Daughter's math homework (1 Viewer)

[Paper Lions]

WOW! I'm glad work kept me busy today... I can't believe this has grown all the way to 27 pages!

I wrote down the equations and gave them to a guy here at work that is the ultimate math geek, so much so that he has actually written a couple of math books for educational use. He quickly, without thinking twice, rattled off the answers as -7 and 18. I gave him a brief synopsis of this thread and he just kind of chuckled. The only way you answer this any other way is to make up your own rules for calculating the value of expressions or making assumptions that aren't in the equation.

No one is disputing that this is the current convention.

A lot of people were disputing it until Shick! and Clayton got here. Some people were ridiculing anybody who went along with the (apparently) standard convention.

I would have been one of them were I here, but seriously, I'm deferring to the math teachers. I should have said no one is currently disputing I guess.

 

[Maurile Tremblay]

He invoked some mysterious corolloray that says two negatives cancel each other out, but he pulled it out of his ### rather than applying it to the elaborate world he's constructed.

How is that mysterious? Negative means opposite and the opposite of a negative is a positive, so a negative negative is a positive.

It was mysterious in the context of the framework world he constructed. He was carefully explaining how everything was represented on his number line model, but then just pulled "oh, and two negatives cancel each other out" from nowhere, without explaining it within the context of his framework.

It makes sense when you think of multiplication as a series of additions.If you unerstand why one minus negative one is positive, and you can think of multiplication in terms of addition (e.g., 2 * 3 = 2 + 2 + 2), then it becomes apparent why negative one times negative one is positive.

 

[Smoo]

Well unless I'm still operating on old math here, we should be able to do the same thing to each side of the equation and still have them be equal. So what the hell, here goes.

-5^2 = -25

(-5^2)^.5 = (-25)^.5

gllll with that.

Actually, never mind, this is a poor example. Parentheses before exponents.

 

[Clayton Gray]

Well unless I'm still operating on old math here, we should be able to do the same thing to each side of the equation and still have them be equal.  So what the hell, here goes.

-5^2 = -25

(-5^2)^.5 = (-25)^.5

gllll with that.

What's the problem?

 

[Smoo]

He invoked some mysterious corolloray that says two negatives cancel each other out, but he pulled it out of his ### rather than applying it to the elaborate world he's constructed.

How is that mysterious? Negative means opposite and the opposite of a negative is a positive, so a negative negative is a positive.

It was mysterious in the context of the framework world he constructed. He was carefully explaining how everything was represented on his number line model, but then just pulled "oh, and two negatives cancel each other out" from nowhere, without explaining it within the context of his framework.

It makes sense when you think of multiplication as a series of additions.If you unerstand why one minus negative one is positive, and you can think of multiplication in terms of addition (e.g., 2 * 3 = 2 + 2 + 2), then it becomes apparent why negative one times negative one is positive.

Guys, I don't misunderstand the concept of two negatives cancelling each other. I'm criticizing the way roly introduced it into his example. If you go back and read my original reply, this is clear.

 

[Maurile Tremblay]

Well unless I'm still operating on old math here, we should be able to do the same thing to each side of the equation and still have them be equal. So what the hell, here goes.

-5^2 = -25

(-5^2)^.5 = (-25)^.5

gllll with that.

Correct.The square root of the opposite of (five squared) = the square root of the opposite of twenty-five.

 

[tommyboy]

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.

wrong. the correct reading is negative 5 squared, since there is no operative assumed when -5 appears at the beginning of the string. If however the writer had said 1-5^2, then you would know that the - in the sentence is an operative. Since this was not the case, the purposely ambiguous question is moot. The real question should be why would anyone assume that - at the beginning of the sentence is assumed to be an operative rather than a primary indicator of the value of the number to be squared. Without any indicator by the asker, the question is void.

 

[dgreen]

If you unerstand why one minus negative one is positive, and you can think of multiplication in terms of addition (e.g., 2 * 3 = 2 + 2 + 2), then it becomes apparent why negative one times negative one is positive.

Can you please write out (-2)*(-3) in addition without just changing the negatives to positives.

 

[jonessed]

WOW! I'm glad work kept me busy today... I can't believe this has grown all the way to 27 pages!

I wrote down the equations and gave them to a guy here at work that is the ultimate math geek, so much so that he has actually written a couple of math books for educational use. He quickly, without thinking twice, rattled off the answers as -7 and 18. I gave him a brief synopsis of this thread and he just kind of chuckled. The only way you answer this any other way is to make up your own rules for calculating the value of expressions or making assumptions that aren't in the equation.

No one is disputing that this is the current convention. Smoo doesn't like the convention.

The thread goes a little beyond that. Even professions that use applied mathematics on a regular basis seem to waffle in its interpretation. That seems odd considering we were all taught in a classroom. All three of my calculators even calculate -5^2 as 25. If the convention is so obvious why doesn't it appear to be utilized much outside of the classroom?

 

[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.

wrong. the correct reading is negative 5 squared, since there is no operative assumed when -5 appears at the beginning of the string. If however the writer had said 1-5^2, then you would know that the - in the sentence is an operative. Since this was not the case, the purposely ambiguous question is moot. The real question should be why would anyone assume that - at the beginning of the sentence is assumed to be an operative rather than a primary indicator of the value of the number to be squared. Without any indicator by the asker, the question is void.

No, to correctly read negative 5 squared, it should be written (-5)^2. This was not how it was written. The correct way to read what was written is the negative of 5 squared.

 

[Maurile Tremblay]

the correct reading is negative 5 squared

That doesn't clear up any ambiguity.(negative five) squared > negative (five squared)

"negative five squared' can be read either way.

 

[The Iguana]

He invoked some mysterious corolloray that says two negatives cancel each other out, but he pulled it out of his ### rather than applying it to the elaborate world he's constructed.

How is that mysterious? Negative means opposite and the opposite of a negative is a positive, so a negative negative is a positive.

It was mysterious in the context of the framework world he constructed. He was carefully explaining how everything was represented on his number line model, but then just pulled "oh, and two negatives cancel each other out" from nowhere, without explaining it within the context of his framework.

It makes sense when you think of multiplication as a series of additions.If you unerstand why one minus negative one is positive, and you can think of multiplication in terms of addition (e.g., 2 * 3 = 2 + 2 + 2), then it becomes apparent why negative one times negative one is positive.

It doesn't really mesh with the world he created above but I always liked the "movie" explanation of why a negative * a negative = a positive:If you film someone running backwards and then play the film backwards, the person will appear to be running forward... It really means little to this discussion but I figure any distraction within this topic has to be a good thing at this point...

 

[Clayton Gray]

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.

wrong. the correct reading is negative 5 squared, since there is no operative assumed when -5 appears at the beginning of the string. If however the writer had said 1-5^2, then you would know that the - in the sentence is an operative. Since this was not the case, the purposely ambiguous question is moot. The real question should be why would anyone assume that - at the beginning of the sentence is assumed to be an operative rather than a primary indicator of the value of the number to be squared. Without any indicator by the asker, the question is void.

Because "negative" means "the opposite of" and "of" means "multiplication". Multiplication is an operation and it occurs after exponents in the order of operations.

 

[Smoo]

If you film someone running backwards and then play the film backwards, the person will appear to be running forward... It really means little to this discussion but I figure any distraction within this topic has to be a good thing at this point...

There was a movie or TV show that actually did this once. It was like Twin Peaks or Naked Lunch or something eually bizarre.They filmed the scene completely backwards, all the actors moved backwards, and I think even memorized their lines as they'd sound backwards. Then they ran it backwards in editing. The final cut was a really creepy scene.

EDIT - A Google search reveals that it was Top Secret! that I was thinking of.

 

[Clayton Gray]

If you unerstand why one minus negative one is positive, and you can think of multiplication in terms of addition (e.g., 2 * 3 = 2 + 2 + 2), then it becomes apparent why negative one times negative one is positive.

Can you please write out (-2)*(-3) in addition without just changing the negatives to positives.

(-2)*(-3)(-1)(2)*(-1)(3)

(-1)(-1)*(2)(3)

(-1)(-1)*(2+2+2)

(-1)(-1)*6

(-1)(-1*6)

(-1)(-6)

6

 

[Paper Lions]

WOW! I'm glad work kept me busy today... I can't believe this has grown all the way to 27 pages!

I wrote down the equations and gave them to a guy here at work that is the ultimate math geek, so much so that he has actually written a couple of math books for educational use. He quickly, without thinking twice, rattled off the answers as -7 and 18. I gave him a brief synopsis of this thread and he just kind of chuckled. The only way you answer this any other way is to make up your own rules for calculating the value of expressions or making assumptions that aren't in the equation.

No one is disputing that this is the current convention. Smoo doesn't like the convention.

The thread goes a little beyond that. Even professions that use applied mathematics on a regular basis seem to waffle in its interpretation. That seems odd considering we were all taught in a classroom. All three of my calculators even calculate -5^2 as 25. If the convention is so obvious why doesn't it appear to be utilized much outside of the classroom?

I didn't say it was obvious. It wasn't obvious to me. I'm just accepting the fact that the actual math here know the actual math convention.And I understand your point on the calculators. But since I don't want to start using parentheticals in every calculator function, I don't have a problem with it. When you punch in "-5" in the calculator and then "^2" it interprets it as "(-5)^2" and not "-(5^2)".



edit to add: Regarding your last question, I suspect it's because HP etc. is catering to the masses, who don't know the actual math convention. If you were HP would you rather cater to the masses or take a stand?

 

[dgreen]

If you unerstand why one minus negative one is positive, and you can think of multiplication in terms of addition (e.g., 2 * 3 = 2 + 2 + 2), then it becomes apparent why negative one times negative one is positive.

Can you please write out (-2)*(-3) in addition without just changing the negatives to positives.

(-2)*(-3)(-1)(2)*(-1)(3)

(-1)(-1)*(2)(3)

(-1)(-1)*(2+2+2)

(-1)(-1)*6

(-1)(-6)

6

That's not what I'm looking for. You can write 2*3 as 2+2+2 because what you are doing is adding 2s three times. Or, you can write 3+3 because you are adding 3s twice. So, how do you add -2 negative three times? How does (-2)(-3) look in addition format?

 

[rolyaTy]

He invoked some mysterious corolloray that says two negatives cancel each other out, but he pulled it out of his ### rather than applying it to the elaborate world he's constructed.

How is that mysterious? Negative means opposite and the opposite of a negative is a positive, so a negative negative is a positive.

It was mysterious in the context of the framework world he constructed. He was carefully explaining how everything was represented on his number line model, but then just pulled "oh, and two negatives cancel each other out" from nowhere, without explaining it within the context of his framework.

It makes sense when you think of multiplication as a series of additions.If you unerstand why one minus negative one is positive, and you can think of multiplication in terms of addition (e.g., 2 * 3 = 2 + 2 + 2), then it becomes apparent why negative one times negative one is positive.

It doesn't really mesh with the world he created above but I always liked the "movie" explanation of why a negative * a negative = a positive:If you film someone running backwards and then play the film backwards, the person will appear to be running forward... It really means little to this discussion but I figure any distraction within this topic has to be a good thing at this point...

That does mesh with the world.I said "left" because we were only dealing with two numbers, one with a positive sign and one with a negative, but when you get more complicated, a negative sign really deals with reversing direction. When you see a negative sign, you think of reversing the direction in which you travel on the number line.

2+-1: Travel from 0 to 2, then reverse your direction and travel 1. Total distance from origin to endpoint =1.

-2*-2: Resolve the overall direction before you start drawing. Reverse once, then reverse again, and travel in that direction 2+2 units.

 

[PC Load Letter]

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?

I really hope this was covered but, I'll do it again just incase-3*-2

(-1*3)*(-1*2)

(-1*-1)*(3*2)

1*6

6

You used the wrong property.

 

[tommyboy]

the correct reading is negative 5 squared

That doesn't clear up any ambiguity.(negative five) squared > negative (five squared)

"negative five squared' can be read either way.

so can -5^2, obviously as this thread has shown. If it was obvious this would never have gone more than about 10 posts.

 

[The Iguana]

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.

wrong. the correct reading is negative 5 squared, since there is no operative assumed when -5 appears at the beginning of the string. If however the writer had said 1-5^2, then you would know that the - in the sentence is an operative. Since this was not the case, the purposely ambiguous question is moot. The real question should be why would anyone assume that - at the beginning of the sentence is assumed to be an operative rather than a primary indicator of the value of the number to be squared. Without any indicator by the asker, the question is void.

I posted this a long time ago but now looks like a good time to do so again. You are actually hurting your case with the 1-5^2 comment. Addition is commutative, meaning that a + b = b + a. So, you can look at the original equation and remove all "ambiguity" by using this property of addition:-5² + 4 x 2³ = 4 x 2³ + -5²

Also most people appear to be in aggreement that 1 + -1 = 1 - 1 so:

4 x 2³ + -5² = 4 x 2³ - 5²

Thus, no ambiguity. For you to claim that it is "negative 5, squared" you are inserting parenthisis around -5 which do not exist in the problem as presented. Likewise,:

-5² = 0 + -5² = 0 - 5² = -25

There is no way to prove any other answer except to introduce the idea of an implied set of parenthesis.

 

[PC Load Letter]

If you unerstand why one minus negative one is positive, and you can think of multiplication in terms of addition (e.g., 2 * 3 = 2 + 2 + 2), then it becomes apparent why negative one times negative one is positive.

Can you please write out (-2)*(-3) in addition without just changing the negatives to positives.

(-2)*(-3)(-1)(2)*(-1)(3)

(-1)(-1)*(2)(3)

(-1)(-1)*(2+2+2)

(-1)(-1)*6

(-1)(-6)

6

That's not what I'm looking for. You can write 2*3 as 2+2+2 because what you are doing is adding 2s three times. Or, you can write 3+3 because you are adding 3s twice. So, how do you add -2 negative three times? How does (-2)(-3) look in addition format?

-(-2-2-2)

 

[The Iguana]

the correct reading is negative 5 squared

That doesn't clear up any ambiguity.(negative five) squared > negative (five squared)

"negative five squared' can be read either way.

so can -5^2, obviously as this thread has shown. If it was obvious this would never have gone more than about 10 posts.

There are lots of things that are obvious in this world that people still argue are wrong for decades, much less a few pages. Just because someone won't admit when they are wrong for a long period of time doesn't make their stance start to be right.

 

[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.

wrong. the correct reading is negative 5 squared, since there is no operative assumed when -5 appears at the beginning of the string. If however the writer had said 1-5^2, then you would know that the - in the sentence is an operative. Since this was not the case, the purposely ambiguous question is moot. The real question should be why would anyone assume that - at the beginning of the sentence is assumed to be an operative rather than a primary indicator of the value of the number to be squared. Without any indicator by the asker, the question is void.

I posted this a long time ago but now looks like a good time to do so again. You are actually hurting your case with the 1-5^2 comment. Addition is commutative, meaning that a + b = b + a. So, you can look at the original equation and remove all "ambiguity" by using this property of addition:-5² + 4 x 2³ = 4 x 2³ + -5²

Also most people appear to be in aggreement that 1 + -1 = 1 - 1 so:

4 x 2³ + -5² = 4 x 2³ - 5²

Thus, no ambiguity. For you to claim that it is "negative 5, squared" you are inserting parenthisis around -5 which do not exist in the problem as presented. Likewise,:

-5² = 0 + -5² = 0 - 5² = -25

There is no way to prove any other answer except to introduce the idea of an implied set of parenthesis.

Where did anyone use "1-5^2"

 

[rolyaTy]

If you unerstand why one minus negative one is positive, and you can think of multiplication in terms of addition (e.g., 2 * 3 = 2 + 2 + 2), then it becomes apparent why negative one times negative one is positive.

Can you please write out (-2)*(-3) in addition without just changing the negatives to positives.

(-2)*(-3)(-1)(2)*(-1)(3)

(-1)(-1)*(2)(3)

(-1)(-1)*(2+2+2)

(-1)(-1)*6

(-1)(-6)

6

That's not what I'm looking for. You can write 2*3 as 2+2+2 because what you are doing is adding 2s three times. Or, you can write 3+3 because you are adding 3s twice. So, how do you add -2 negative three times? How does (-2)(-3) look in addition format?

-(-(2+2+2))You can't start drawing on the number line though until you've resolved both magnitude and direction. So reading this you think:

I start off by default going right, I see the first - sign and that means I should start drawing left when I draw, then I see the next - sign and that says I should reverse directions again and start going right when I draw, and then 2+2+2=6 so I draw 6 units to the right.

Think of negatives like the reverse cards in UNO. Lay two down in a row and you're back to the direction you were going before the first card was laid.

 

[Clayton Gray]

If you unerstand why one minus negative one is positive, and you can think of multiplication in terms of addition (e.g., 2 * 3 = 2 + 2 + 2), then it becomes apparent why negative one times negative one is positive.

Can you please write out (-2)*(-3) in addition without just changing the negatives to positives.

(-2)*(-3)(-1)(2)*(-1)(3)

(-1)(-1)*(2)(3)

(-1)(-1)*(2+2+2)

(-1)(-1)*6

(-1)(-6)

6

That's not what I'm looking for. You can write 2*3 as 2+2+2 because what you are doing is adding 2s three times. Or, you can write 3+3 because you are adding 3s twice. So, how do you add -2 negative three times? How does (-2)(-3) look in addition format?

Negative means "the opposite of". "Of" means multiplication and can not be added so it is factored out twice (that's why "-1" shows up twice).

 

[Buckychudd]

 

[FatMax]

Well unless I'm still operating on old math here, we should be able to do the same thing to each side of the equation and still have them be equal.  So what the hell, here goes.

-5^2 = -25

(-5^2)^.5 = (-25)^.5

gllll with that.

Actually, never mind, this is a poor example. Parentheses before exponents.

:kickrock:I tried.

 

[tommyboy]

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.

wrong. the correct reading is negative 5 squared, since there is no operative assumed when -5 appears at the beginning of the string. If however the writer had said 1-5^2, then you would know that the - in the sentence is an operative. Since this was not the case, the purposely ambiguous question is moot. The real question should be why would anyone assume that - at the beginning of the sentence is assumed to be an operative rather than a primary indicator of the value of the number to be squared. Without any indicator by the asker, the question is void.

Because "negative" means "the opposite of" and "of" means "multiplication". Multiplication is an operation and it occurs after exponents in the order of operations.

you do not need parentheses to depict negative 5 as -5

it is a number all by itself. There is no operative property when depicting -5 by itself. Assuming your order of operations is moot since -5 is not an operation, it is a single entity, a number. You can see this in these equal equations

93-5=88

93+ -5= 88

in the first equation it is assumed you are using the subtraction definition of -

in the 2nd equation it is known you are using - to depict negativity.

in this equation -5^2= 25 it is assumed that negative 5 is being squared which produces +25 as an answer, but it could also be assumed the correct answer should be -25. That is why in math when multiple assumptions could logically come about, when only one correct answer is possible, the writer must specify exactly what he intends to mean. In this case, if the writer intended to produce negative 25 as the answer the writer should have forced the reader to assume that positive 5 was being squared and then made negative by depicting the question as -(5^2).

 

[The Iguana]

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.

wrong. the correct reading is negative 5 squared, since there is no operative assumed when -5 appears at the beginning of the string. If however the writer had said 1-5^2, then you would know that the - in the sentence is an operative. Since this was not the case, the purposely ambiguous question is moot. The real question should be why would anyone assume that - at the beginning of the sentence is assumed to be an operative rather than a primary indicator of the value of the number to be squared. Without any indicator by the asker, the question is void.

I posted this a long time ago but now looks like a good time to do so again. You are actually hurting your case with the 1-5^2 comment. Addition is commutative, meaning that a + b = b + a. So, you can look at the original equation and remove all "ambiguity" by using this property of addition:-5² + 4 x 2³ = 4 x 2³ + -5²

Also most people appear to be in aggreement that 1 + -1 = 1 - 1 so:

4 x 2³ + -5² = 4 x 2³ - 5²

Thus, no ambiguity. For you to claim that it is "negative 5, squared" you are inserting parenthisis around -5 which do not exist in the problem as presented. Likewise,:

-5² = 0 + -5² = 0 - 5² = -25

There is no way to prove any other answer except to introduce the idea of an implied set of parenthesis.

Where did anyone use "1-5^2"

In the post that starts out "wrong. the correct reading ...", Tommyboy goes on to say that if the equation had been written 1-5^2 then it would be obvious. Along that line of thinking you can always add a 0+ to any equation and not change the value, X + 0 == X. Thus leading to the point that 0 + -5^2 clearly shows that -5^2 = -25 unless you claim an implied () around -5, which doesn't exist.

 

[Sam Quentin]

29 ####ing pages:shakeshead:I'm gonna get drunk 29 pages on this:Shukeshead:

 

[dgreen]

If you unerstand why one minus negative one is positive, and you can think of multiplication in terms of addition (e.g., 2 * 3 = 2 + 2 + 2), then it becomes apparent why negative one times negative one is positive.

Can you please write out (-2)*(-3) in addition without just changing the negatives to positives.

(-2)*(-3)(-1)(2)*(-1)(3)

(-1)(-1)*(2)(3)

(-1)(-1)*(2+2+2)

(-1)(-1)*6

(-1)(-6)

6

That's not what I'm looking for. You can write 2*3 as 2+2+2 because what you are doing is adding 2s three times. Or, you can write 3+3 because you are adding 3s twice. So, how do you add -2 negative three times? How does (-2)(-3) look in addition format?

-(-(2+2+2))You can't start drawing on the number line though until you've resolved both magnitude and direction. So reading this you think:

I start off by default going right, I see the first - sign and that means I should start drawing left when I draw, then I see the next - sign and that says I should reverse directions again and start going right when I draw, and then 2+2+2=6 so I draw 6 units to the right.

Think of negatives like the reverse cards in UNO. Lay two down in a row and you're back to the direction you were going before the first card was laid.

Multiplication is easily explained by breaking it down to addition only. I'm only trying to figure out how to break down (-2)(-3) to only addition. Your breakdown still has multiplication with the negative signs at the beginning.I'm in agreement with your number line view and figuring out the direction then moving and its implications on how to view -5^2.

 

[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.

wrong. the correct reading is negative 5 squared, since there is no operative assumed when -5 appears at the beginning of the string. If however the writer had said 1-5^2, then you would know that the - in the sentence is an operative. Since this was not the case, the purposely ambiguous question is moot. The real question should be why would anyone assume that - at the beginning of the sentence is assumed to be an operative rather than a primary indicator of the value of the number to be squared. Without any indicator by the asker, the question is void.

Because "negative" means "the opposite of" and "of" means "multiplication". Multiplication is an operation and it occurs after exponents in the order of operations.

you do not need parentheses to depict negative 5 as -5

it is a number all by itself. There is no operative property when depicting -5 by itself. Assuming your order of operations is moot since -5 is not an operation, it is a single entity, a number. You can see this in these equal equations

93-5=88

93+ -5= 88

in the first equation it is assumed you are using the subtraction definition of -

in the 2nd equation it is known you are using - to depict negativity.

in this equation -5^2= 25 it is assumed that negative 5 is being squared which produces +25 as an answer, but it could also be assumed the correct answer should be -25. That is why in math when multiple assumptions could logically come about, when only one correct answer is possible, the writer must specify exactly what he intends to mean. In this case, if the writer intended to produce negative 25 as the answer the writer should have forced the reader to assume that positive 5 was being squared and then made negative by depicting the question as -(5^2).

This is where you would be wrong.

 

[dgreen]

If you unerstand why one minus negative one is positive, and you can think of multiplication in terms of addition (e.g., 2 * 3 = 2 + 2 + 2), then it becomes apparent why negative one times negative one is positive.

Can you please write out (-2)*(-3) in addition without just changing the negatives to positives.

(-2)*(-3)(-1)(2)*(-1)(3)

(-1)(-1)*(2)(3)

(-1)(-1)*(2+2+2)

(-1)(-1)*6

(-1)(-6)

6

That's not what I'm looking for. You can write 2*3 as 2+2+2 because what you are doing is adding 2s three times. Or, you can write 3+3 because you are adding 3s twice. So, how do you add -2 negative three times? How does (-2)(-3) look in addition format?

Negative means "the opposite of". "Of" means multiplication and can not be added so it is factored out twice (that's why "-1" shows up twice).

So it would be incorrect to say that all multiplication can be rewritten in addition terms only?

 

[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.

wrong. the correct reading is negative 5 squared, since there is no operative assumed when -5 appears at the beginning of the string. If however the writer had said 1-5^2, then you would know that the - in the sentence is an operative. Since this was not the case, the purposely ambiguous question is moot. The real question should be why would anyone assume that - at the beginning of the sentence is assumed to be an operative rather than a primary indicator of the value of the number to be squared. Without any indicator by the asker, the question is void.

I posted this a long time ago but now looks like a good time to do so again. You are actually hurting your case with the 1-5^2 comment. Addition is commutative, meaning that a + b = b + a. So, you can look at the original equation and remove all "ambiguity" by using this property of addition:

-5² + 4 x 2³ = 4 x 2³ + -5²

Also most people appear to be in aggreement that 1 + -1 = 1 - 1 so:

4 x 2³ + -5² = 4 x 2³ - 5²

Thus, no ambiguity. For you to claim that it is "negative 5, squared" you are inserting parenthisis around -5 which do not exist in the problem as presented. Likewise,:

-5² = 0 + -5² = 0 - 5² = -25

There is no way to prove any other answer except to introduce the idea of an implied set of parenthesis.

Where did anyone use "1-5^2"

In the post that starts out "wrong. the correct reading ...", Tommyboy goes on to say that if the equation had been written 1-5^2 then it would be obvious. Along that line of thinking you can always add a 0+ to any equation and not change the value, X + 0 == X. Thus leading to the point that 0 + -5^2 clearly shows that -5^2 = -25 unless you claim an implied () around -5, which doesn't exist.

 

[The Iguana]

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.

wrong. the correct reading is negative 5 squared, since there is no operative assumed when -5 appears at the beginning of the string. If however the writer had said 1-5^2, then you would know that the - in the sentence is an operative. Since this was not the case, the purposely ambiguous question is moot. The real question should be why would anyone assume that - at the beginning of the sentence is assumed to be an operative rather than a primary indicator of the value of the number to be squared. Without any indicator by the asker, the question is void.

Because "negative" means "the opposite of" and "of" means "multiplication". Multiplication is an operation and it occurs after exponents in the order of operations.

you do not need parentheses to depict negative 5 as -5

it is a number all by itself. There is no operative property when depicting -5 by itself. Assuming your order of operations is moot since -5 is not an operation, it is a single entity, a number. You can see this in these equal equations

93-5=88

93+ -5= 88

in the first equation it is assumed you are using the subtraction definition of -

in the 2nd equation it is known you are using - to depict negativity.

in this equation -5^2= 25 it is assumed that negative 5 is being squared which produces +25 as an answer, but it could also be assumed the correct answer should be -25. That is why in math when multiple assumptions could logically come about, when only one correct answer is possible, the writer must specify exactly what he intends to mean. In this case, if the writer intended to produce negative 25 as the answer the writer should have forced the reader to assume that positive 5 was being squared and then made negative by depicting the question as -(5^2).

wrong, wrong wrong... you prove it in your own freaking example...by your example:

93 - 5 = 93 + -5 = 88

so obviously:

93 - 5^2 = 93 + -5^2 = 118

 

[The Iguana]

By the way, 93 - 5^2 != 118...

 

[Clayton Gray]

you do not need parentheses to depict negative 5 as

-5

When "-5" is a stand alone, I agree.

it is a number all by itself. There is no operative property when depicting -5 by itself.

Yes there is. To get "-5", you must first take the opposite of 5.

Assuming your order of operations is moot since -5 is not an operation, it is a single entity, a number. You can see this in these equal equations

93-5=88

93+ -5= 88

in the first equation it is assumed you are using the subtraction definition of -

in the 2nd equation it is known you are using - to depict negativity.

Actually, in the 2nd you have operated on five and changed it to the opposite of five - more commonly known as "-5".

in this equation -5^2= 25 it is assumed that negative 5 is being squared which produces +25 as an answer, but it could also be assumed the correct answer should be -25. That is why in math when multiple assumptions could logically come about, when only one correct answer is possible, the writer must specify exactly what he intends to mean. In this case, if the writer intended to produce negative 25 as the answer the writer should have forced the reader to assume that positive 5 was being squared and then made negative by depicting the question as -(5^2).

Nope.

 

[tommyboy]

This is where you would be wrong.

me and about 4 billion other people on this planet

 

[rolyaTy]

-(-(2+2+2))

You can't start drawing on the number line though until you've resolved both magnitude and direction. So reading this you think:

I start off by default going right, I see the first - sign and that means I should start drawing left when I draw, then I see the next - sign and that says I should reverse directions again and start going right when I draw, and then 2+2+2=6 so I draw 6 units to the right.

Think of negatives like the reverse cards in UNO. Lay two down in a row and you're back to the direction you were going before the first card was laid.

Multiplication is easily explained by breaking it down to addition only. I'm only trying to figure out how to break down (-2)(-3) to only addition. Your breakdown still has multiplication with the negative signs at the beginning.I'm in agreement with your number line view and figuring out the direction then moving and its implications on how to view -5^2.

Well, -(-(2+2+2)) can be simplified to --(2+2+2) and those negatives aren't really multiplication necessarily, anymore than -(2) is multiplication. I'd say in it's barest form, -2 is not (-1)*(2) but merely a directional guide for the quantity two. But multiplication, which is advanced addition, allows for shortcuts, and thinking of -2 as (-1)*(2) allows for much easier manipulation.

so,

Think of it like --(2+2+2) where the -- is not multiplication, but direction. Reverse, Reverse, result of sum.

 

[Maurile Tremblay]

Multiplication is easily explained by breaking it down to addition only. I'm only trying to figure out how to break down (-2)(-3) to only addition. Your breakdown still has multiplication with the negative signs at the beginning.

(-2) is defined as (-1)(2).It's not as easy as I made it sound before to add negative 2 to itself negative three times.

But Clayton gave the correct way to manipulate the expression. (-2) is defined as (-1)(2) and (-3) is defined as (-1)(3).

So we've got (-1)(-1)(2)(3).

(2)(3) means to add two to itself three times, which gives us six.

So we've got (-1)(-1)(6).

Now here's where the definitional stuff comes back in. (-6) is defined as (-1)(6), which means that (-1)(6) is defined as (-6).

So we've got (-1)(-6).

Doing the definition thing again, (-1)(-6) is defined as -(-6). Which is 6.

 

[Smoo]

Okay, I am now convinced that Maurile et al's way of interpreting things is no longer inherently illogical. I still don't prefer it as an interpretation, but I can no longer call it logically inconsistent. And when neither method is logically inconsistent, convention wins the day.I still disagree with convention, but arguing against it is now in the arena of thought exercise rather than legitimate complaint.

 

[Clayton Gray]

If you unerstand why one minus negative one is positive, and you can think of multiplication in terms of addition (e.g., 2 * 3 = 2 + 2 + 2), then it becomes apparent why negative one times negative one is positive.

Can you please write out (-2)*(-3) in addition without just changing the negatives to positives.

(-2)*(-3)(-1)(2)*(-1)(3)

(-1)(-1)*(2)(3)

(-1)(-1)*(2+2+2)

(-1)(-1)*6

(-1)(-6)

6

That's not what I'm looking for. You can write 2*3 as 2+2+2 because what you are doing is adding 2s three times. Or, you can write 3+3 because you are adding 3s twice. So, how do you add -2 negative three times? How does (-2)(-3) look in addition format?

Negative means "the opposite of". "Of" means multiplication and can not be added so it is factored out twice (that's why "-1" shows up twice).

So it would be incorrect to say that all multiplication can be rewritten in addition terms only?

I define "multiplying by -1" as "taking the opposite", so I would say it is correct to say "Except for multiplying by -1, all multiplication can be rewritten in addition terms only."

 

[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.

wrong. the correct reading is negative 5 squared, since there is no operative assumed when -5 appears at the beginning of the string. If however the writer had said 1-5^2, then you would know that the - in the sentence is an operative. Since this was not the case, the purposely ambiguous question is moot. The real question should be why would anyone assume that - at the beginning of the sentence is assumed to be an operative rather than a primary indicator of the value of the number to be squared. Without any indicator by the asker, the question is void.

Because "negative" means "the opposite of" and "of" means "multiplication". Multiplication is an operation and it occurs after exponents in the order of operations.

you do not need parentheses to depict negative 5 as

-5

it is a number all by itself. There is no operative property when depicting -5 by itself. Assuming your order of operations is moot since -5 is not an operation, it is a single entity, a number. You can see this in these equal equations

93-5=88

93+ -5= 88

in the first equation it is assumed you are using the subtraction definition of -

in the 2nd equation it is known you are using - to depict negativity.

in this equation -5^2= 25 it is assumed that negative 5 is being squared which produces +25 as an answer, but it could also be assumed the correct answer should be -25. That is why in math when multiple assumptions could logically come about, when only one correct answer is possible, the writer must specify exactly what he intends to mean. In this case, if the writer intended to produce negative 25 as the answer the writer should have forced the reader to assume that positive 5 was being squared and then made negative by depicting the question as -(5^2).

This is where you would be wrong.

me and about 4 billion other people on this planet

and?

 

[tommyboy]

if you are going to unteach 4 billion people on this planet that -5 isn't -5 if you square it, then have at it.

 

[Clayton Gray]

Okay, I am now convinced that Maurile et al's way of interpreting things is no longer inherently illogical. I still don't prefer it as an interpretation, but I can no longer call it logically inconsistent. And when neither method is logically inconsistent, convention wins the day.

I still disagree with convention, but arguing against it is now in the arena of thought exercise rather than legitimate complaint.

What's wrong with this explanation?

In class, I would always describe "integers" as the counting numbers (1, 2, 3, etc) and their opposites (-1, -2, -3, etc) and zero.

I hope that no one here will argue that -1 is not the opposite of 1.

We say "negative 1" because it is the opposite of "positive 1". We write -1 because it is far easier than continuously writing "negative 1".

Clearly the "-" symbol is defined as "the opposite of" in mathematics. Does anyone argue against this?

----------------------------------------------------------

The word "of" in mathematics means "multiply".

If the problem reads "What is 1/2 of 6?", you multiply 1/2 times 6 and get an answer of 3.

If the problem reads, "30 is 50% of what number?", we set up the following equation:

30 is 50% of what number30 = 0.5 * XWe divide 30 by 0.5 and get 60.Clearly, the word "of" is defined as "multiplication" in mathematics. Does anyone argue against this?

----------------------------------------------------------

The order of operations tells us that exponents are performed before any multiplication.

If the problem reads 2*5^2, we square 5 before multiplying because multiplying clearly means multiplication.

2*5^2

2*25

50

If the problem reads -5^2, we square before taking the opposite of because the word of clearly means multiplication.

-5^2

-25

These two problems are performed consistent with one another. Does anyone argue against this?

 

[dgreen]

Doing the definition thing again, (-1)(-6) is defined as -(-6). Which is 6.

And is using rolyaty's directional idea the way to get there? How else can you jump from -(-6) to 6? Again, I know that's the answer, I'm just wondering how math explains that jump.

 

[IanTucker]

You guys still haven't convinced me that -5 should not be considered a self-contained number. You've come close, but not there yet.

-5 could be looked at as a self-contained number.But when we put it with ^2 as in the problem: -5^2, it is no longer a self-contained number unless it was written: (-5)^2.

Since it wasn't written that way I would assume the "-" is an operator and we get

0-5^2=-25

 

[dgreen]

Think of it like --(2+2+2) where the -- is not multiplication, but direction. Reverse, Reverse, result of sum.

I'm fine with that. I'm just wondering if "direction" is how mathematics first reached the conclusion that negative x negative = positive.

 

[Maurile Tremblay]

if you are going to unteach 4 billion people on this planet that -5 isn't -5 if you square it, then have at it.

Negative five is still negative five even if you square it (in which case the answer is twenty-five). It's just that when you write "-5^2", you're not squaring negative five. You're squaring five.You're making up an inconsistency that doesn't exist.