Daughter's math homework (1 Viewer)

[rick6668]

Ok, I don't know about all this no negative number crap, but can someone please answer me this!

What =(-1)^(1/2)

i

 

[Kleck]

Ok, I don't know about all this no negative number crap, but can someone please answer me this!

What =(-1)^(1/2)

i

imagine that.

 

[Kleck]

Ok, I don't know about all this no negative number crap, but can someone please answer me this!

What =(-1)^(1/2)

i

On a similar note...-1^(1/2)=-1

 

[Smoo]

-1 is not a numeral. 1 is a numeral expressing the value of one. -1 is an expression containing two characters (an operator and a numeral) that means the opposite of the value of one.

So it actually means (-1)(1). That's exactly what you're saying.Infinite regression.

It means the opposite of one, which can also (but doesn't have to be) written as (-1)(1).Similarly, 1 can be written as (1)(1) -- but that doesn't turn it into an infinite regression, either.

We're not talking about how something can be written.It has been said that -5^2 = -25 because -5 isn't really -5, it's actually (-1)(5).

If that is a rule that must apply to all negative representations, then it must also apply to -1. -1 = (-1)(1) = (-1)(1)(1) = (-1)(1)(1)(1) = etc...

 

[Buckychudd]

Ok, I don't know about all this no negative number crap, but can someone please answer me this!

What =(-1)^(1/2)

i

On a similar note...-1^(1/2)=-1

So it's not imaginary after all?

 

[Rooster]

An interesting side note.

I'm an engineer (about 7 years removed from college) and answered 25 (to the solution of -5^2 in the first problem), but I don't really practive traditional engineering so I figured I might be a bit out of the loop. I tossed a friend of mine who currently works for a civil engineering firm this problem and he also answered 25. We then became curious as to how the other practicing engineers (in this case civil and structural) would answer given the same notation and he sent out an email to his peers. He had 18 replies and all of them were 25 as well. (Update: it's 17 and 3 now; someone changed their first answer and two new ones came in )

That's not to say that this is the proper arithmetic nor is this obviously a satisfactory sampling of industry (given the confusion in the thread a large sampling would undoubtedly give answers going both ways), BUT I think it is important to note that if it doesn't translate to where people will likely be using it then it doesn't mean much. At the very least there is some serious confusion and it doesn't stop with just Mom and Dad remembering High School math .

Edit: The answer of -25 seems accurate to me now given the logic. I don't think it's most people's off the cuff answer though unless they are teaching it. Just a guess.

This post needs more love. Add one more engineer to your list that got it wrong the first time.

I submit that because so many people you would expect to get this problem right disagree, the "standard" has failed and parentheses should be used to avoid confusion. Nobody arguing is an idiot, we all understand the technical part of math (order of operations, - * - = +, etc.) as it relates to this problem. Where this problem is failing is communicating the intent of the expression.

 

[Paper Lions]

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.

I've been convinced, largely on rolyaTy's discussion. I read your stuff too Smoo. Interesting debate.

 

[tommyboy]

I think I have proved beyond a reasonable doubt in just 2 little posts how ####### stupid math has become without parenthesis to clearly indicate what the operative of the - sign.

 

[Rooster]

I think I have proved beyond a reasonable doubt in just 2 little posts how ####### stupid math has become without parenthesis to clearly indicate what the operative of the - sign.

[sidetrack]What we have here is failure to communicate.

Some men you just can't reach.

[/sidetrack]

 

[MrPack]

STOP THE MADNESS!!!!!WHAT HAVE I CREATED!!!!!!! :headache:

 

[(HULK)]

An interesting side note. 

I'm an engineer (about 7 years removed from college) and answered 25 (to the solution of -5^2 in the first problem), but I don't really practive traditional engineering so I figured I might be a bit out of the loop.  I tossed a friend of mine who currently works for a civil engineering firm this problem and he also answered 25.  We then became curious as to how the other practicing engineers (in this case civil and structural) would answer given the same notation and he sent out an email to his peers.  He had 18 replies and all of them were 25 as well.  (Update:  it's 17 and 3 now; someone changed their first answer and two new ones came in )

That's not to say that this is the proper arithmetic nor is this obviously a satisfactory sampling of industry (given the confusion in the thread a large sampling would undoubtedly give answers going both ways), BUT I think it is important to note that if it doesn't translate to where people will likely be using it then it doesn't mean much.  At the very least there is some serious confusion and it doesn't stop with just Mom and Dad remembering High School math .

Edit:  The answer of -25 seems accurate to me now given the logic.  I don't think it's most people's off the cuff answer though unless they are teaching it.  Just a guess.

This post needs more love. Add one more engineer to your list that got it wrong the first time.

I submit that because so many people you would expect to get this problem right disagree, the "standard" has failed and parentheses should be used to avoid confusion. Nobody arguing is an idiot, we all understand the technical part of math (order of operations, - * - = +, etc.) as it relates to this problem. Where this problem is failing is communicating the intent of the expression.

It seems to me that this "standard" was set not because it followed good logic, but rather as a lazy way to save time.

 

[Smoo]

It seems to me that this "standard" was set not because it followed good logic, but rather as a lazy way to save time.

 

[Kleck]

It seems to me that this "standard" was set not because it followed good logic, but rather as a lazy way to save time.

The standard isn't to write it this way: -5^2The standard is to interpret it in a specific way when its encountered.

 

[Smoo]

It seems to me that this "standard" was set not because it followed good logic, but rather as a lazy way to save time.

The standard isn't to write it this way: -5^2The standard is to interpret it in a specific way when its encountered.

A specific way that defies logic. That doesn't sound very mathematical.

 

[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.

 

[Kleck]

It seems to me that this "standard" was set not because it followed good logic, but rather as a lazy way to save time.

The standard isn't to write it this way: -5^2The standard is to interpret it in a specific way when its encountered.

A specific way that defies logic. That doesn't sound very mathematical.

The order of operations defy logic? Exponents before multiplication.

 

[tommyboy]

on a simple question there were two different answers given, one was - and one was +.

 

[Smoo]

It seems to me that this "standard" was set not because it followed good logic, but rather as a lazy way to save time.

The standard isn't to write it this way: -5^2The standard is to interpret it in a specific way when its encountered.

A specific way that defies logic. That doesn't sound very mathematical.

The order of operations defy logic? Exponents before multiplication.

Exactly. You have to square the -5 before you can factor out the -1.

 

[shuke]

If -5 cannot be -5 and has to be (-1)(5) then all you've done is create an infinite regression which makes solving the expression impossible. Because then -1 cannot be -1, it has to be (-1)(1). But then the -1 in that expression cannot be -1, it also has to be (-1)(1). And so on. It is clear at some point that you have to accept that a number can be inherently negative. And once you've made that assumption, there's no need for the initial factoring, the -5 can be inherently negative.

I don't know about everyone else, but I'm not trying to argue that -5 isn't an inherently negative integer.The point, when negative numbers are used in algebraic expressions, they should have parentheses around them otherwise you would not know which of the following is implied:

-(5^2)=x

(-5)^2=x

HOW MOTHER ####ING DIFFICULT IS THIS TO UNDERSTAND?

 

[bryhamm]

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.

####, that's the way it is, too bad

 

[Swapster]

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.

I've been convinced, largely on rolyaTy's discussion. I read your stuff too Smoo. Interesting debate.

This answer convinced me, too because I'm picturing a graph, with -5, then -25, then -125 - it should all go in the downward direction like 5, 25, 125. It isn't intuitive for it to go -5, 25, -125 on a graph. BUT

A negative times a negative is a positive, so the graph should jump above and below the x line.

I can't see how you can factor out the - as in:

-5^2

-1(5*5)

-1(25)

-25

But not then have to apply this to all negative multiplied by negatives as in:

-3*-2

-1(3*2)

-1(6)

-6

Does this not follow the same logic?

 

[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.

####, that's the way it is, too bad

 

[Smoo]

If -5 cannot be -5 and has to be (-1)(5) then all you've done is create an infinite regression which makes solving the expression impossible. Because then -1 cannot be -1, it has to be (-1)(1). But then the -1 in that expression cannot be -1, it also has to be (-1)(1). And so on. It is clear at some point that you have to accept that a number can be inherently negative. And once you've made that assumption, there's no need for the initial factoring, the -5 can be inherently negative.

I don't know about everyone else, but I'm not trying to argue that -5 isn't an inherently negative integer.The point, when negative numbers are used in algebraic expressions, they should have parentheses around them otherwise you would not know which of the following is implied:

-(5^2)=x

(-5)^2=x

HOW MOTHER ####ING DIFFICULT IS THIS TO UNDERSTAND?

If the number is inherently negative, it should not require parentheses.HOW MOTHER ####ING DIFFICULT IS THAT TO UNDERSTAND?

 

[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.

I've been convinced, largely on rolyaTy's discussion. I read your stuff too Smoo. Interesting debate.

This answer convinced me, too because I'm picturing a graph, with -5, then -25, then -125 - it should all go in the downward direction like 5, 25, 125. It isn't intuitive for it to go -5, 25, -125 on a graph. BUT

A negative times a negative is a positive, so the graph should jump above and below the x line.

I can't see how you can factor out the - as in:

-5^2

-1(5*5)

-1(25)

-25

But not then have to apply this to all negative multiplied by negatives as in:

-3*-2

-1(3*2)

-1(6)

-6

Does this not follow the same logic?

 

[Kleck]

If -5 cannot be -5 and has to be (-1)(5) then all you've done is create an infinite regression which makes solving the expression impossible.  Because then -1 cannot be -1, it has to be (-1)(1).  But then the -1 in that expression cannot be -1, it also has to be (-1)(1).  And so on.  It is clear at some point that you have to accept that a number can be inherently negative.  And once you've made that assumption, there's no need for the initial factoring, the -5 can be inherently negative.

I don't know about everyone else, but I'm not trying to argue that -5 isn't an inherently negative integer.The point, when negative numbers are used in algebraic expressions, they should have parentheses around them otherwise you would not know which of the following is implied:

-(5^2)=x

(-5)^2=x

HOW MOTHER ####ING DIFFICULT IS THIS TO UNDERSTAND?

:VisionsOfShukeTeachingMathToHighSchoolKids:

 

[Smoo]

HOW MOTHER ####ING DIFFICULT IS THIS TO UNDERSTAND?

:VisionsOfShukeTeachingMathToHighSchoolKids:

 

[shuke]

If -5 cannot be -5 and has to be (-1)(5) then all you've done is create an infinite regression which makes solving the expression impossible.  Because then -1 cannot be -1, it has to be (-1)(1).  But then the -1 in that expression cannot be -1, it also has to be (-1)(1).  And so on.  It is clear at some point that you have to accept that a number can be inherently negative.  And once you've made that assumption, there's no need for the initial factoring, the -5 can be inherently negative.

I don't know about everyone else, but I'm not trying to argue that -5 isn't an inherently negative integer.The point, when negative numbers are used in algebraic expressions, they should have parentheses around them otherwise you would not know which of the following is implied:

-(5^2)=x

(-5)^2=x

HOW MOTHER ####ING DIFFICULT IS THIS TO UNDERSTAND?

If the number is inherently negative, it should not require parentheses.HOW MOTHER ####ING DIFFICULT IS THAT TO UNDERSTAND?

Then how do you know if -5^2 refers to the negative of 5 squared or negative five squared?I don't understand why you are arguing about this. It's just a matter of convention that has been established.

 

[Kleck]

But not then have to apply this to all negative multiplied by negatives as in:

-3*-2

-1(3*2)

-1(6)

-6

Does this not follow the same logic?

Factoring involves dividing a common item from two separate terms. (3*2) is one term, not two. Terms are separated by addition or subtration operators.

 

[bryhamm]

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.

I've been convinced, largely on rolyaTy's discussion. I read your stuff too Smoo. Interesting debate.

This answer convinced me, too because I'm picturing a graph, with -5, then -25, then -125 - it should all go in the downward direction like 5, 25, 125. It isn't intuitive for it to go -5, 25, -125 on a graph. BUT

A negative times a negative is a positive, so the graph should jump above and below the x line.

I can't see how you can factor out the - as in:

-5^2

-1(5*5)

-1(25)

-25

But not then have to apply this to all negative multiplied by negatives as in:

-3*-2

-1(3*2)

-1(6)

-6

Does this not follow the same logic?

No, you cannot factor multiplication out from other multiplication. The bolded section is wrong. Using your process, then the following would be true:8*4

2(4*2)

Obviously you get 2 different answers.

A proper way of factoring out -1 would be:

3 - 2

-1(-3 + 2)

 

[Maurile Tremblay]

An interesting side note.

I'm an engineer (about 7 years removed from college) and answered 25 (to the solution of -5^2 in the first problem), but I don't really practive traditional engineering so I figured I might be a bit out of the loop. I tossed a friend of mine who currently works for a civil engineering firm this problem and he also answered 25. We then became curious as to how the other practicing engineers (in this case civil and structural) would answer given the same notation and he sent out an email to his peers. He had 18 replies and all of them were 25 as well. (Update: it's 17 and 3 now; someone changed their first answer and two new ones came in )

That's not to say that this is the proper arithmetic nor is this obviously a satisfactory sampling of industry (given the confusion in the thread a large sampling would undoubtedly give answers going both ways), BUT I think it is important to note that if it doesn't translate to where people will likely be using it then it doesn't mean much. At the very least there is some serious confusion and it doesn't stop with just Mom and Dad remembering High School math .

Edit: The answer of -25 seems accurate to me now given the logic. I don't think it's most people's off the cuff answer though unless they are teaching it. Just a guess.

This post needs more love. Add one more engineer to your list that got it wrong the first time.

I agree -- that's an important post. I think it proves pretty persuasively that, no matter what the people on either side say, the original expression was ambiguous.It's not ambiguous under either proposed convention. But which convention is preferred is itself ambiguous.

 

[Smoo]

If -5 cannot be -5 and has to be (-1)(5) then all you've done is create an infinite regression which makes solving the expression impossible. Because then -1 cannot be -1, it has to be (-1)(1). But then the -1 in that expression cannot be -1, it also has to be (-1)(1). And so on. It is clear at some point that you have to accept that a number can be inherently negative. And once you've made that assumption, there's no need for the initial factoring, the -5 can be inherently negative.

I don't know about everyone else, but I'm not trying to argue that -5 isn't an inherently negative integer.The point, when negative numbers are used in algebraic expressions, they should have parentheses around them otherwise you would not know which of the following is implied:

-(5^2)=x

(-5)^2=x

HOW MOTHER ####ING DIFFICULT IS THIS TO UNDERSTAND?

If the number is inherently negative, it should not require parentheses.HOW MOTHER ####ING DIFFICULT IS THAT TO UNDERSTAND?

Then how do you know if -5^2 refers to the negative of 5 squared or negative five squared?I don't understand why you are arguing about this. It's just a matter of convention that has been established.

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

 

[Kleck]

I don't understand why you are arguing about this.

Because he can, and he wants to understand the convention in some way that makes sense to him. Is it not the job of math teachers everywhere to 1) establish a standard and 2) clearly explain its meaning?

 

[Maurile Tremblay]

It seems to me that this "standard" was set not because it followed good logic, but rather as a lazy way to save time.

The standard isn't to write it this way: -5^2The standard is to interpret it in a specific way when its encountered.

A specific way that defies logic. That doesn't sound very mathematical.

The order of operations defy logic? Exponents before multiplication.

Exactly. You have to square the -5 before you can factor out the -1.

You're not "factoring out" the -1. What you're doing is applying the "-" sign, which has the same effect as multiplying by -1. There is no factoring.Moreover, the application of the "-" sign comes after the application of the "^" sign.

 

[Maurile Tremblay]

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.

 

[munga30]

I am *El Cyclone,* from... Bolivia. One-man gang. This classroom is *my* domain. Don't give me no gas, or I'll jump on your face and tattoo your chromosomes... If the only thing you know how to do is add and subtract, you will only be prepared to do one thing: Pump gas.

Nobody??? Aw....

 

[Maurile Tremblay]

If -5 cannot be -5 and has to be (-1)(5) then all you've done is create an infinite regression which makes solving the expression impossible. Because then -1 cannot be -1, it has to be (-1)(1). But then the -1 in that expression cannot be -1, it also has to be (-1)(1). And so on. It is clear at some point that you have to accept that a number can be inherently negative. And once you've made that assumption, there's no need for the initial factoring, the -5 can be inherently negative.

I don't know about everyone else, but I'm not trying to argue that -5 isn't an inherently negative integer.The point, when negative numbers are used in algebraic expressions, they should have parentheses around them otherwise you would not know which of the following is implied:

-(5^2)=x

(-5)^2=x

HOW MOTHER ####ING DIFFICULT IS THIS TO UNDERSTAND?

I agree with shuke. Negative five is an inherently negative number. But there is no negative five expressed in "-5^2".

 

[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.

 

[Swapster]

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.

I've been convinced, largely on rolyaTy's discussion. I read your stuff too Smoo. Interesting debate.

This answer convinced me, too because I'm picturing a graph, with -5, then -25, then -125 - it should all go in the downward direction like 5, 25, 125. It isn't intuitive for it to go -5, 25, -125 on a graph. BUT

A negative times a negative is a positive, so the graph should jump above and below the x line.

I can't see how you can factor out the - as in:

-5^2

-1(5*5)

-1(25)

-25

But not then have to apply this to all negative multiplied by negatives as in:

-3*-2

-1(3*2)

-1(6)

-6

Does this not follow the same logic?

No, you cannot factor multiplication out from other multiplication. The bolded section is wrong. Using your process, then the following would be true:8*4

2(4*2)

Obviously you get 2 different answers.

A proper way of factoring out -1 would be:

3 - 2

-1(-3 + 2)

Exactly!I knew the logic was flawed, but I'm asserting the logic in the first part is the same as the 2nd. You're taking -1 out of (5*5) when you say the -5^2 is really -1 * 5^2. 5^2 is 5x5. 5^2 is really just shorthand for 5*5, is it not? So, why can you factor multiplication out of a multiplication problem written in shorthand but not one that is written out explicitly?

 

[Smoo]

I agree with shuke. Negative five is an inherently negative number. But there is no negative five expressed in "-5^2".

Sure there is.

 

[Maurile Tremblay]

If -5 cannot be -5 and has to be (-1)(5) then all you've done is create an infinite regression which makes solving the expression impossible. Because then -1 cannot be -1, it has to be (-1)(1). But then the -1 in that expression cannot be -1, it also has to be (-1)(1). And so on. It is clear at some point that you have to accept that a number can be inherently negative. And once you've made that assumption, there's no need for the initial factoring, the -5 can be inherently negative.

I don't know about everyone else, but I'm not trying to argue that -5 isn't an inherently negative integer.The point, when negative numbers are used in algebraic expressions, they should have parentheses around them otherwise you would not know which of the following is implied:

-(5^2)=x

(-5)^2=x

HOW MOTHER ####ING DIFFICULT IS THIS TO UNDERSTAND?

If the number is inherently negative, it should not require parentheses.HOW MOTHER ####ING DIFFICULT IS THAT TO UNDERSTAND?

What shouldn't require parentheses? The number negative five doesn't require parentheses.But in the expression "-5^2", the number negative five does not appear.

 

[shuke]

If -5 cannot be -5 and has to be (-1)(5) then all you've done is create an infinite regression which makes solving the expression impossible.  Because then -1 cannot be -1, it has to be (-1)(1).  But then the -1 in that expression cannot be -1, it also has to be (-1)(1).  And so on.  It is clear at some point that you have to accept that a number can be inherently negative.  And once you've made that assumption, there's no need for the initial factoring, the -5 can be inherently negative.

I don't know about everyone else, but I'm not trying to argue that -5 isn't an inherently negative integer.The point, when negative numbers are used in algebraic expressions, they should have parentheses around them otherwise you would not know which of the following is implied:

-(5^2)=x

(-5)^2=x

HOW MOTHER ####ING DIFFICULT IS THIS TO UNDERSTAND?

If the number is inherently negative, it should not require parentheses.HOW MOTHER ####ING DIFFICULT IS THAT TO UNDERSTAND?

Then how do you know if -5^2 refers to the negative of 5 squared or negative five squared?I don't understand why you are arguing about this. It's just a matter of convention that has been established.

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

BECAUSE WE WOULDN'T KNOW WHAT -5^2 MEANT UNLESS WE ESTABLISHED A WAY TO DIFFERENTIATE THE DIFFERENT INTERPRETATIONS. NOW STMFU.

 

[Maurile Tremblay]

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.

 

[Swapster]

If -5 cannot be -5 and has to be (-1)(5) then all you've done is create an infinite regression which makes solving the expression impossible.  Because then -1 cannot be -1, it has to be (-1)(1).  But then the -1 in that expression cannot be -1, it also has to be (-1)(1).  And so on.  It is clear at some point that you have to accept that a number can be inherently negative.  And once you've made that assumption, there's no need for the initial factoring, the -5 can be inherently negative.

I don't know about everyone else, but I'm not trying to argue that -5 isn't an inherently negative integer.The point, when negative numbers are used in algebraic expressions, they should have parentheses around them otherwise you would not know which of the following is implied:

-(5^2)=x

(-5)^2=x

HOW MOTHER ####ING DIFFICULT IS THIS TO UNDERSTAND?

If the number is inherently negative, it should not require parentheses.HOW MOTHER ####ING DIFFICULT IS THAT TO UNDERSTAND?

What shouldn't require parentheses? The number negative five doesn't require parentheses.But in the expression "-5^2", the number negative five does not appear.

That's crazy talk... I can see it right there.

 

[Maurile Tremblay]

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.

 

[shuke]

I agree with shuke. Negative five is an inherently negative number. But there is no negative five expressed in "-5^2".

Sure there is.

Not according to convention.But let's have it your way and say that for now on the parentheses were not required. How would you ####### know what -5^2 referred to? Are you just going to ####### guess?

 

[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. It is clearly there as an inherent part of the negative number and all operations on said number are also to be applied to the negative.

 

[bryhamm]

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.

I've been convinced, largely on rolyaTy's discussion. I read your stuff too Smoo. Interesting debate.

This answer convinced me, too because I'm picturing a graph, with -5, then -25, then -125 - it should all go in the downward direction like 5, 25, 125. It isn't intuitive for it to go -5, 25, -125 on a graph. BUT

A negative times a negative is a positive, so the graph should jump above and below the x line.

I can't see how you can factor out the - as in:

-5^2

-1(5*5)

-1(25)

-25

But not then have to apply this to all negative multiplied by negatives as in:

-3*-2

-1(3*2)

-1(6)

-6

Does this not follow the same logic?

No, you cannot factor multiplication out from other multiplication. The bolded section is wrong. Using your process, then the following would be true:8*4

2(4*2)

Obviously you get 2 different answers.

A proper way of factoring out -1 would be:

3 - 2

-1(-3 + 2)

Exactly!I knew the logic was flawed, but I'm asserting the logic in the first part is the same as the 2nd. You're taking -1 out of (5*5) when you say the -5^2 is really -1 * 5^2. 5^2 is 5x5. 5^2 is really just shorthand for 5*5, is it not? So, why can you factor multiplication out of a multiplication problem written in shorthand but not one that is written out explicitly?

No, we're not taking the -1 out. If we were to write:-1*5^2

I don't think anyone here would disagree that the answer is -25. But, since the expression was written:

-5^2

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

 

[Maurile Tremblay]

If -5 cannot be -5 and has to be (-1)(5) then all you've done is create an infinite regression which makes solving the expression impossible. Because then -1 cannot be -1, it has to be (-1)(1). But then the -1 in that expression cannot be -1, it also has to be (-1)(1). And so on. It is clear at some point that you have to accept that a number can be inherently negative. And once you've made that assumption, there's no need for the initial factoring, the -5 can be inherently negative.

I don't know about everyone else, but I'm not trying to argue that -5 isn't an inherently negative integer.The point, when negative numbers are used in algebraic expressions, they should have parentheses around them otherwise you would not know which of the following is implied:

-(5^2)=x

(-5)^2=x

HOW MOTHER ####ING DIFFICULT IS THIS TO UNDERSTAND?

If the number is inherently negative, it should not require parentheses.HOW MOTHER ####ING DIFFICULT IS THAT TO UNDERSTAND?

What shouldn't require parentheses? The number negative five doesn't require parentheses.But in the expression "-5^2", the number negative five does not appear.

That's crazy talk... I can see it right there.

That's like saying three squared appears in the expression "23^2". It doesn't, because the pairing of the 2 and the 3 to form the number 23 comes before the operation of the exponent. By the same token, five squared does not appear in "-5^2" since the exponent comes before the operation of the minus sign.

 

[Paper Lions]

Let me see if I can recap here.In an algebra equation, it should technically read either:-(5^2) or(-5)^2If 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.

 

[Maurile Tremblay]

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.