (-X), and this can clearly be reduced to -X = -X, then using the said rule of Fields can be reduced further to -1X = -1X.

Once again, it's been a while and coming off the top of my head, so please correct me where I am wrong.

 

[rsaavedra]

No it wasn't really all BS, I said mostly BS

Hope it didn't offend you, didn't mean to, just really wanted to catch your attention

Don't see why you get into "Fields" when the sum is one of the most basic mathematical operations, if not the most basic. The multiplication after all is just an application of the sum several times over.

So simply put, by saying it was BS I mostly was pointing to the fact that you don't need multiplication to define things at the level of sums or substractions. And this bit of folk-wisdom sounding statement can actually be backed up by group theory, as I showed

 

[manofmathematics]

No, you didn't offend me at all, but if I am wrong, please do correct me.

 

[XGJFilmsX]

pemdas is the way to go

 

[manofmathematics]

No, you are absolutely right. I just wanted to try and provide the best evidence I could for why -5^2 = -25, and really, the best I could think of was to refer to Fields and the structure's relevant properties and rules.