Evaluating -3^2

Factual Questions
61

It’s interesting how quickly a person can overthrow his own position. My neighbor’s son is taking arithmetic and they are up to powers and roots. So he asked me how to work out the answer to 20 - 4[sup]2[/sup]. Without any thought whatever, I took 4[sup]2[/sup] from 20 and got the answer 4 which is correct and completely opposite to my previous posts which can now be safely ignored.

My posts were correct but completely irrelevant to the question that was asked. The negative sign in - 3[sup]2[/sup] is an instruction (or operator) telling us to subtract the square of 3 from something.

62

Well, that is kind of the whole point of the problem.

63

I don’t see how there can be a popular parsing of -3[sup]2[/sup] except by way of analogy to -x[sup]2[/sup]. I’m not arguing there isn’t, and I agree it is -9, but I’ve never seen exponents like that (without variables) not already evaluated. It seems to be a pretty unique situation, IMO, not a standard one.

64

erislover: see equation 4 or 17 of the page on Sine to get an example of (-1) raised to a power, n-1 and n respectively. I would say that this sort of formula comes up all the time.

65

Ah drat, I understand now that that doesn’t work, because n is a variable. Okay, sometimes, we will write a series like this:

1 + 1 / 2[sup]2[/sup] + 1 / 3[sup]2[/sup] + 1 / 4[sup]2[/sup] + …

but this situtaion does not come up all that often. I can’t even find a cite for it.

66

Yeah, I was just thinking that I’ve seen examples of that when demonstrating an evaluated series expansion.

67

This thread has me convinced that the proper convention for evaluating -3^2 is -9. But, all this talk about unary negation gives me a problem.

How do you define unary negation for purposes of this notational operation precedence that’s being discussed? I can think of two ways, both of which give me a problem. Bear in mind that it’s not just the formal definition of negation that I’m challenging, but also the way the notation is used.

The easiest way is just to say that , for any number a, -a = (-1)*a. That’s clear enough. But, now would you tell me what you mean by writing “-1”? If you’re going to tell me that it means (-1)*1, well, I hope you can see that there’s a problem.

The other way is to invoke the algebra of numbers and say someting like “the unary negation of a is the additive inverse of a”. Or “for any number a, it’s unary negation is the number b such that a+b=0”. This is also a fine definition, but doesn’t describe the notation we all use. So, you might say that “we indicate the additive inverse of any positive number a with the notation -a, the additive inverses of negative numbers do not require any special notation”. The problem with this definition is that the notation “-3” becomes a representation of a number all unto itself, without any concept of being parsed into (-1)*3. Under this application, then -3^2 = 9 in the same way that 1.2^2 = 1.44, and not 1.2^2 = 1.04.

Where’d I get that? Well, 1.2 just really means 1 + 2/10 doesn’t it? Then, since exponentiation supercedes addition, shouldn’t I square the 2/10 before adding the 1? Of course, nobody thinks so, but that’s because we’re thinking of the notation 1.2 as describing a number all by itself, and not some operation involving 1 and 2 that can be broken down.

Likewise, I think the people here that lean towards -3^2 = 9 (myself included, initally) are thinking of “-3” not as some kind of operation, but as a way of denoting a number.

68

Understood. But that argument works against you. If you want to use 1 + 2/10 as a representation for 1.2, it works fine as it is. However, if you want to add an exponent to the expression, you have to include the 1 + 2/10 in parentheses. Same thing for -3.

If there were a numeral that represented -3 by itself, there’d be no problem. For instance, the golden ratio is (1+sqrt[5))/2 is often represented by the letter phi. So, phi^2 is unambiguous, and it doesn’t mean the same thing as (1+sqrt[5))/2^2

BTW, astro, I’ve saved a couple of your posts too–just in case someone wants to set up a democracy. :slight_smile:

69

I don’t see how

-1 == (-1)1

is a problem. For the sake of applying exponents, this issue is resolved:

-1^2 == (-1)1^2 == -1.

It effectively gets the negative out of the way of the exponentiation which we are agreeing is the big pickle here. This is different from:

(-1)^2

which is, if you bother to expand uselessly:

((-1)1)^2

which is, of course, 1.

I suppose the argument could be that the -1 expands out ‘indefinitely’, but that’s just an excuse. If you are in a situation where you’re using an exponent, and there is a ‘bare’ minus sign in front of the thing being raised to a power other than 1, it can be thought of as a (-1) factor in front of the thing being raised to a power that itself is not being raised to that power.

That is, if you have:

-y^k

for some number y and some exponent k, then to remember the order of operations, you effectively have:

xy^k

where x is -1.

Since x is not being raised to a power (or at least, its being raised to the first power) any further expansion is useless.

So I’m afraid I don’t see Race’s point at all.

Having said this, none of this defines the order of operations, really. Properly, it seems that unary negation should occur after exponentiation. Period. It doesn’t matter how you look at it. I just used the (-1) factor as a convenient memory tool.

Brackets
Exponents
Division and
Multiplication
Addition and
Subtraction

Negation could be slotted between exponents and division for BENDMAS. (It should properly go after Multiplication, I suppose, but it doesn’t seem to matter if you negate before or after multiplication: -3*5 and -3/5 is the same regardless of whether you negate first or perform the Multiplication/Divison first).

70

Sounds like a wickedly perverse holiday. Put your stockings up by the chimney and grab your ankles!

Anyway, back to the regular thread.

71

Or something straight out of Futurama…

72

The point is that it’s a circular definition. You’re saying that -1 == (-1)1. You’re defining a term on the left side by using it again on the right side. You need to tell me what the typographical characters “-1” mean.

Is it a number? Just like “12” is two characters that we use to identify twelve? So, 12^2 is 144. Likewise if “-1” is a two character representation of a number, then if we assume we know what “-1” means, then -1^2 should be 1.

73

It is a mnemonic, and nothing else.

If you want a more rigorous definition in that vein, then:

-x^1 == -x
-x^k == (-1)x^k where k!=1

There, that disambiguates it. The expansion that you’re annoyed with is gone.

But it’s just a memory trick. Its not a strict definition on what you should do.

You want it rigorous? Adhere to the BENDMAS order of operations. It is an accepted mathematical convention (and nothing more). Which says do unary negation (as an operation) after exponentiation. (Before multiplication/division, subject to philosophical speculation at this point).

A complete grammatical syntax can be developed in Backus-Naur form if you’re absolutely picky about this.

You can define unary negation as a negative sign that is not a subtraction sign, preceeding an expression, that turns the expression into its additive inverse. And you perform this action after exponentiation, not before.

74

I think you missed my point. I don’t want to “add” an exponent to (1+2/10), I just want to understand why, if exponents take precedence over addition, don’t we do something with the “2^2” part of “1.2^2” before considering the “1.” part.

Of course, the answer is that “1.2” is intended to represent a certain number that we all understand, according to a convention. We must understand what “1.2” means before squaring it. I’m just suggesting that you could also understand that “-3” is a representation of a number that we understand, and not an operation on the number 3.

75

Well, the fact of the matter is, Race, that at SOME point mathematicians decided that while 1.2 represents a value in and of itself as a whole, -3 does not. Whether this was the best idea or not at this point is immaterial.

76

I think the biggest problem with the argument is what I pointed out in my first post in this mess. We’re confusing the mathematics with the syntax that represents the mathematics.

If what you want is the value -3, squared, the notation is:

(-3)^2

If what you want instead is the square of 3, negated, the notation is:

-3^2
The notation is arbitrary, even if the mathematics is not. You can define the notation any way you like. Consider a simple postfix notation for the opening problem. What is the value of:

3 ^ -

The answer, of course, is the additive inverse of 9 (that is, -9). There’s no easy way to have a unary minus in postfix notation, so we might have to come up with a symbology for representing this. We could use _9 to represent the additive inverse of 9. This is syntactically equivalent to saying that _ is a unary negation operator that has the highest possible precedence. So an alternate syntax for postfix could be:

_3 ^

Alternately we could abandon the _ syntax, and just say that the answer to 3 ^ - is:

0 3 -

Anyway, the point is, the syntax can be invented, entirely arbitrarily, provided that we can unambiguously convey to someone else the meaning of the underlying mathematical concept that is universal.

Having said that, the chosen mathematical convention is that a unary minus is an operator that is below and not above exponentiation, but the decimal point, if you want to consider it an operator, is above exponentiation.

Is there a reason?

Nope.

Its entirely arbitrary.

But it is convention which is crucial to conveying ideas to other people. Otherwise you’d have to lay out the ground rules of how to discuss mathematics before any conversation about mathematics.

77

Damn.
Damn, a few goofs. Have go fix my notation.

In the notation I chose:

0 3 2 ^ -

is equivalent to

-3^2
whereas

0 3 - 2 ^

is equivalent to

(-3)^2

Further, if we use _x to represent a high-priority negation of x as Race wants, then:

_3 2 ^

is the same as

(-3)^2

whereas

0 3 2 ^ -

is the same as

-3^2
So my apologies for blowing a lobe in the math there, but the text part of the previous message is what I meant to convey.

78

Go back and read my first couple posts in this thread. There, I defined exactly what “-1” means.

William, your definition doesn’t work for x = 1; it’s circular then.

79

Yes, it is convention, and it is somewhat arbitrary, but it wasn’t defined that way just out of the blue. It was defined that way because it was found to be convenient in many applications to interpret it that way.

80

ultrafilter:

Sigh.

Fine; :smiley:

-1^k == -1 for all k
-x^1 == -x where x!=1
-x^k == (-1)x^k where x!=1 and k!=1

Again I stress that this is pointless as its just a memory mnemonic. Defer to the BENDMAS order of operations, wherein the above ‘rules’ fall out by consequence. yeesh. :smiley:
Richard:
Yes, that’s sort of what I meant. The usefulness of the convention is somewhat inverse to the amount of ambiguity and proportional to the ease of communication. Whatever that means.

Suffice it to say the way we do it isn’t the only way, but it works pretty darn well.