Should be easy math

[CaptainChewbacca]

I was helping my cousin go over her math homework, and she was doing true-false with regards to math problems. One question was:

1. |x| is always positive.

She marked true, and got marked wrong. The teacher explained that x could be +3 or -3, and then went onto the next. Am I insane, or did this teacher make an incredibly obvious mistake. My cousin was trying to figure out how it could be right, as she pointed it out to the teacher and was quickly dismissed.

[Yogi]

The teacher is wrong, but x could be 0 which technically isn't a positive number.

[SCRawl]

I'll second Yogi's analysis: while x can be negative or positive, |x| is certainly nonnegative. The teacher might be buffaloed into giving a correct mark if the "zero" possibility is somehow missed.

[SpacedTeddyBear]

Actually, the teacher is right. If you graph the function F(x)= abs(x), you'd have two lines with the slope of x and -x respectively diverging from the origin. So abs(x)= x for x>0
= -x for x<0

[CaptainChewbacca]

Actually, the teacher is right. If you graph the function F(x)= abs(x), you'd have two lines with the slope of x and -x respectively diverging from the origin. So abs(x)= x for x>0
= -x for x<0

Uh.. the question was "is abs(x) always positive". They weren't talking about the slope of abs(x).

[Molyneux]

Actually, the teacher is right. If you graph the function F(x)= abs(x), you'd have two lines with the slope of x and -x respectively diverging from the origin. So abs(x)= x for x>0
= -x for x<0

You're talking about the derivative of abs(x), not abs(x) itself.

The teacher is right, but for the wrong reason; abs(x) is non-negative, but not necessarily positive.

[Surlethe]

Actually, the teacher is right. If you graph the function F(x)= abs(x), you'd have two lines with the slope of x and -x respectively diverging from the origin. So abs(x)= x for x>0
= -x for x<0

You're thinking that d|x|/dx is not always positive; however, the sign of the derivative is not an indicator of the sign of the function. For example, f(x) = -1/x has a derivative which is always positive, but the value of the function }itself is always less than zero. In this case, |x| is, in fact, always positive for nonzero x.

However, CaptainChewbacca, your cousin did miss the problem, because |x| is not strictly positive; for x zero, it's not the case that |x|>0. The teacher's justification strikes me as incorrect, though; in her explanation, she gave the domain as including negatives (which it does), when the question asked about the range of the function.

[SpacedTeddyBear]

Woah guys, I wasn't talking about the derivative. But I did made a bad oopsie on my explaination. I made the same arguement that the teacher made on that the domain can be a non-negative number. Again my bad.

[slebetman]

Woah guys, I wasn't talking about the derivative. But I did made a bad oopsie on my explaination. I made the same arguement that the teacher made on that the domain can be a non-negative number. Again my bad.

The domain of x can be negative, true. But the domain of |x| cannot be negative. That is to say, if y = |x| then x may be negative but y can never be negative.

[Kuroneko]

As others noted, |x| may be zero, and hence is not always positive.

The domain of x can be negative, true. But the domain of |x| cannot be negative. That is to say, if y = |x| then x may be negative but y can never be negative.

You're confusing domain and range. The domain of |x| is the set of all real numbers (or some other seminormed space), but the range of |x| is the set of all nonnegative reals.

[Adrian Laguna]

The teacher is right, but the justification is wrong.

|x| is always positive.

The teacher explained that x could be +3 or -3

Yes, the x itself could be a negative number, but that is not what the statement is saying. The statement is saying that the absolute value of the x is always positive. The teacher is right because 0 is not a negative or a positve, zero is quite literally nothing.

[SyntaxVorlon]

Yeah. Tell her to ask her teacher

[SyntaxVorlon]

whoops,...
Tell her to ask again.

[CaptainChewbacca]

ARG!

The teacher's explanation MAKES NO SENSE! She says "its false because X is a variable". Unless she's talking about zero and not communicating, she's off her nut.

[Shadow WarChief]

Put abs(X) into a graphing calculator, either handheld or online and just show it to her beyond that, I can't think of anything else you could do to show her that she's abs(fucking wrong). If it actually gets to that point, you should probably bring it to the attention of the teacher's higher ups because this kind of stupidity should NOT be allowed to progress further.

[Surlethe]

ARG!

The teacher's explanation MAKES NO SENSE! She says "its false because X is a variable". Unless she's talking about zero and not communicating, she's off her nut.

That's bullshit; by that logic, "exp(x) is always positive" is false because x runs through R.

[Exonerate]

ARG!

The teacher's explanation MAKES NO SENSE! She says "its false because X is a variable". Unless she's talking about zero and not communicating, she's off her nut.

I think your cousin's math teacher is just incompetent and doesn't understand the problem.

[Kuroneko]

If for the moment you're more concerned about your cousing learning which answer is correct and why rather than divining the meaning of the teacher's explanation (however alarming its unintelligibility may be), just explain that absolutely value is "distance from zero". It then becomes quite obvious that this can be zero, and hence not always positive. This way of conceptualizing absolute value is also more useful than "gets rid of negatives", since that way she'll have a smoother transition to absolutely value in other contexts later on (e.g., complex numbers, which is a fairly standard high school topic).

[Master of Cards]

I was helping my cousin go over her math homework, and she was doing true-false with regards to math problems. One question was:

1. |x| is always positive.

She marked true, and got marked wrong. The teacher explained that x could be +3 or -3, and then went onto the next. Am I insane, or did this teacher make an incredibly obvious mistake. My cousin was trying to figure out how it could be right, as she pointed it out to the teacher and was quickly dismissed.

x by it self can be either l l those lines mean it turns to a + number no matter what

[Surlethe]

x by it self can be either l l those lines mean it turns to a + number no matter what

That's a rather simplistic way of looking at it, and it lends itself to misconceptions, in the special case of zero in the real numbers, as well as when "absolute value" is extended in vector and complex algebra. Better to use Kuroneko's explanation of absolute value as distance from zero.

Interestingly enough, as an extension of that explanation, |x-y| represents the distance between x and y, and this way of looking at it holds in more general cases.

[Kuroneko]

Interestingly enough, as an extension of that explanation, |x-y| represents the distance between x and y, and this way of looking at it holds in more general cases.

Indeed. As a bonus, the inequality |x-y|≤|x|+|y| then expresses the same thing as the triangle inequality AC≤AB+BC commonly taught in geometry: going straight from one place to another covers no more distance than making a detour through some point (C or zero) first. Realizing the same concepts recur in many different guises is very important to building problem-solving skills.

P.S. Not to ruin a witty quote, but you do realize that the pH scale is logarithmic, right?

[Surlethe]

Interestingly enough, as an extension of that explanation, |x-y| represents the distance between x and y, and this way of looking at it holds in more general cases.

Indeed. As a bonus, the inequality |x-y|≤|x|+|y| then expresses the same thing as the triangle inequality AC≤AB+BC commonly taught in geometry: going straight from one place to another covers no more distance than making a detour through some point (C or zero) first. Realizing the same concepts recur in many different guises is very important to building problem-solving skills.

Indeed. Actually, I didn't hear about the "triangle inequality" until I got into Calculus III, where we experienced it with vector addition. The algebraic representation looks intimidating at first blush, but when you think about it, it makes sense, and is nicely generalizable from the reals into more complicated settings (at least the nice stuff (R^n, C) which I've studied so far).

P.S. Not to ruin a witty quote, but you do realize that the pH scale is logarithmic, right?

Yes. That's what I found funny about the quote, so I sigged it.

[Kuroneko]

Actually, I didn't hear about the "triangle inequality" until I got into Calculus III, where we experienced it with vector addition. The algebraic representation looks intimidating at first blush, but when you think about it, it makes sense, and is nicely generalizable from the reals into more complicated settings (at least the nice stuff (R^n, C) which I've studied so far).

That's strange. You probably encountered it in the form of determining whether or not there exists a triangle with sides of three given lengths, even if the name itself was not covered. Speaking of vectors, did your class prove it? If so, I'm curious of the method of showing the Cauchy-Schwarz inequality x·y ≤ |x||y| (the triangle inequality for R^n is a direct consequence).

Yes. That's what I found funny about the quote, so I sigged it.

Ah.

[Havok]

This is the most confusing thing I have read on here to date. Does anyone know if there is like a Math for Dummies (i.e. ME) or something that might help me not feel so much like, well, a dummy when I read these threads?

[Hawkwings]

It's only confusing because of Chewie's cousin's stupid math teacher. |x| cannot be negative. End of story. Graph it if she doesn't believe it.

Other than that, what Kuroneko and Surlethe are talking about is probably beyond the grasp of most people on the board, not to mention in real life.

To get the best foundation though, college classes really are the best way to go.

I'd really say that High School classes are the foundation of math. You know, good old basic manipulation of numbers, all the way up to basic calculus.