Daughter's math homework (1 Viewer)

[Nigel Tufnel]

How in the name of Odin's raven has this thread gone 36 pages?

 

[Psychopav]

How in the name of Odin's raven has this thread gone 36 pages?

It's only 26 pages.

I didn't know Odin had a raven. What's his name?

 

[Nigel Tufnel]

How in the name of Odin's raven has this thread gone 36 pages?

It's only 26 pages.

I didn't know Odin had a raven. What's his name?

36 here. Honest.

36 Pages « < 34 35 36

The Odin's raven reference was a movie quote gone awry.

 

[tommyboy]

maybe this will settle it:

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Negative Squared, or Squared Negative?

Date: 10/18/2002 at 22:57:56

From: Tom

Subject: Negative Numbers Squared.

After reading your answer in

Exponents and Negative numbers

http://mathforum.org/library/drmath/view/55709.html

it seems to me that you're ignoring an important fact: -3 isn't just

-1*3, but a number in its own right, i.e., the number 3 units to the

left of zero. If that's the case, then shouldn't -3^2 have the value

-3*-3, or 9?

If -3 was intended to mean -1*3, then shouldn't it be written that way

and not implied?

Thank you for your time.

Tom

Date: 10/19/2002 at 20:44:50

From: Doctor Peterson

Subject: Re: Negative Numbers Squared.

Hi, Tom.

I do recognize that it is possible to disagree on -3^2. Dr. Rick's

answer to a similar question,

Squaring Negative Numbers

http://mathforum.org/library/drmath/view/55713.html

mentions this disagreement. (Dr. Rick is my twin brother, by the way!)

Like you, he notes that if you think of -3 as a single number, it

makes sense for the negation to bind more tightly to the 3 than any

operation. That reasoning makes some sense, though I think other

arguments are stronger. But I do agree that since there _is_ some

reason to read it either way, it is prudent always to include

parentheses one way or the other, to clarify your intent, i.e., to

write either -(3^2) or (-3)^2.

Occasionally people will try to argue the point based on the behaviors

of particular calculators or spreadsheet programs. However, these are

really irrelevant, since they all define their own input formats, and

programmers (of which I am one) are notorious for choosing what's

easiest for them, rather than what is most appropriate for the user.

I've noted in several answers in our archives that some calculators,

and Excel, use non-standard orders of operation without apology. But

calculators in particular just don't use standard algebraic notation

in the first place.

There also seems to be a generational difference, with older people

(including some teachers) claiming that they were taught to interpret

-3^2 as (-3)^2.

I suspect that what has changed is not the rules governing "order of

operation" (operation precedence), but that schools are introducing

the issue earlier, before students get into algebra proper. That means

that they start by looking at expressions for which it is less clear

why the rules make sense. I think you will rarely find examples of

"-3^2" in practice, because there is no need for mathematicians to

write it. You will find "-x^2" frequently.

If you approach the idea starting with numerical expressions like

-3^2, you are thinking of -3 as a number and assuming that the

expression says to square it. If you approach it first using

variables, having first discovered that "-" in a negative number is

actually an operator, then it is easier to see why -x^2 should be

taken as the negative of the square. So I'll start with the latter,

and then it becomes natural to treat numbers the same way we treat

variables.

Now, in an expression like -x, clearly "-" is a (unary) operator,

which takes a value "x" and converts it to its opposite, or negative.

The expression "-x" is not just a single symbol, but a statement that

something is to be done to a value. As soon as we start combining

symbols like this, as in -x^2 or -x*y, we have to decide what order to

use in evaluating them.

The trouble is that the "order of operations" rules as commonly taught

(PEMDAS) don't mention negatives. So if we are going to go by the

rules, we have to figure out how a negative relates to them. Well,

there are two ways to express a negative in terms of binary

operations. One is as multiplication by -1:

-x = -1 * x

Treating it this way, clearly

-x^2 = -1 * x^2 = -(x^2)

That is, since -x means a product, we have to do the exponentiation

first.

The other way to talk about negation is as the additive inverse,

subtracting x from 0:

-x = 0 - x

(This is why the "-" sign is used for both negation and subtraction.)

Using this view, we see that

-x^2 = 0 - x^2 = -(x^2)

So both views of negation produce the same interpretation, which does

exponents first, and it is logical to put negation here in the order

of precedence.

But the fact is that there is no authority decreeing these rules;

just as in the grammar of English, we get the "rules" by observing how

the language is actually used, not by deducing them from some first

principles. The order of operations is just the grammar of algebra.

So the real question is, how do mathematicians really interpret

negatives and exponents combined in an expression?

If you look in books, you will rarely find "-3^2" written out, but

you will often find polynomials with negative coefficients. And you

will find that

-x^2 + 3x - 2

is read as the negative of the square of x, plus three times x, minus

2. I have come to believe that the order of operations is what it is

largely so that polynomials can be written efficiently. If "-x^2"

meant the square of -x, then we would have to write this as

-(x^2) + 3x - 2

to make it mean what we intend. Since powers are the core of a

polynomial, we ensure that powers are evaluated first, followed by

products and negatives (the two ways to write a coeffient) and then

sums (adding the terms).

Since we can easily see that this is how -x^2 is universally

interpreted, it makes sense to treat -3^2 the same way.

- Doctor Peterson, The Math Forum

http://mathforum.org/dr.math/

Associated Topics:

Middle School Algebra

Middle School Negative Numbers

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[Psychopav]

Great! So you're finally backing off?Glad you see things our way...

 

[Maurile Tremblay]

maybe this will settle it:

You tell us. Did it convince you?

 

[popeye.]

With this many posts, shouldn't we be talking about little Melvin?

 

[Clayton Gray]

nice.  You can drop the superiority complex.  Those were some pretty unnecessary words there since I'm a father of 2 with one more on the way. I was an advanced math student through high school and college, got 720 on the math part of the SAT and I know for a fact that -5^2 is 25.  Maybe you should quit drinking.  In fact, if you are a teacher and you don't know how to express a proper math question, I feel bad for you students.  If you are teaching your kids that -5^2 = -25 then I fear for god the direction this country is headed.

FYI, chalk up another vote for "you're very wrong about the convention."

Also, just going out on a limb here, but did you go to (a) the Chicago public schools or (b) a rural school district?

But he got a *720*!!!(I got a 770 on that part, and I've been lost at times in this thread. The ability to understand basic algebra means nothing when it comes to mathematical standards, and advanced mathematics.)

I only got a 730.I would have got a 720, but I got that -5^2 question right.

I took the ACT and made a 36 on the math section (aka not missing any ).

 

[Clayton Gray]

I was an advanced math student through high school . . .

BTW, this really isn't math so much as syntax. The best mathematician in the world would get this wrong if he wasn't taught it properly, just like the smartest person in the world wouldn't know the Chinese word for bathtub if he didn't speak Chinese. It has nothing to do with being smart. It has to do with memorizing an arbitrary rule.

浴缸

 

[Gamblor]

Who knows, maybe you were too busy dreaming about Mary Jane Rottencrotch during math class.

 

[PC Load Letter]

I was an advanced math student through high school . . .

BTW, this really isn't math so much as syntax. The best mathematician in the world would get this wrong if he wasn't taught it properly, just like the smartest person in the world wouldn't know the Chinese word for bathtub if he didn't speak Chinese. It has nothing to do with being smart. It has to do with memorizing an arbitrary rule.

浴缸

yù pén.Wait.. sorry. -yù pén

 

[NCCommish]

I was an advanced math student through high school . . .

BTW, this really isn't math so much as syntax. The best mathematician in the world would get this wrong if he wasn't taught it properly, just like the smartest person in the world wouldn't know the Chinese word for bathtub if he didn't speak Chinese. It has nothing to do with being smart. It has to do with memorizing an arbitrary rule.

浴缸

yù pén.Wait.. sorry. -yù pén

shouldn't there be a negative one or is that assumed?

 

[Mjolnirs]

just like the smartest person in the world wouldn't know the Chinese word for bathtub if he didn't speak Chinese.

yù pén.Wait.. sorry. -yù pén

So is that the opposite of a bathtub, or a negative bathtub Oh and,

The "-" you are referring to in your reply is commonly called a "subtraction" sign. By definition, subtracting is merely "adding the opposite". In your "3 - 2" example, we can rewrite it as "3 + (-2)" which can be read as "three plus negative two" or "three plus the opposite of two".

So when earlier it was said that we should assume-5^2 = (-1)(5)^2 and to get there we could think of it as

0-5^2 then we can do this

0+(-5)^2 which brings us back to

25

 

[Clayton Gray]

Oh and,

The "-" you are referring to in your reply is commonly called a "subtraction" sign.  By definition, subtracting is merely "adding the opposite".  In your "3 - 2" example, we can rewrite it as "3 + (-2)" which can be read as "three plus negative two" or "three plus the opposite of two".

So when earlier it was said that we should assume-5^2 = (-1)(5)^2 and to get there we could think of it as

0-5^2 then we can do this

0+(-5)^2 which brings us back to

25

0-5^2 then we can do this 0+(-5)^2You can't do that. Read this thread for reasons why.

 

[Mjolnirs]

0-5^2 then we can do this 0+(-5)^2

You can't do that. Read this thread for reasons why.

Wow, it's Deja Vu all over again

 

[flipflop1970]

1) 57

2) 54

Parentheses

Exponents

Multiplication

Division

Addition

Subdtraction

you're done

Please Excuse My Dear Aunt SallyThat was the acronym my teachers used to help us remember what oredr of operations

I would bet the farm on

1. 57

2. 54

 

[rolyaTy]

1) 57

2) 54

Parentheses

Exponents

Multiplication

Division

Addition

Subdtraction

you're done

Please Excuse My Dear Aunt SallyThat was the acronym my teachers used to help us remember what oredr of operations

I would bet the farm on

1. 57

2. 54

I'll take that bet. Where is this farm?

 

[flipflop1970]

1) 57

2) 54

Parentheses

Exponents

Multiplication

Division

Addition

Subdtraction

you're done

Please Excuse My Dear Aunt SallyThat was the acronym my teachers used to help us remember what oredr of operations

I would bet the farm on

1. 57

2. 54

I'll take that bet. Where is this farm?

Upstate New York

 

[Psychopav]

Oh and,

The "-" you are referring to in your reply is commonly called a "subtraction" sign.  By definition, subtracting is merely "adding the opposite".  In your "3 - 2" example, we can rewrite it as "3 + (-2)" which can be read as "three plus negative two" or "three plus the opposite of two".

So when earlier it was said that we should assume-5^2 = (-1)(5)^2 and to get there we could think of it as

0-5^2 then we can do this

0+(-5)^2 which brings us back to

25

This is actually a descent way of illustrating why -5^2 = -25.0 - 5^2 can be simplified to 0 - 25, or -25 (order of operations dictates the exponent should be executed first).

 

[Brock Middlebrook]

I can't believe this thing is still alive

 

[Smoo]

Oh and,

The "-" you are referring to in your reply is commonly called a "subtraction" sign. By definition, subtracting is merely "adding the opposite". In your "3 - 2" example, we can rewrite it as "3 + (-2)" which can be read as "three plus negative two" or "three plus the opposite of two".

So when earlier it was said that we should assume-5^2 = (-1)(5)^2 and to get there we could think of it as

0-5^2 then we can do this

0+(-5)^2 which brings us back to

25

This is actually a descent way of illustrating why -5^2 = -25.0 - 5^2 can be simplified to 0 - 25, or -25 (order of operations dictates the exponent should be executed first).

That's a terrible way of explaining it. Yes, it's accurate, but it's a terrible way of explaining it.

 

[Psychopav]

Oh and,

The "-" you are referring to in your reply is commonly called a "subtraction" sign.  By definition, subtracting is merely "adding the opposite".  In your "3 - 2" example, we can rewrite it as "3 + (-2)" which can be read as "three plus negative two" or "three plus the opposite of two".

So when earlier it was said that we should assume-5^2 = (-1)(5)^2 and to get there we could think of it as

0-5^2 then we can do this

0+(-5)^2 which brings us back to

25

This is actually a descent way of illustrating why -5^2 = -25.0 - 5^2 can be simplified to 0 - 25, or -25 (order of operations dictates the exponent should be executed first).

That's a terrible way of explaining it. Yes, it's accurate, but it's a terrible way of explaining it.

I didn't say it was a good way of explaining it. I said it was a good way of illustrating it.

 

[culdeus]

Just sent this nonsense to my wife's collegue who is the head MathCounts guy in the region and has a PhD from Stanford.

His answer-25+(4x8)

-25+32

7

-36+(2x9)

-36+18

-18



Not reading the rest of the thread.

 

[sirelfman]

We should have the FBG's equivalent of nobel prizes.

I'd nominate this thread for the FBG's nobel prize for mathematics.

I'd also like to nominate VD posthumously for the FBG's nobel prize for literature.

Well, actually, it would have to be the FBG Fields Prize, cuz there ain't no Nobel prize in mathematics. We usually only win for physics or economics.

 

[Armando]

1) 57

2) 54

Parentheses

Exponents

Multiplication

Division

Addition

Subdtraction

you're done

Please Excuse My Dear Aunt SallyThat was the acronym my teachers used to help us remember what oredr of operations

I would bet the farm on

1. 57

2. 54

Damn wish I was around then to take that bet.

 

[MrPack]

I can't believe this thing is still alive

 

[adonis]

We should have the FBG's equivalent of nobel prizes.

I'd nominate this thread for the FBG's nobel prize for mathematics.

I'd also like to nominate VD posthumously for the FBG's nobel prize for literature.

Well, actually, it would have to be the FBG Fields Prize, cuz there ain't no Nobel prize in mathematics. We usually only win for physics or economics.

The Fields Medal

THat's what good will hunting's mentor won right?<---------------- Knows things.

 

[sirelfman]

Okay, if there is anybody still listening, this thing kept me awake last night. This is what came to me early this morning.

(Caveat: I'm not arguing for one convention over the other, I'm simply trying to address a concern Smoo had with the 'accepted' convention. I believe both sides have merit, and I, as a math teacher, teach the 'accepted' convention.)

Smoo (who now accepts the "standard" as at least not illogical (different from logical) and at best plausible) wanted some basis for the convention that gave it some logical grounds. (I hadn't read far enough to see he had accepted it when I finally went to bed last night). For all who seem to still be wondering about a logical arguement (or perhaps a reasonable, thought out thinking process that leads to a convention), let me try something on you:

There is a older (I believe, the multiplication convention preceded the exponent convention we are discussing, chronologically) convention, which seems to be fairly accepted by all involved because both sides have used it, that if two "values" are written side-by-side, it means multiplication, i.e. ab means a * b, 47x means 47 * x, and 12(39+48) means 12 * (39+48). With me? Okay, lets move on.

The expression in question is -5^2, with one side saying this should the square of -5, and thus 25. Let's go a little further for demonstrations purposes. By definition of exponents, this is repeated multiplication. But if we write -5-5, 'sans operator' (per above convention) we have another ambiguous expression, because is this -5 multiplied by itself, or is it subtraction. If we glue the "-" and the "5" together and use a already estabished convention, we arrive at an even more ambiguous statement. Now, is -5^2=25, -25, or -10. So, to interpret this statement in this way, and apply another well known and accepted convention, we arrive at a MORE ambiguous statement.

But, if we interpret the expression -5^2 as the other side is saying, that it is the negative of 5^2, it has a single, solitary possible evaluation. Everyone here agrees that 5^2 is 25 and the negative of that is -25.

Because of this, we establish, by convention, that -5^2 should be -25. And since we would write (-5)(-5)=(-5)^2, this is the convention for symbolically writing "the square of negative five". This also leaves only the negative of five squared interpretation for -5^2.

Comments welcome.

 

[Mjolnirs]

Why, oh why did you revive this thread?

 

[BlueOnion]

Okay, if there is anybody still listening, this thing kept me awake last night. This is what came to me early this morning.

(Caveat: I'm not arguing for one convention over the other, I'm simply trying to address a concern Smoo had with the 'accepted' convention. I believe both sides have merit, and I, as a math teacher, teach the 'accepted' convention.)

Smoo (who now accepts the "standard" as at least not illogical (different from logical) and at best plausible) wanted some basis for the convention that gave it some logical grounds. (I hadn't read far enough to see he had accepted it when I finally went to bed last night). For all who seem to still be wondering about a logical arguement (or perhaps a reasonable, thought out thinking process that leads to a convention), let me try something on you:

There is a older (I believe, the multiplication convention preceded the exponent convention we are discussing, chronologically) convention, which seems to be fairly accepted by all involved because both sides have used it, that if two "values" are written side-by-side, it means multiplication, i.e. ab means a * b, 47x means 47 * x, and 12(39+48) means 12 * (39+48). With me? Okay, lets move on.

The expression in question is -5^2, with one side saying this should the square of -5, and thus 25. Let's go a little further for demonstrations purposes. By definition of exponents, this is repeated multiplication. But if we write -5-5, 'sans operator' (per above convention) we have another ambiguous expression, because is this -5 multiplied by itself, or is it subtraction. If we glue the "-" and the "5" together and use a already estabished convention, we arrive at an even more ambiguous statement. Now, is -5^2=25, -25, or -10. So, to interpret this statement in this way, and apply another well known and accepted convention, we arrive at a MORE ambiguous statement.

But, if we interpret the expression -5^2 as the other side is saying, that it is the negative of 5^2, it has a single, solitary possible evaluation. Everyone here agrees that 5^2 is 25 and the negative of that is -25.

Because of this, we establish, by convention, that -5^2 should be -25. And since we would write (-5)(-5)=(-5)^2, this is the convention for symbolically writing "the square of negative five". This also leaves only the negative of five squared interpretation for -5^2.

Comments welcome.

Regardless, when I raise negative five to the second power, I am always going to come up with 25. If somebody wants to say negative five raised to the second power is -25, more power to them.

 

[shahekee]

I'm not going to read this whole crazy thing again, but with negatives, I think I was taught to assume that unless specified, they were to be treated as -1*5. So in this case, it should read as such for order of operations -1*5^2.

 

[John 14:6]

Regardless, when I raise negative five to the second power, I am always going to come up with 25. If somebody wants to say negative five raised to the second power is -25, more power to them.

Yep. Raising negative five to the second power = (-5)^2 = 25.-5^2 = The negative of raising five to the second power = -25.

 

[Smoo]

Writing the words "negative five squared" isn't solving anything for either side, because it's the exact same problem. It's ambiguous. It could mean "(negative five) squared" or i could mean "negative (five squared)".

 

[BlueOnion]

Regardless, when I raise negative five to the second power, I am always going to come up with 25.    If somebody wants to say negative five raised to the second power is -25, more power to them.

Yep. Raising negative five to the second power = (-5)^2 = 25.-5^2 = The negative of raising five to the second power = -25.

[brittishAccent]

New math says the expression 5^2 shall now be read as the positive of raising five to the second power.

[/brittishAccent]

 

[Armando]

I'm not going to read this whole crazy thing again, but with negatives, I think I was taught to assume that unless specified, they were to be treated as -1*5. So in this case, it should read as such for order of operations -1*5^2.

You are correct sir. So puzzling why this has went so long. It's as simple as you put above.

 

[John 14:6]

Regardless, when I raise negative five to the second power, I am always going to come up with 25.    If somebody wants to say negative five raised to the second power is -25, more power to them.

Yep. Raising negative five to the second power = (-5)^2 = 25.-5^2 = The negative of raising five to the second power = -25.

[brittishAccent]

New math says the expression 5^2 shall now be read as the positive of raising five to the second power.

[/brittishAccent]

Or just raising five to the second power. Positive is assumed unless a negative sign is present.

 

[Maurile Tremblay]

Regardless, when I raise negative five to the second power, I am always going to come up with 25. If somebody wants to say negative five raised to the second power is -25, more power to them.

Yep. Raising negative five to the second power = (-5)^2 = 25.-5^2 = The negative of raising five to the second power = -25.

[brittishAccent]

New math says the expression 5^2 shall now be read as the positive of raising five to the second power.

[/brittishAccent]

I'm not sure why it's funny, but John 14:6 has it exactly right.

 

[Smoo]

Regardless, when I raise negative five to the second power, I am always going to come up with 25. If somebody wants to say negative five raised to the second power is -25, more power to them.

Yep. Raising negative five to the second power = (-5)^2 = 25.-5^2 = The negative of raising five to the second power = -25.

[brittishAccent]

New math says the expression 5^2 shall now be read as the positive of raising five to the second power.

[/brittishAccent]

I'm not sure why it's funny, but John 14:6 has it exactly right.

For hate's sake, I spit my last breath at thee.

 

[BlueOnion]

Regardless, when I raise negative five to the second power, I am always going to come up with 25.    If somebody wants to say negative five raised to the second power is -25, more power to them.

Yep. Raising negative five to the second power = (-5)^2 = 25.-5^2 = The negative of raising five to the second power = -25.

[brittishAccent]

New math says the expression 5^2 shall now be read as the positive of raising five to the second power.

[/brittishAccent]

I'm not sure why it's funny, but John 14:6 has it exactly right.

The math is still the same, however, it seems to me (and I could be mistaken) that some literary snobs learned how to do exponential math and decided to change the literary representation of negative five and call it 'new math'. For whatever reason, this ‘new math’ has had an empowering effect on others and I find it humorous.

 

[Maurile Tremblay]

Regardless, when I raise negative five to the second power, I am always going to come up with 25. If somebody wants to say negative five raised to the second power is -25, more power to them.

Yep. Raising negative five to the second power = (-5)^2 = 25.-5^2 = The negative of raising five to the second power = -25.

[brittishAccent]

New math says the expression 5^2 shall now be read as the positive of raising five to the second power.

[/brittishAccent]

I'm not sure why it's funny, but John 14:6 has it exactly right.

For hate's sake, I spit my last breath at thee.

I'm assuming 'last' here means 'most recent' rather than 'final'?

 

[Smoo]

Regardless, when I raise negative five to the second power, I am always going to come up with 25. If somebody wants to say negative five raised to the second power is -25, more power to them.

Yep. Raising negative five to the second power = (-5)^2 = 25.-5^2 = The negative of raising five to the second power = -25.

[brittishAccent]

New math says the expression 5^2 shall now be read as the positive of raising five to the second power.

[/brittishAccent]

I'm not sure why it's funny, but John 14:6 has it exactly right.

For hate's sake, I spit my last breath at thee.

I'm assuming 'last' here means 'most recent' rather than 'final'?

It's a metaphor breath.

 

[Psychopav]

This thread makes me laugh and cry.

If had I kissed 3 bucks goodbye as well, I'd feel like I just watched the trailer for Hardware Wars.

 

[bryhamm]

Well, since N.Y. Shreks decided to mention it ... I thought I would bump it.

 

[Topes]

That's like saying 12^2 is 14 (drop off the 1, compute the 2, add the one back in).

 

[MrPack]

Finally, an

 

[Bevo]

Finally, an

 

[UOFI_316]

Finally, an

So -5^2 is really -14??

 

[MikeMan]

Another vote for 57/54. Adding stuff that is not there by changing to "-1 X.." or "0 +.." etc.. is crazy.

6 is really an assumed 3*2, therefore 6^2 = 3*2^2 = 12. Insane!

It's not an operator b/c there's nothing in front to operate on! -5 is the number negative 5.

 

[Sid]

Finally, an

NO! It is 14, not -14, and I'll defend this with my last breath!!!

 

[shuke]

Another vote for 57/54. Adding stuff that is not there by changing to "-1 X.." or "0 +.." etc.. is crazy.

6 is really an assumed 3*2, therefore 6^2 = 3*2^2 = 12. Insane!

It's not an operator b/c there's nothing in front to operate on! -5 is the number negative 5.

Huh?