Daughter's math homework (1 Viewer)

[Maurile Tremblay]

at some point the convention of -5^2 was accepted that it is -(5^2)?as opposed to (-5)^2?...when did that become the convention out of curiosity?

According to the MathGuys around here, it's always been that way.

 

[Ilov80s]

You are just picking and choosing here. The - infront of x should not be considered any different than the - infront of x^2.

However, (-1)(5^2) is not the notational equivalent of -5^2. If you'd like a detailed explanation of the reason why they are not the same, please reread my previous posts on the matter. I've been fairly clear about it, but I'd be happy to answer specific questions.

If x = 5, -x^2 = -25.

However, -5^2 =25, NOT -25.

If this is true, then the burden on me is to prove why -x is distinct from -5 if they both have a minus in front of them. I am doing so by showing that -x and -5 are different in the same way that 2x and 25 are different and -1x and -1x are different. The minus sign has a different meaning when placed in front of a variable than when placed in front of a number.

A minus sign in front of a number is an adjective. It's a way of describing a complete concept. A minus sign in front of a variable is an operation. It's a way of describing the number by which we multiply the variable.

You may replace -5 with (-1*5) or even ((-1)*(5)) if you prefer. But you can't replace it without using parentheses around it, anymore than you can replace it with 17-22 without using parentheses.

-1x and -1x are different? Yeah, you are proving something here, but it isn't related to math.

 

[Mr. Pickles]

I can't tell, is fred shticking it up in here, or is he serious? Ever since Dawn/Don, I can't tell when he's wrong and when he's just pretending to be wrong.

It's a rich premium blend.

 

[Ilov80s]

1/10 .1 1/100 is .011/infinity is .00000...11/infinity plus .99999... = 1therefore 1/infinity = 0therefore 0 * infinity = 1therefore 1/0 = infinity

At least we now know this is all shtick.

 

[Short Corner]

1/10 .1 1/100 is .011/infinity is .00000...11/infinity plus .99999... = 1therefore 1/infinity = 0therefore 0 * infinity = 1therefore 1/0 = infinity

At least we now know this is all shtick.

I'm ok with it. He could have skipped lines 3 and 4. 1/0 could also be -inf depending on the expression you are evaluating

 

[Ignoratio Elenchi]

It's not like fred's wrong about anything here. He knows that the convention is essentially arbitrary, he just thinks we should have picked the other way to be the standard. Part of me agrees. "-5" is an integer, it shouldn't require parentheses. On the other hand, much of math becomes easier to understand and manipulate when we treat the - sign as a separate operator. I think ultimately we're better off with -52 = -25 than the alternative, but perhaps that's just because it's always been that way.

 

[bostonfred]

It's not like fred's wrong about anything here. He knows that the convention is essentially arbitrary, he just thinks we should have picked the other way to be the standard. Part of me agrees. "-5" is an integer, it shouldn't require parentheses. On the other hand, much of math becomes easier to understand and manipulate when we treat the - sign as a separate operator. I think ultimately we're better off with -52 = -25 than the alternative, but perhaps that's just because it's always been that way.

 

[Ilov80s]

It's not like fred's wrong about anything here. He knows that the convention is essentially arbitrary, he just thinks we should have picked the other way to be the standard. Part of me agrees. "-5" is an integer, it shouldn't require parentheses. On the other hand, much of math becomes easier to understand and manipulate when we treat the - sign as a separate operator. I think ultimately we're better off with -52 = -25 than the alternative, but perhaps that's just because it's always been that way.

Even when he said -1x is different from -1x? Or when he said 0 * infinity is 1?

 

[Ignoratio Elenchi]

The symbols we use, and the rules we have for manipulating them, are matters of convention. But the underlying math is, I think, objectively real. Integers exist as mathematical objects apart from our ability to conceive of them, and 2 + 2 = 4 would be a fundamental mathematical truth even if it tended to be false as a matter of physics. (Meaning that, in an alternate universe, whenever we took two apples and put them next to another two apples, if we somehow ended up with three apples, or five, because of some screwy alternate law of physics, it wouldn't change the fact that the integer two, when added to itself, would make four. In our universe, physics and math seem completely aligned — and indeed a universe in which physics and math are aligned may be the only type of universe that can possibly exist. But even if that weren't the case, I think math is real in and of itself; it doesn't depend on any agreement from physics to be real.)

I think I disagree with this. I believe 2 + 2 = 4 is a fundamental mathematical truth precisely because when you put two apples next to two other apples, you end up with four apples. And when you lay down two meter sticks end to end, and then lay down two more, they measure out four meters. That's how we define concepts like "integers" and "addition". If we lived in a universe where you ended up with five every time you put two and two together, there would be nothing fundamentally true about the expression 2 + 2 = 4.

 

[Ignoratio Elenchi]

It's not like fred's wrong about anything here. He knows that the convention is essentially arbitrary, he just thinks we should have picked the other way to be the standard. Part of me agrees. "-5" is an integer, it shouldn't require parentheses. On the other hand, much of math becomes easier to understand and manipulate when we treat the - sign as a separate operator. I think ultimately we're better off with -52 = -25 than the alternative, but perhaps that's just because it's always been that way.

Even when he said -1x is different from -1x? Or when he said 0 * infinity is 1?

For the former, I think you were missing his point there. One of them represents "negative one times x" and the other represents "the additive inverse of (ten plus x)". But I thought that whole bit about "notational equivalence" was sort of boring and pointless, so I wasn't paying close attention.Ditto for the latter, except I skimmed over because it looked like one of those shticky "proofs". The concept of dividing by zero (or multiplying by infinity) is undefined. But the limit of 1/x as x goes to zero is is infinity.

 

[bostonfred]

Even when he said -1x is different from -1x?

That was a typo that would have been clear from context in the previous post. It should have said -1x is different from -15. The context was that the minus signs in -x and -5 are treated differently. I contend that if x=5, then -x^2 is equal to -25, even though I also contend that -5^2 is equal to 25. That's because -x is literally nothing more than a shorthand way of expressing -1x. -5 is a fully defined number that is five less than zero, and five more than -10. If it seems odd that we treat -x differently from the way we treat -5, then consider that we treat -1x and -2x differently from the way we treat -15 and -25. It's a consistent way of reading things, and I believe that it's the correct way. Treating -5^2 as though it's the negative result of 5^2 means that you cannot square a wholly expressed negative number without using parentheses, and that's wrong to me.

Or when he said 0 * infinity is 1?

Go get an infinite supply of zeroes and start adding them up. It might take a while but I think you'll see I'm right.

 

[videoguy505]

I can't tell, is fred shticking it up in here, or is he serious? Ever since Dawn/Don, I can't tell when he's wrong and when he's just pretending to be wrong.

Deadly serious. I understand the convention, and I believe it's clearly wrong and creates ambiguity. They should have gone the other way when deciding on a convention.

OK. But upthread you said

If x = 5, -x^2 = -25,However, -5^2 =25, NOT -25.

I get that you think the integer should be treated differently than the variable.My issue with your position is that you can't solve -x^2 for x=5 and get -25, if you show your work, and maintain that your second statement is also true. At some point, when solving -x^2, you substitute in 5 and write down -5^2, at which point you'd halt because you've already given that answer as +25. So you'd never get that first statement.In more complicated equations, say when these kids get to quadratic formulas, or solving for two variables simultaneously, or any other more involved set of operations, the need for a standard convention that treats both integers and variables the same is needed. At some point, when solving these more complicated equations, they'll have to substitute in 5 for x, and if they forget a few steps later that the "5" they wrote into the equation has to be treated differently from a "5" elsewhere in the equation, because the substituted 5 is standing in for an "x" that's been solved for, then the process breaks down. They might forget that the minus sign in front of that "5" is an operation, and not an adjective, even though in other cases in the same formula, a minus sign in front of a five is an adjective.Like, if a kid is given:"Solve (-x^2 + -5) if x = 5. Show your work."They'll write:Known = 5-x^2 + -5 = ?therefore (substituting 5 for x)-5^2 + -5 = ? (now we have a minus sign acting as an operation in front of the first 5 and as an adjective in front of the second)-5^2 = +25 (your statement #2) therefore (substituting +25 for "-5^2" as per #2)+25 + -5 = ?? = 20... and that's inconsistent with your first statement, where you said "If x = 5, -x^2 = -25". The answer should be -30, not +20. I think it's more ambiguous, and less clear, to have one sign acting in two different ways. Especially since it requires people to track which numbers are, and always have been, numbers, and which have been substituted in from another part of the problem, or the result from a different problem, or what have you. It's far more clear, and less ambiguous to say "If you see parentheses, do those first. Then exponents. Then operators like + and -."

 

[Ilov80s]

Even when he said -1x is different from -1x?

That was a typo that would have been clear from context in the previous post. It should have said -1x is different from -15. The context was that the minus signs in -x and -5 are treated differently. I contend that if x=5, then -x^2 is equal to -25, even though I also contend that -5^2 is equal to 25. That's because -x is literally nothing more than a shorthand way of expressing -1x. -5 is a fully defined number that is five less than zero, and five more than -10. If it seems odd that we treat -x differently from the way we treat -5, then consider that we treat -1x and -2x differently from the way we treat -15 and -25. It's a consistent way of reading things, and I believe that it's the correct way. Treating -5^2 as though it's the negative result of 5^2 means that you cannot square a wholly expressed negative number without using parentheses, and that's wrong to me.

Or when he said 0 * infinity is 1?

Go get an infinite supply of zeroes and start adding them up. It might take a while but I think you'll see I'm right.

How does -5*-5= 25 if the negative is treated as a part of a unique natural number where the negative does not act as an operator indicating the opposite of?

 

[Ignoratio Elenchi]

Even when he said -1x is different from -1x?

That was a typo that would have been clear from context in the previous post. It should have said -1x is different from -15. The context was that the minus signs in -x and -5 are treated differently. I contend that if x=5, then -x^2 is equal to -25, even though I also contend that -5^2 is equal to 25. That's because -x is literally nothing more than a shorthand way of expressing -1x. -5 is a fully defined number that is five less than zero, and five more than -10. If it seems odd that we treat -x differently from the way we treat -5, then consider that we treat -1x and -2x differently from the way we treat -15 and -25. It's a consistent way of reading things, and I believe that it's the correct way. Treating -5^2 as though it's the negative result of 5^2 means that you cannot square a wholly expressed negative number without using parentheses, and that's wrong to me.

Or when he said 0 * infinity is 1?

Go get an infinite supply of zeroes and start adding them up. It might take a while but I think you'll see I'm right.

How does -5*-5= 25 if the negative is treated as a part of a unique natural number where the negative does not act as an operator indicating the opposite of?

The integer -5, when multiplied by itself, equals 25. It's not about operators and "opposite of", that's just simple shorthand for what's really happening. When you multiply -5 * -5, it's not like the negative signs are breaking off from the fives and canceling each other out. That's just an easy way for people to conceptualize and remember that when you multiply two negative integers together, the result is a positive integer. But that's not why the result is a positive integer.

 

[Ilov80s]

Even when he said -1x is different from -1x?

That was a typo that would have been clear from context in the previous post. It should have said -1x is different from -15. The context was that the minus signs in -x and -5 are treated differently. I contend that if x=5, then -x^2 is equal to -25, even though I also contend that -5^2 is equal to 25. That's because -x is literally nothing more than a shorthand way of expressing -1x. -5 is a fully defined number that is five less than zero, and five more than -10. If it seems odd that we treat -x differently from the way we treat -5, then consider that we treat -1x and -2x differently from the way we treat -15 and -25. It's a consistent way of reading things, and I believe that it's the correct way. Treating -5^2 as though it's the negative result of 5^2 means that you cannot square a wholly expressed negative number without using parentheses, and that's wrong to me.

Or when he said 0 * infinity is 1?

Go get an infinite supply of zeroes and start adding them up. It might take a while but I think you'll see I'm right.

How does -5*-5= 25 if the negative is treated as a part of a unique natural number where the negative does not act as an operator indicating the opposite of?

The integer -5, when multiplied by itself, equals 25. It's not about operators and "opposite of", that's just simple shorthand for what's really happening. When you multiply -5 * -5, it's not like the negative signs are breaking off from the fives and canceling each other out. That's just an easy way for people to conceptualize and remember that when you multiply two negative integers together, the result is a positive integer. But that's not why the result is a positive integer.

I would be interested to read your explanation.

 

[chet]

does mr pack have an alias?

i miss the big guy

 

[bostonfred]

My issue with your position is that you can't solve -x^2 for x=5 and get -25, if you show your work, and maintain that your second statement is also true. At some point, when solving -x^2, you substitute in 5 and write down -5^2, at which point you'd halt because you've already given that answer as +25. So you'd never get that first statement.

Sure you can. If x was equal to y-2, you would replace x with (y-2) if it was necessary in the formula. If you put parentheses around the 5, that works, too. I do agree - and have said as much in the thread - that there is some ambiguity for new learners, which is why this is a reasonable standard. But that doesn't change the fact that it's wrong.

 

[Ignoratio Elenchi]

I would be interested to read your explanation.

Negative integers aren't natural to us. They are a useful mathematical construction. There's no such thing as a bowl full of negative five apples. Negative integers are defined as the additive inverses of the positive integers. That is, given some positive integer (let's say 5), we define it's additive inverse (-5) as the number that, when added to 5, results in zero.Now we can wonder what happens when we multiply with these negative integers. The answer isn't obvious; well, it is to us, of course, because we've already been taught how to multiply with negative integers. But imagine we were trying to figure it out from scratch. Multiplication is like repeated addition. We can make sense of 2 * 5 because we can imagine five different piles of twos, and putting them all together. But how do we make negative 5 piles of 2? What does that even mean?

We know that 2 * 5 = 10. We also know that 5 + -5 = 0. We can use those facts and the distributive property of multiplication as follows:

0 = 2 * 0 = 2 * (5 + -5) = (2 * 5) + (2 * -5) = 10 + (2 * -5) = 0

Therefore by definition, the quantity (2 * -5) is the additive inverse of 10. In other words, 2 * -5 = -10. We could repeat this procedure to show that -2 * -5 = 10. But at no point was the - sign treated as an operator that acted on any integers. We never separated the - sign from the integers. It's not like we "multiplied both sides" by -1. That's just how we think of it, because it's easy to think of it that way. But that's not what's really happening. That's not why a negative times a negative equals a positive.

 

[Juxtatarot]

If only we required all negative integers to be written in red. Everyone would agree. 5^2 = 25

 

[jon_mx]

If only we required all negative integers to be written in red. Everyone would agree. 5^2 = 25

If we could only square the deficit.

 

[the moops]

If only we required all negative integers to be written in red. Everyone would agree. 5^2 = 25

5^2 = 25

 

[Juxtatarot]

If only we required all negative integers to be written in red. Everyone would agree. 5^2 = 25

5^2 = 25

Really?

 

[jon_mx]

If only we required all negative integers to be written in red. Everyone would agree. 5^2 = 25

5^2 = 25

Really?

Moops lives in the imaginary world.

 

[Mr. Pickles]

If only we required all negative integers to be written in red. Everyone would agree. 5^2 = 25

5^2 = 25

Really?

20 more pages.

 

[bostonfred]

How does -5*-5= 25 if the negative is treated as a part of a unique natural number where the negative does not act as an operator indicating the opposite of?

Here are several explanations. 1) -5 is equal to (0-5). Cross multiplying (0-5) * (0-5) gives us 0 - 0 - 0 + 25, for a total of positive twenty five. 2) Multiplication is just shorthand for addition. If I multiply three times three, I'm really saying three plus three plus three. You're adding the first number to itself a certain number of times indicated by the second number. When you multiply something by a negative number, you're adding the first number to itself a negative number of times, which is to say that you're subtracting it a certain number of times instead of adding it a certain number of times. So -3 times 3 is really a way of subtracting three three times, as opposed to adding it three times. -3 times -3 is similarly a way of subtracting -3 three times, which is identical to saying you're adding three three times. 3) Two wrongs make a right. It's pretty well known. Some people will try to tell you otherwise, but they have an agenda.

 

[jon_mx]

How does -5*-5= 25 if the negative is treated as a part of a unique natural number where the negative does not act as an operator indicating the opposite of?

Here are several explanations. 1) -5 is equal to (0-5). Cross multiplying (0-5) * (0-5) gives us 0 - 0 - 0 + 25, for a total of positive twenty five.

2) Multiplication is just shorthand for addition. If I multiply three times three, I'm really saying three plus three plus three. You're adding the first number to itself a certain number of times indicated by the second number. When you multiply something by a negative number, you're adding the first number to itself a negative number of times, which is to say that you're subtracting it a certain number of times instead of adding it a certain number of times. So -3 times 3 is really a way of subtracting three three times, as opposed to adding it three times. -3 times -3 is similarly a way of subtracting -3 three times, which is identical to saying you're adding three three times.

3) Two wrongs make a right. It's pretty well known. Some people will try to tell you otherwise, but they have an agenda.

So you are saying Bush should have invaded Pakistan too.

 

[Ignoratio Elenchi]

If we could only square the deficit.

So you are saying Bush should have invaded Pakistan too.

Really?

 

[jon_mx]

If we could only square the deficit.

So you are saying Bush should have invaded Pakistan too.

Really?

Sorry if I disturbed this otherwise so interesting 32-page thread which was beaten to death after the first two pages.

 

[tommyboy]

I would be interested to read your explanation.

Negative integers aren't natural to us. They are a useful mathematical construction. There's no such thing as a bowl full of negative five apples. Negative integers are defined as the additive inverses of the positive integers. That is, given some positive integer (let's say 5), we define it's additive inverse (-5) as the number that, when added to 5, results in zero.Now we can wonder what happens when we multiply with these negative integers. The answer isn't obvious; well, it is to us, of course, because we've already been taught how to multiply with negative integers. But imagine we were trying to figure it out from scratch. Multiplication is like repeated addition. We can make sense of 2 * 5 because we can imagine five different piles of twos, and putting them all together. But how do we make negative 5 piles of 2? What does that even mean?

We know that 2 * 5 = 10. We also know that 5 + -5 = 0. We can use those facts and the distributive property of multiplication as follows:

0 = 2 * 0 = 2 * (5 + -5) = (2 * 5) + (2 * -5) = 10 + (2 * -5) = 0

Therefore by definition, the quantity (2 * -5) is the additive inverse of 10. In other words, 2 * -5 = -10. We could repeat this procedure to show that -2 * -5 = 10. But at no point was the - sign treated as an operator that acted on any integers. We never separated the - sign from the integers. It's not like we "multiplied both sides" by -1. That's just how we think of it, because it's easy to think of it that way. But that's not what's really happening. That's not why a negative times a negative equals a positive.

seems like a support of why -5^2 would equal 25

 

[bostonfred]

Thank goodness jon dumbed up the math thread so I wouldn't feel so dirty agreeing with tommyboy.

 

[jon_mx]

Thank goodness jon dumbed up the math thread so I wouldn't feel so dirty agreeing with tommyboy.

I did provide the definitive answer a while back, so anything else I contributed was just gravy....

but =0-5^2 = -25....which is why i say it is ambiguous and depends upon the context. As a stand alone number, -5^2 is 25. But embedded in an equation as an operation it represents -25. That is the problem with the "-" sign in math. It is the same symbol for two different concepts, a negative and a minus. Not always the same outcome.

 

[the moops]

If only we required all negative integers to be written in red. Everyone would agree. 5^2 = 25

5^2 = 25

Really?

20 more pages.

There is too much shtick available to let this thread die.

 

[((Morpheus))]

This thread is awesome

 

[videoguy505]

My issue with your position is that you can't solve -x^2 for x=5 and get -25, if you show your work, and maintain that your second statement is also true. At some point, when solving -x^2, you substitute in 5 and write down -5^2, at which point you'd halt because you've already given that answer as +25. So you'd never get that first statement.

Sure you can. If x was equal to y-2, you would replace x with (y-2) if it was necessary in the formula. If you put parentheses around the 5, that works, too. I do agree - and have said as much in the thread - that there is some ambiguity for new learners, which is why this is a reasonable standard. But that doesn't change the fact that it's wrong.

Passing parens is problematic.Sometimes it'll just be messy, they'll end up with a final step that looks like "3 + 4 - (5) + 6 - (6) + (4) = ?" and they'll get the "what's up with all those parentheses?" question from a confused parent, and they'll have to answer "We keep the parentheses after substitution so that just in case later on in the problem there's an order-of-operations issue, we remember which numbers are number and which are variables." At which point the parent gives up on trying to understand 4th grade math.But later on, it could worse. The kids will move on to 5th grade, and if they're like me (and my co-worker's kid, who has to turn in her history reports as Powerpoint presentations) they might start messing around with computers, and maybe programming, and BASIC or whatever, and then the issue pops up again. You can't pass parens in a computer program (easily enough for the 5th graders, at least) in the simpler programming languages. If a variable is declared as an INT (or FLOAT as may be needed), it stays an INT, to pass parens would require it being a STRING or something.... a text-based variable instead of a numeric one, which limits the operations that can be performed upon it. While it's one thing to get a kid to remember he has to keep track of his parentheses, getting that kid a year later to write a simple calculator program on a PC will trip him up. Or if things move along, and the kid takes some computer classes, he might have one program feed its answer to another program, and he'd risk losing track of whether the original program took into account whether the - was an adjective or operator, and there'd be no way to pass that info to the second program, and things get really fouled up.If we're going to prepare these kids for engineering and science careers, thinking like a computer is a crucial step. Simple, consistent hierarchies for order-of-operations should make it easier (& always consistent) for them later.ETA: I don't care about what the order-of-operations is, or should be, or is more natural, or whatever. I'm just against treating variables and numbers differently, and mostly because it makes doing substitutions in solving equations, or constructing proofs, or writing programs and algorithms more difficult.

 

[jon_mx]

Let's put it this way...if B = -5.....then B^2 = 25. But if B= 5....then -B^2 = -25. It really matters if the "-" sign represents a negative or if it represents a minus. They are really not the same thing.

 

[Ignoratio Elenchi]

I would be interested to read your explanation.

Negative integers aren't natural to us. They are a useful mathematical construction. There's no such thing as a bowl full of negative five apples. Negative integers are defined as the additive inverses of the positive integers. That is, given some positive integer (let's say 5), we define it's additive inverse (-5) as the number that, when added to 5, results in zero.Now we can wonder what happens when we multiply with these negative integers. The answer isn't obvious; well, it is to us, of course, because we've already been taught how to multiply with negative integers. But imagine we were trying to figure it out from scratch. Multiplication is like repeated addition. We can make sense of 2 * 5 because we can imagine five different piles of twos, and putting them all together. But how do we make negative 5 piles of 2? What does that even mean?

We know that 2 * 5 = 10. We also know that 5 + -5 = 0. We can use those facts and the distributive property of multiplication as follows:

0 = 2 * 0 = 2 * (5 + -5) = (2 * 5) + (2 * -5) = 10 + (2 * -5) = 0

Therefore by definition, the quantity (2 * -5) is the additive inverse of 10. In other words, 2 * -5 = -10. We could repeat this procedure to show that -2 * -5 = 10. But at no point was the - sign treated as an operator that acted on any integers. We never separated the - sign from the integers. It's not like we "multiplied both sides" by -1. That's just how we think of it, because it's easy to think of it that way. But that's not what's really happening. That's not why a negative times a negative equals a positive.

seems like a support of why -5^2 would equal 25

To the extent that you could provide support for an essentially arbitrary convention, sure. I've said several times that it would make sense if the world agreed to say -52 = 25, since -5 is an integer. But there are also compelling reasons to say -52 = -25. The latter is how it's actually done. If you want to square a negative integer, you put it in parentheses. If you don't include the parentheses, the - sign is assumed to operate on the quantity (52). There's no would or should involved, there's just the way it is. It's like arguing that when networks put up their colored maps on election night, Republicans should be blue and Democrats should be red. One doesn't make any more or less sense than the other, but everyone agrees to do it one way, and that's the way it's done. 30 years from now, maybe -52 will be 25, but in all the time I've spent studying math it's always been -25.

 

[The Iguana]

The problem isn't that she would think 25 is an acceptable answer, its that she'd believe -5^2 is an acceptable question.

-5^2 is acceptable... it's sloppy but it is acceptable.This is an acceptable equation^2 - 15x + 25 = -5^2 there are a number of ways to solve it, one of which is:1. add 5^2 to both sides^2 - 15x + 25 + 5^2 = -5^2 + 5^2equates to^2 - 15x + 25 + 25 = 0from there you eventuall get that x = 5 or x = 10if you even think of trying to say that you have to subract (-5)^2 from both sides of this equation, you have no mathematical case or presidence to stand on.

I find it amazing that your bias is so ingrained that you utterly fail to recognize it.-5^2 would only be acceptable for personal notes, and then only if you were consistent. It's unacceptable notation to present as an answer or to another person because it is ambiguous - an ambiguity that can easily be fixed. And the equation you presented is just as equally valid to be x^2 - 15x = 0 as it is to be x^2 - 15x + 50 = 0.

no, it is not. Unless you believe in invisible ()'s. It is not the same.

 

[The Iguana]

So some here are honestly telling me that 5^2 - 5^2= 50?

yes, they typed that above. it's great, isn't it?

 

[The Iguana]

So some here are honestly telling me that 5^2 - 5^2= 50?

No, but what is 5^2+-5^2? I would say that is 50, because to me that says plus negative 5 squared, not minus 5 squared.

that would be wrong as well.

 

[culdeus]

wow, this again?

 

[Mjolnirs]

Seriously one of my favorite threads.

 

[Maelstrom]

Nothing to add except this (and I didn't read the whole thread to see if this had been shared before) cause I think Wolfram|Alpha is pretty cool:

http://www.wolframalpha.com/input/?i=-5² + 4 x 2³

http://www.wolframalpha.com/input/?i=-6² + 2 x 3²

 

[bostonfred]

How does -5*-5= 25 if the negative is treated as a part of a unique natural number where the negative does not act as an operator indicating the opposite of?

Here are several explanations. 1) -5 is equal to (0-5). Cross multiplying (0-5) * (0-5) gives us 0 - 0 - 0 + 25, for a total of positive twenty five.

2) Multiplication is just shorthand for addition. If I multiply three times three, I'm really saying three plus three plus three. You're adding the first number to itself a certain number of times indicated by the second number. When you multiply something by a negative number, you're adding the first number to itself a negative number of times, which is to say that you're subtracting it a certain number of times instead of adding it a certain number of times. So -3 times 3 is really a way of subtracting three three times, as opposed to adding it three times. -3 times -3 is similarly a way of subtracting -3 three times, which is identical to saying you're adding three three times.

3) Two wrongs make a right. It's pretty well known. Some people will try to tell you otherwise, but they have an agenda.

That explanation doesn't work at all.

I like how you said explanation 1 doesn't work at all, but didn't say a word about explanation 3

 

[Mr. Pickles]

Nothing to add except this (and I didn't read the whole thread to see if this had been shared before) cause I think Wolfram|Alpha is pretty cool:

http://www.wolframalpha.com/input/?i=-5² + 4 x 2³

http://www.wolframalpha.com/input/?i=-6² + 2 x 3²

Alpha is great, but you have to format it correctly.Steve Wolfram is also a fan of -5^2 = -25.

 

[Maurile Tremblay]

The symbols we use, and the rules we have for manipulating them, are matters of convention. But the underlying math is, I think, objectively real. Integers exist as mathematical objects apart from our ability to conceive of them, and 2 + 2 = 4 would be a fundamental mathematical truth even if it tended to be false as a matter of physics. (Meaning that, in an alternate universe, whenever we took two apples and put them next to another two apples, if we somehow ended up with three apples, or five, because of some screwy alternate law of physics, it wouldn't change the fact that the integer two, when added to itself, would make four. In our universe, physics and math seem completely aligned — and indeed a universe in which physics and math are aligned may be the only type of universe that can possibly exist. But even if that weren't the case, I think math is real in and of itself; it doesn't depend on any agreement from physics to be real.)

I think I disagree with this. I believe 2 + 2 = 4 is a fundamental mathematical truth precisely because when you put two apples next to two other apples, you end up with four apples. And when you lay down two meter sticks end to end, and then lay down two more, they measure out four meters. That's how we define concepts like "integers" and "addition". If we lived in a universe where you ended up with five every time you put two and two together, there would be nothing fundamentally true about the expression 2 + 2 = 4.

Cool. We disagree.That's fun!

 

[Maelstrom]

Nothing to add except this (and I didn't read the whole thread to see if this had been shared before) cause I think Wolfram|Alpha is pretty cool:

http://www.wolframalpha.com/input/?i=-5² + 4 x 2³

http://www.wolframalpha.com/input/?i=-6² + 2 x 3²

Alpha is great, but you have to format it correctly.Steve Wolfram is also a fan of -5^2 = -25.

Yah, the URL got mangled...I'll try to fix, but pretty sure the URL encoding is screwing up the link. Copy/Paste works though

 

[gonzobill5]

-5^2 is read "the opposite of 5 squared"

(-5)^2 is read "the square of the opposite of 5"

If you can communicate well there is much less confusion. Unfortunately many math teachers just write them on the board, read them the exact same way, and tell their students to remember which is which.

 

[bostonfred]

-5^2 is read "the opposite of 5 squared"

If you can communicate well there is much less confusion.

Sure, but "the opposite of 5 squared" can be read as, "the opposite of five, squared" or "the opposite of five-squared".It's like that thing with the girl spinning and you can't tell which direction but once you see it one way you can't see how someone could see it the other way. http://www.moillusions.com/2007/06/spinnin...l-illusion.html

 

[bostonfred]

After watching a while, I'm fairly certain she's just waving her leg back and forth from left to right. Now if I can just figure out which way she's facing.

 

[Hooper31]

I remember the day when my mind was blown when I first learned that "0.9999999..... (infinitely repeating)" is the same as "1".

There are a number of proofs for that; here's a simple one that's kind of cool.It should be evident that 0.999999... is equal to 9 * 0.111111....Ever since the invention of calculators, we've known that 1/9 is equal to 0.111111....Since 9 * 1/9 = 1, and since 0.999999... is 9 * 1/9, it must also be true that 0.999999... = 1.

Can you 'prove' 1 = 2?

Only if you let me divide by 0.