Daughter's math homework (1 Viewer)

[Smoo]

I continue to be unsurprised at the complete lack of attempt to resolve the -1 debacle here. I eagerly await the next "####, that's the way it is, too bad" post.

I thought I already resolved it.

Nope.

Doesn't your -1 debacle rely on pretending that we're factoring? We're not factoring.

Don't call it factoring, then. Call it multiplying.To make -5 = (-1)(5), you're either doing a multiplicative operation or you're stating that -5 is inherently defined as (-1)(5).

1. If you're doing a multiplicative operation, that cannot be done until after the exponent is resolved, which leads to the answer being 25.

2. If you're inherently defining it, then you have to explain why -5 has to be inherently defined as such, but -1 doesn't. Clearly defining -1 in the same way leads to an infinite regression. Why is -1 afforded the exemption that -5 is not?

 

[Smoo]

and there are no parenthesis, the 5^2 is computed first before the "inherent" -1 is applied.

Tense disagreement.There is no parenthesis.

There are no parentheses.

We're having enough trouble with math here, don't bring English errors into it as well.

 

[Boffo]

-a^b is always evaluated as -(a^b). I havent read all 20+ pages of this thread, but I think the main problem with people is the grouping.

"If I evaluate negative 2 squared, I get negative 4."

"If I take negative 2 and square it, I get 4."

"If I square negative 2 , I get 4."

"If I take the opposite of 2 squared, I get negative 4."

All of the above statements are correct. "Negative three squared" is the negative value of the square of three. "The square of negative three" is the square of a negative number, and since a that means it is a neg * neg, the answer must be a pos.

I dont think the confusion here is about mathematical notation, but rather about how to translate English into mathematical notation.

When "-5^2" is typed into a calculator, the calculator does order of operations --> First the exponent "5^2" and then the multiplication by "-1". If you type "(-5)^2" you get the parentheses evaluated first [ which is just -5 or (-1)(5)] and then the result (-5) is squared. That's how it has been taught since I've been in school ... i dont see why it would ever have been different.

 

[Maurile Tremblay]

So Smoo says (possibly rightfully) that this established convention defies logic. I haven't seen anyone address this point, I think, because no one knows the answer. I mean, at some point it became understood that in an algebra equation the term "-5^2" is "(5^2)", and not "-(5^2)".

The convention could go either way. Neither way defies logic.

 

[Smoo]

I want to know why convention was established in a way that defies logic and is inconsistent.

It doesn't defy logic, and it isn't inconsistent."5" is a numeral. "-" is not a numeral. "-" is an operator. That's perfectly consistent.

"-" isn't an operator when it immediately precedes a numeral at the beginning of an expression.

That convention has been rejected. The fact that you disapprove of the rejection doesn't make it inconsistent or illogical.

The convention has been rejected for illogical reasons. The fact that I disapprove of the rejection makes me a super-hero.

 

[Buckychudd]

and there are no parenthesis, the 5^2 is computed first before the "inherent" -1 is applied.

Tense disagreement.There is no parenthesis.

There are no parentheses.

We're having enough trouble with math here, don't bring English errors into it as well.

Who care, thats just a convention that defy logic.

 

[rolyaTy]

Here's the most concise logical explanation I can muster:

The answer is that it doesn't matter that -5 is an integer or not, the negative sign is a direction that modifies the value of 5. Five exists, negative five does not. Negative five is only Five in a specific direction. The negative sign is an operator on the value of five. As it's an operator, it's subject to the order of operations, and comes UNDER the priority of exponents, unless parentheses say otherwise.

roly has provided the only argument so far which has any chance of swaying me.

DON'T DO IT!!

He won't be able to resolve the -1 issue.

I don't see the issue with -1. It's just 1 step in the negative direction. It's a unit vector, used to scale other magnitudes in the same direction.

It's being used as an inherently negative number. All other negative numbers seem to require factoring. This is inconsistent. Math isn't inconsistent.

If you like the vector stuff, -1 is a scaler that keeps the vector being scaled's original magnitude, but changes the direction. It's called a unit vector. The same is being done with numbers. -1 multiplied by another number retains the numbers magnitude, but changes the direction. You have two -1's. -1 as a "number" is one unit in the direction to the left of 0 on the number line. And then you have the -1 as a scaler, which is used to reverse the direction of another number/vector due to the property that 1 times any number is that number, and 1 times and number in the opposite direction is that same number in the opposite direction.

It seems confusing when you break it down to the basics, but it's totally consistant. Negative numbers, even -1, only differ from positive numbers by their signs...and that just denotes direction. If you have no problem with (1)*(5) you should have no problem with (-1)*(5) because it's just saying "The answer is five times one in the negative direction"

 

[Mjolnirs]

Let me see if I can recap here.

In an algebra equation, it should technically read either:

-(5^2) or

(-5)^2

If we wanted to avoid any confusion anyway.

Absent that, there's an established convention that says you read -5^2 as -(5^2). Sounds like this is how it has been since everyone here can remember (those in the know anyway)

And it's this convention some have a problem with. I would be curious as to when and where this convention was established. It seems some of us were taught that without the parens you assume:-5^2 = (-5)^2

while others were taught

-5^2 = -(5^2)

Maybe there wasn't a consensus school of thought on this and some time in the past a decree was established for one over the other.

 

[Bottomfeeder Sports]

By convention does -5 = -(5) or (-5) ?Isn't this the real question?

 

[Swapster]

Let me see if I can recap here.

In an algebra equation, it should technically read either:

-(5^2) or

(-5)^2

If we wanted to avoid any confusion anyway.

Absent that, there's an established convention that says you read -5^2 as -(5^2). Sounds like this is how it has been since everyone here can remember (those in the know anyway)

So Smoo says (possibly rightfully) that this established convention defies logic. I haven't seen anyone address this point, I think, because no one knows the answer. I mean, at some point it became understood that in an algebra equation the term "-5^2" is "(5^2)", and not "-(5^2)".

So why are we still discussing this? At this point, I think it just is what it is. I'm not sure this is all that different from my question earlier this week about one or two spaces after the comma. It's just convention. I'm not sure we always need to find logic in convention.

Thank you.I will now quit thinking about this. It is what it is and it is convention, possibly logical and possibly not.

However, I still think computers are calculating wrong in respect to this convention since if you do not include parenthesis they are assuming negative first (the sign bit is integral to the integer).

I will now quit thinking about this until one of my kids has a test on this and gets it wrong, then I can print this out and send it to school with them.

And I obviously still put 2 spaces after a sentence.

I'm such a loser.

 

[rolyaTy]

I do find it ironic that the person perhaps the most insistant upon consistant grammatical rules, finds fault with a clearly stated math rule. What is there logically consistant about "I before E except after C"? That doesn't seem to keep you up at night Smoo

 

[shuke]

Seriously, this is the dumbest ####ing argument/conversation I've ever been involved in. Later morons.

 

[Paper Lions]

Let me see if I can recap here.

In an algebra equation, it should technically read either:

-(5^2) or

(-5)^2

If we wanted to avoid any confusion anyway.

Absent that, there's an established convention that says you read -5^2 as -(5^2).  Sounds like this is how it has been since everyone here can remember (those in the know anyway)

And it's this convention some have a problem with. I would be curious as to when and where this convention was established. It seems some of us were taught that without the parens you assume:-5^2 = (-5)^2

while others were taught

-5^2 = -(5^2)

Maybe there wasn't a consensus school of thought on this and some time in the past a decree was established for one over the other.

I'm deferring to the actual math teachers on what the actual convention is.

 

[Paper Lions]

Seriously, this is the dumbest ####ing argument/conversation I've ever been involved in. Later morons.

I found it pretty interesting.

 

[Maurile Tremblay]

Don't call it factoring, then. Call it multiplying.

To make -5 = (-1)(5), you're either doing a multiplicative operation or you're stating that -5 is inherently defined as (-1)(5).

No, -5 already is (-1)(5). You don't have to multiply anything to make -5 equivalent to (-1)(5). It already is.

1. If you're doing a multiplicative operation, that cannot be done until after the exponent is resolved, which leads to the answer being 25.

The operation of the "-" is the multiplication. That comes after the exponent is resolved.

2. If you're inherently defining it, then you have to explain why -5 has to be inherently defined as such, but -1 doesn't. Clearly defining -1 in the same way leads to an infinite regression. Why is -1 afforded the exemption that -5 is not?

I wouldn't bother explaining this (at this point) to many other people. But you've got the capacity to get it, so I'll keep trying.-5 and -1 are inherently defined in the exact same way. -5 means the opposite of 5, and -1 means the opposite of 1.

In both cases, the minus sign acts as an operator. The operation it is performing is to make the operand negative (which is an operation of multiplication).

So -5 means "take the value of five, and convert it to its opposite." -1 means "take the value of one, and convert it to its opposite."

The minus sign, as an operator converting the operand to its opposite, comes before addition or subtraction, but after exponentiation.

Thus in the expression "-5^2", the operand of the minus sign is 5^2.

 

[Smoo]

I do find it ironic that the person perhaps the most insistant upon consistant grammatical rules, finds fault with a clearly stated math rule. What is there logically consistant about "I before E except after C"? That doesn't seem to keep you up at night Smoo

Oh, I've been known to fight illogical grammar rules, too. The way I have deemed logical to punctuate around quotations would make every convention-oriented grammaticist cringe.

 

[whoknew]

Seriously, this is the dumbest ####ing argument/conversation I've ever been involved in.  Later morons.

I found it pretty interesting.

Me too. And I agree with Smoo.

 

[Maurile Tremblay]

By convention does

-5 = -(5)

or (-5) ?

Isn't this the real question?

-5 = (-5) = -(5)but -5^2 = -(5^2) < (-5)^2

 

[chet]

I want to know why convention was established in a way that defies logic and is inconsistent.

It doesn't defy logic, and it isn't inconsistent."5" is a numeral. "-" is not a numeral. "-" is an operator. That's perfectly consistent.

If 22 and counting pages can be devoted to figuring out the right answer to a simple question, then the convention isn't logical.

 

[KTM]

22 pages

 

[shuke]

Seriously, this is the dumbest ####ing argument/conversation I've ever been involved in.  Later morons.

I found it pretty interesting.

I find it frustrating. It's like trying to teach to someone that the color of a banana is "yellow" who thinks that's not logical, and that it should be referred to as "purple".

 

[Smoo]

Don't call it factoring, then. Call it multiplying.

To make -5 = (-1)(5), you're either doing a multiplicative operation or you're stating that -5 is inherently defined as (-1)(5).

No, -5 already is (-1)(5). You don't have to multiply anything to make -5 equivalent to (-1)(5). It already is.

1. If you're doing a multiplicative operation, that cannot be done until after the exponent is resolved, which leads to the answer being 25.

The operation of the "-" is the multiplication. That comes after the exponent is resolved.

2. If you're inherently defining it, then you have to explain why -5 has to be inherently defined as such, but -1 doesn't. Clearly defining -1 in the same way leads to an infinite regression. Why is -1 afforded the exemption that -5 is not?

I wouldn't bother explaining this (at this point) to many other people. But you've got the capacity to get it, so I'll keep trying.-5 and -1 are inherently defined in the exact same way. -5 means the opposite of 5, and -1 means the opposite of 1.

In both cases, the minus sign acts as an operator. The operation it is performing is to make the operand negative (which is an operation of multiplication).

So -5 means "take the value of five, and convert it to its opposite." -1 means "take the value of one, and convert it to its opposite."

The minus sign, as an operator converting the operand to its opposite, comes before addition or subtraction, but after exponentiation.

Thus in the expression "-5^2", the operand of the minus sign is 5^2.

So now we're just back to you saying there are no negative numbers, which you tried to deny earlier.

 

[Maurile Tremblay]

Seriously, this is the dumbest ####ing argument/conversation I've ever been involved in. Later morons.

Sixth-grade teacher material for sure.

 

[Maurile Tremblay]

Don't call it factoring, then. Call it multiplying.

To make -5 = (-1)(5), you're either doing a multiplicative operation or you're stating that -5 is inherently defined as (-1)(5).

No, -5 already is (-1)(5). You don't have to multiply anything to make -5 equivalent to (-1)(5). It already is.

1. If you're doing a multiplicative operation, that cannot be done until after the exponent is resolved, which leads to the answer being 25.

The operation of the "-" is the multiplication. That comes after the exponent is resolved.

2. If you're inherently defining it, then you have to explain why -5 has to be inherently defined as such, but -1 doesn't. Clearly defining -1 in the same way leads to an infinite regression. Why is -1 afforded the exemption that -5 is not?

I wouldn't bother explaining this (at this point) to many other people. But you've got the capacity to get it, so I'll keep trying.-5 and -1 are inherently defined in the exact same way. -5 means the opposite of 5, and -1 means the opposite of 1.

In both cases, the minus sign acts as an operator. The operation it is performing is to make the operand negative (which is an operation of multiplication).

So -5 means "take the value of five, and convert it to its opposite." -1 means "take the value of one, and convert it to its opposite."

The minus sign, as an operator converting the operand to its opposite, comes before addition or subtraction, but after exponentiation.

Thus in the expression "-5^2", the operand of the minus sign is 5^2.

So now we're just back to you saying there are no negative numbers, which you tried to deny earlier.

There are negative numbers. Negative five, for example, is a negative number.

There are no negative numerals (although, as I mentioned before, I think that Prince symbol is available for cooption).

 

[Paper Lions]

A couple of posts from Smoo in the "one space or two space" after a period thread. Smoo, I'm not posting this to dog on you at all, I just really am seeing the similarity here. Convention is convention. Why bother looking for the logic?

1 is correct, 2 looks better, and nobody cares.

behold.

Give me a break with these MLA Zealots.They make me want to puke with their grandstanding, and strongarm tactics.

I'm picturing a small, bespectacled bald man in a tweed coat looking in a mirror and saying "Are you addressing me?" over and over again. Language goons...

At least someone got it.Grammar geeks are WAY too serious.

And people who can't follow simple rules like grammar are mostly stupid.

 

[Boffo]

Don't call it factoring, then. Call it multiplying.

To make -5 = (-1)(5), you're either doing a multiplicative operation or you're stating that -5 is inherently defined as (-1)(5).

No, -5 already is (-1)(5). You don't have to multiply anything to make -5 equivalent to (-1)(5). It already is.

1. If you're doing a multiplicative operation, that cannot be done until after the exponent is resolved, which leads to the answer being 25.

The operation of the "-" is the multiplication. That comes after the exponent is resolved.

2. If you're inherently defining it, then you have to explain why -5 has to be inherently defined as such, but -1 doesn't. Clearly defining -1 in the same way leads to an infinite regression. Why is -1 afforded the exemption that -5 is not?

I wouldn't bother explaining this (at this point) to many other people. But you've got the capacity to get it, so I'll keep trying.-5 and -1 are inherently defined in the exact same way. -5 means the opposite of 5, and -1 means the opposite of 1.

In both cases, the minus sign acts as an operator. The operation it is performing is to make the operand negative (which is an operation of multiplication).

So -5 means "take the value of five, and convert it to its opposite." -1 means "take the value of one, and convert it to its opposite."

The minus sign, as an operator converting the operand to its opposite, comes before addition or subtraction, but after exponentiation.

Thus in the expression "-5^2", the operand of the minus sign is 5^2.

This is correct. End thread

 

[Paper Lions]

Seriously, this is the dumbest ####ing argument/conversation I've ever been involved in.  Later morons.

Sixth-grade teacher material for sure.

and :X all at the same time.It's funny that I'm going to have to re-learn this #### at some point when the little dudes get a little older.

 

[Smoo]

There are negative numbers. Negative five, for example, is a negative number.

Yes, and if you square it, you get 25, not -25.

 

[shuke]

A couple of posts from Smoo in the "one space or two space" after a period thread. Smoo, I'm not posting this to dog on you at all, I just really am seeing the similarity here. Convention is convention. Why bother looking for the logic?

1 is correct, 2 looks better, and nobody cares.

behold.

Give me a break with these MLA Zealots.They make me want to puke with their grandstanding, and strongarm tactics.

I'm picturing a small, bespectacled bald man in a tweed coat looking in a mirror and saying "Are you addressing me?" over and over again. Language goons...

At least someone got it.Grammar geeks are WAY too serious.

And people who can't follow simple rules like grammar are mostly stupid.

 

[Ignoramus]

Seriously, this is the dumbest ####ing argument/conversation I've ever been involved in. Later morons.

I found it pretty interesting.

I find it frustrating. It's like trying to teach to someone that the color of a banana is "yellow" who thinks that's not logical, and that it should be referred to as "purple".

 

[Smoo]

This is correct. End thread

We know it's correct by convention. We established that on page 3. Now explain why it's correct by reason. That's taken much longer and is still not resolved.

 

[Smoo]

A couple of posts from Smoo in the "one space or two space" after a period thread. Smoo, I'm not posting this to dog on you at all, I just really am seeing the similarity here. Convention is convention. Why bother looking for the logic?

1 is correct, 2 looks better, and nobody cares.

behold.

Give me a break with these MLA Zealots.They make me want to puke with their grandstanding, and strongarm tactics.

I'm picturing a small, bespectacled bald man in a tweed coat looking in a mirror and saying "Are you addressing me?" over and over again. Language goons...

At least someone got it.Grammar geeks are WAY too serious.

And people who can't follow simple rules like grammar are mostly stupid.

 

[rolyaTy]

Here's the most concise logical explanation I can muster:

The answer is that it doesn't matter that -5 is an integer or not, the negative sign is a direction that modifies the value of 5. Five exists, negative five does not. Negative five is only Five in a specific direction. The negative sign is an operator on the value of five. As it's an operator, it's subject to the order of operations, and comes UNDER the priority of exponents, unless parentheses say otherwise.

roly has provided the only argument so far which has any chance of swaying me.

DON'T DO IT!!

He won't be able to resolve the -1 issue.

I don't see the issue with -1. It's just 1 step in the negative direction. It's a unit vector, used to scale other magnitudes in the same direction.

It's being used as an inherently negative number. All other negative numbers seem to require factoring. This is inconsistent. Math isn't inconsistent.

If you like the vector stuff, -1 is a scaler that keeps the vector being scaled's original magnitude, but changes the direction. It's called a unit vector. The same is being done with numbers. -1 multiplied by another number retains the numbers magnitude, but changes the direction. You have two -1's. -1 as a "number" is one unit in the direction to the left of 0 on the number line. And then you have the -1 as a scaler, which is used to reverse the direction of another number/vector due to the property that 1 times any number is that number, and 1 times and number in the opposite direction is that same number in the opposite direction.

It seems confusing when you break it down to the basics, but it's totally consistant. Negative numbers, even -1, only differ from positive numbers by their signs...and that just denotes direction. If you have no problem with (1)*(5) you should have no problem with (-1)*(5) because it's just saying "The answer is five times one in the negative direction"

If you want to compare it to grammar, it's like a "Not". Not by itself doesn't have meaning, but when you couple it to another word, it signifies the opposite of that meaning, like "Not happy" or "not good at math". If you take "not good at math" and break it down, what does "not" modify? Does it simply modify "good", as in , maybe their behavior is "not good" while in math class? Or does it mean "not (good at math)". In grammar, the mistake being discussed here is called a misplaced modifier I think, or a dangling something or another, you'd know better than I would, but the same concept holds true here.

Not is like a negative sign. Neither have meaning of their own, but when applied to a word or a value, they have meaning. What words or values they operate on, depend on the sentence structure and on the mathematical formula's structure, respectively.

If I were to say "You are always not right about it", it's clear that I'm saying you're always not right ABOUT "it", because of whatever grammar rules state that is what the sentence is saying.

Anyways, I"m losing my train of thought here, but for the same reason grammar has an order of operations, math does too. And just like the word "not", a negative sign has no meaning on its own. It's only a modifier, or an operator upon a magnitude.

 

[shuke]

This is correct.  End thread 

We know it's correct by convention. We established that on page 3. Now explain why it's correct by reason. That's taken much longer and is still not resolved.

There isn't one. If it was correct by reason, there wouldn't be a need to establish a convention.

 

[Paper Lions]

This is correct.  End thread 

We know it's correct by convention. We established that on page 3. Now explain why it's correct by reason. That's taken much longer and is still not resolved.

I agree with you here...but why bother? Leaving out the parentheticals causes confusion, right? You and MT disagree on the logic of it, right? So you have to pick a single convention so that one will be correct, right?I think it pretty much is what it is here.

 

[Buckychudd]

This is correct.  End thread 

We know it's correct by convention. We established that on page 3. Now explain why it's correct by reason. That's taken much longer and is still not resolved.

Please explain why we multiply before we subtract. I know it's by convention, but I don't understand the reasoning.

 

[jwvdcw]

Did we ever get a ruling here? Did Shick ever stop in?

 

[Maurile Tremblay]

There are negative numbers. Negative five, for example, is a negative number.

Yes, and if you square it, you get 25, not -25.

Exactly right. If you square negative five, you get 25.But in the expression -5^2, you are not squaring negative five.

In the expression 23^2, you are not squaring three.

The operation of the minus sign comes after the operation of the exponent. The operation of converting "23" into "(2*10 + 3*1)" comes before the operation of the exponent.

This is not inconsistent or illogical. "2" and "3" are numerals, as is "5". "-" is not a numeral.

Converting paired numerals into a single value comes before any operators are applied.

Then operators are applied in (arbitrary) order. According to that order, exponentiation comes before multiplication. "^" is exponentiation. "-" is multiplication.

 

[IvanKaramazov]

Have I told anybody about the kid in my daughter's preschool who's allergic to peanuts and everybody else has to rearrange their lives to accomodate him? I think I should start a thread about that.

 

[Smoo]

Here's the most concise logical explanation I can muster:

The answer is that it doesn't matter that -5 is an integer or not, the negative sign is a direction that modifies the value of 5. Five exists, negative five does not. Negative five is only Five in a specific direction. The negative sign is an operator on the value of five. As it's an operator, it's subject to the order of operations, and comes UNDER the priority of exponents, unless parentheses say otherwise.

roly has provided the only argument so far which has any chance of swaying me.

DON'T DO IT!!

He won't be able to resolve the -1 issue.

I don't see the issue with -1. It's just 1 step in the negative direction. It's a unit vector, used to scale other magnitudes in the same direction.

It's being used as an inherently negative number. All other negative numbers seem to require factoring. This is inconsistent. Math isn't inconsistent.

If you like the vector stuff, -1 is a scaler that keeps the vector being scaled's original magnitude, but changes the direction. It's called a unit vector. The same is being done with numbers. -1 multiplied by another number retains the numbers magnitude, but changes the direction. You have two -1's. -1 as a "number" is one unit in the direction to the left of 0 on the number line. And then you have the -1 as a scaler, which is used to reverse the direction of another number/vector due to the property that 1 times any number is that number, and 1 times and number in the opposite direction is that same number in the opposite direction.

It seems confusing when you break it down to the basics, but it's totally consistant. Negative numbers, even -1, only differ from positive numbers by their signs...and that just denotes direction. If you have no problem with (1)*(5) you should have no problem with (-1)*(5) because it's just saying "The answer is five times one in the negative direction"

If you want to compare it to grammar, it's like a "Not". Not by itself doesn't have meaning, but when you couple it to another word, it signifies the opposite of that meaning, like "Not happy" or "not good at math". If you take "not good at math" and break it down, what does "not" modify? Does it simply modify "good", as in , maybe their behavior is "not good" while in math class? Or does it mean "not (good at math)". In grammar, the mistake being discussed here is called a misplaced modifier I think, or a dangling something or another, you'd know better than I would, but the same concept holds true here.

Not is like a negative sign. Neither have meaning of their own, but when applied to a word or a value, they have meaning. What words or values they operate on, depend on the sentence structure and on the mathematical formula's structure, respectively.

If I were to say "You are always not right about it", it's clear that I'm saying you're always not right ABOUT "it", because of whatever grammar rules state that is what the sentence is saying.

Anyways, I"m losing my train of thought here, but for the same reason grammar has an order of operations, math does too. And just like the word "not", a negative sign has no meaning on its own. It's only a modifier, or an operator upon a magnitude.

It's not an equivalent issue."not (good at math)" is the same as "(not good) at math"

 

[Paper Lions]

I was not trying to be an ###. Do you see a similarity Smoo?

 

[Buckychudd]

Have I told anybody about the kid in my daughter's preschool who's allergic to peanuts and everybody else has to rearrange their lives to accomodate him?

I think I should start a thread about that.

Pffft, wouldn't even get one response.

 

[Maurile Tremblay]

Did we ever get a ruling here? Did Shick ever stop in?

Yes. Several math teachers (including Shick! and Clayton) stopped in. Among the math teachers, it's unanimous that -5^2 = -25. But among engineers and other people who use math in practical applications, there was no unanimity, and the bulk of them seemed to favor the notion that -5^2 = 25.

 

[jonessed]

When "-5^2" is typed into a calculator, the calculator does order of operations --> First the exponent "5^2" and then the multiplication by "-1".  If you type "(-5)^2" you get the parentheses evaluated first [ which is just -5 or (-1)(5)] and then the result (-5) is squared.  That's how it has been taught since I've been in school ... i dont see why it would ever have been different.

My HP 48G evaluates -5^2 (no parantheses) as 25. Apparently HPs mathematicians got it wrong too? I don't know of a more commonly used calculator amongst engineering students. The argument that this is somehow the standard is ludicrous. It *may* be correct, but that interpretation is apparently stuck in the classroom.

Edit: And not college classroom? Strange. Most engineers I know are not that far removed from college (4-10 years). I don't think it's being taught that way there either because the few I have talked to evaluated it as 25. It's not like we don't take a ton of math courses. Physics and engineering are more practical applications though so maybe I just don't remember. Even something as simple as vector notations are different.

Just thinking out loud.

 

[rolyaTy]

Here's the most concise logical explanation I can muster:

The answer is that it doesn't matter that -5 is an integer or not, the negative sign is a direction that modifies the value of 5. Five exists, negative five does not. Negative five is only Five in a specific direction. The negative sign is an operator on the value of five. As it's an operator, it's subject to the order of operations, and comes UNDER the priority of exponents, unless parentheses say otherwise.

roly has provided the only argument so far which has any chance of swaying me.

DON'T DO IT!!

He won't be able to resolve the -1 issue.

I don't see the issue with -1. It's just 1 step in the negative direction. It's a unit vector, used to scale other magnitudes in the same direction.

It's being used as an inherently negative number. All other negative numbers seem to require factoring. This is inconsistent. Math isn't inconsistent.

If you like the vector stuff, -1 is a scaler that keeps the vector being scaled's original magnitude, but changes the direction. It's called a unit vector. The same is being done with numbers. -1 multiplied by another number retains the numbers magnitude, but changes the direction. You have two -1's. -1 as a "number" is one unit in the direction to the left of 0 on the number line. And then you have the -1 as a scaler, which is used to reverse the direction of another number/vector due to the property that 1 times any number is that number, and 1 times and number in the opposite direction is that same number in the opposite direction.

It seems confusing when you break it down to the basics, but it's totally consistant. Negative numbers, even -1, only differ from positive numbers by their signs...and that just denotes direction. If you have no problem with (1)*(5) you should have no problem with (-1)*(5) because it's just saying "The answer is five times one in the negative direction"

If you want to compare it to grammar, it's like a "Not". Not by itself doesn't have meaning, but when you couple it to another word, it signifies the opposite of that meaning, like "Not happy" or "not good at math". If you take "not good at math" and break it down, what does "not" modify? Does it simply modify "good", as in , maybe their behavior is "not good" while in math class? Or does it mean "not (good at math)". In grammar, the mistake being discussed here is called a misplaced modifier I think, or a dangling something or another, you'd know better than I would, but the same concept holds true here.

Not is like a negative sign. Neither have meaning of their own, but when applied to a word or a value, they have meaning. What words or values they operate on, depend on the sentence structure and on the mathematical formula's structure, respectively.

If I were to say "You are always not right about it", it's clear that I'm saying you're always not right ABOUT "it", because of whatever grammar rules state that is what the sentence is saying.

Anyways, I"m losing my train of thought here, but for the same reason grammar has an order of operations, math does too. And just like the word "not", a negative sign has no meaning on its own. It's only a modifier, or an operator upon a magnitude.

It's not an equivalent issue."not (good at math)" is the same as "(not good) at math"

Yeah, I can't come up with an example right now but grammar has its own order of operations too, and if you don't follow those, you get screwy results. I was mainly going with the "not" doesn't have meaning itself, and equating that with a negative sign, when I got sidetracked by grammatical order of operations. My head hurts.

 

[Smoo]

There are negative numbers. Negative five, for example, is a negative number.

Yes, and if you square it, you get 25, not -25.

Exactly right. If you square negative five, you get 25.But in the expression -5^2, you are not squaring negative five.

In the expression 23^2, you are not squaring three.

The operation of the minus sign comes after the operation of the exponent. The operation of converting "23" into "(2*10 + 3*1) comes before the operation of the exponent.

This is not inconsistent or illogical. "2" and "3" are numerals, as is "5". "-" is not a numeral.

Converting paired numerals into a single value comes before any operators are applied.

Then operators are applied in (arbitrary) order. According to that order, exponentiation comes before multiplication. "^" is exponentiation. "-" is multiplication.

You're saying that negative numbers do not exist inherently, only as the result of a multiplicative operation performed on a numeral.

 

[Paper Lions]

Seriously, this is the dumbest ####ing argument/conversation I've ever been involved in.  Later morons.

I found it pretty interesting.

I find it frustrating. It's like trying to teach to someone that the color of a banana is "yellow" who thinks that's not logical, and that it should be referred to as "purple".

BTW, I've seriously always wondered about this. What if my yellow is really your red? I've come to know my yellow as "yellow" because I've been taught that, but you've come to know your red as "yellow" because you've been taught that. We could be seeing two totally different colors in our brains and both be referring to them as yellow.

 

[Smoo]

I was not trying to be an ###. Do you see a similarity Smoo?

No.

 

[Bottomfeeder Sports]

By convention  does

-5 = -(5)

or    (-5)  ?

Isn't this the real question?

-5 = (-5) = -(5)but -5^2 = -(5^2) < (-5)^2

Aren't you really arguing that -5^2 =-(5)^2

and Smoo is arguing that

(-5)^2

is more intuitive, more concise, more logical?

If you are arguing that only in the case of -5^2 is -5 really -(5) then the convention is stupid. But if -5 always is -(5), like I think you are arguing then the convention has nothing to do with exponents, but how to interpret the -5.

 

[Maurile Tremblay]

My HP 48G evaluates -5^2 (no parantheses) as 25. Apparently HPs mathematicians got it wrong too? I don't know of a more commonly used calculator amongst engineering students.

MS Excel does the same thing, but their programmers specifically acknowledged that they were flouting convention.