I Why does the answer key sometimes have a different form compared to my solution?

[NODARman]

TL;DR Summary

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When I solve the equation sometimes the answer in the answer keys is different but the same. Why do they do that?
For example:
After solving the equation I got 1/√2 which is the same as √2/2 because we multiplied it by √2/√2. Is there any good explanation why the book writer mathematicians like to do that thing?

 

[topsquark]

TL;DR Summary: .

When I solve the equation sometimes the answer in the answer keys is different but the same. Why do they do that?
For example:
After solving the equation I got 1/√2 which is the same as √2/2 because we multiplied it by √2/√2. Is there any good explanation why the book writer mathematicians like to do that thing?

Usually the standard way to write a fraction that has roots in it is to "rationalize" the denominator to remove any roots from the denominator. It's just the standard way of writing it.

-Dan

 

[Orodruin]

That said, these two are obviously equivalent. There are cases, in particular in relativity, where two answers may look significantly more different than this on the surface yet still be equivalent.

 

[symbolipoint]

TL;DR Summary: .

When I solve the equation sometimes the answer in the answer keys is different but the same. Why do they do that?
For example:
After solving the equation I got 1/√2 which is the same as √2/2 because we multiplied it by √2/√2. Is there any good explanation why the book writer mathematicians like to do that thing?

You may already understand. No matter - the answer key is to help you to know if you handled the problem solving properly or not. The actual form of the answer in the key is usually far less important.

 

[PeroK]

Usually the standard way to write a fraction that has roots in it is to "rationalize" the denominator to remove any roots from the denominator. It's just the standard way of writing it.

-Dan

I would write $\sin \frac \pi 4 = \frac 1 {\sqrt 2}$. I don't think I've seen it written as $\frac {\sqrt 2} 2$.

PS now I have!

 

[PeroK]

TL;DR Summary: .

When I solve the equation sometimes the answer in the answer keys is different but the same. Why do they do that?
For example:
After solving the equation I got 1/√2 which is the same as √2/2 because we multiplied it by √2/√2. Is there any good explanation why the book writer mathematicians like to do that thing?

There is no good reason that I can see.

 

[symbolipoint]

I would write $\sin \frac \pi 4 = \frac 1 {\sqrt 2}$. I don't think I've seen it written as $\frac {\sqrt 2} 2$.

That value for sine has USUALLY been written in the rationalized form, both in the textbooks and in lecture instruction. ( $\frac {\sqrt 2} 2$ ) Some occasions happened, best I can recall, that drawing some right triangles were done using the non-rationalized form because doing so was more convenient at that time.

 

[Orodruin]

I don't think I've seen it written as 22.

https://en.wikipedia.org/wiki/Sine_and_cosine#Special_values

I mean, I don’t see any point in quibbling over this. I could write it $3\sqrt 2/ 2\sqrt 9$ if I wanted to and it would still be correct. There is no one unique way of representing this.

 

[PeroK]

https://en.wikipedia.org/wiki/Sine_and_cosine#Special_values

I mean, I don’t see any point in quibbling over this.

I genuinely thought I'd never seen that before.

To the best of my knowledge, no one writes $\frac{\sqrt \pi}{\pi}$ instead of $\frac 1 {\sqrt \pi}$.

Square roots appear in the denominator all over statistics, quantum mechanics, and the gamma factor in SR!

I'm not quibbling about one being right or wrong. But, I am disputing the claim that square roots should not appear in the denominator.

 

[PeroK]

Who's going to revise all the Clebsch-Gordan tables? I nominate @symbolipoint.

 

[symbolipoint]

Who's going to revise all the Clebsch-Gordan tables? I nominate @symbolipoint.

Clebsch-Gordan not my areaMy statement (post #7) was about Trigonometry instruction; not other topics.

 

[topsquark]

I would write $\sin \frac \pi 4 = \frac 1 {\sqrt 2}$. I don't think I've seen it written as $\frac {\sqrt 2} 2$.

PS now I have!

I'm not saying that I understand why the convention is what it is. It makes no sense to me. I completely agree that $\frac{1}{\sqrt{2}}$ or $\dfrac{1}{i}$ are more natural answers in this case.

-Dan

 

[Mondayman]

I can recall some of my classmates losing marks because they didn't rationalize their denominators. If it was specified to do so I would agree, but it wasn't at all in this case. I thought it was absurd.

 

[Orodruin]

I can recall some of my classmates losing marks because they didn't rationalize their denominators. If it was specified to do so I would agree, but it wasn't at all in this case. I thought it was absurd.

That’s not absurd, it is idiotic imho.

 

[MidgetDwarf]

That’s not absurd, it is idiotic imho.

high school algebra text are based of Euler's Elements of Algebra. Where Euler does not rationalize the denominator. Who are we to argue wit Euler.

 

[pasmith]

I can recall some of my classmates losing marks because they didn't rationalize their denominators.

Possibly the marker could not recognise that the answers were equivalent, or the mark scheme was poorly drafted and said "\sqrt{2}/2" rather than "\sqrt{2}/2 or equivalent".

 

[Orodruin]

Possibly the marker could not recognise that the answers were equivalent, or the mark scheme was poorly drafted and said "\sqrt{2}/2" rather than "\sqrt{2}/2 or equivalent".

Someone not able to recognise that $1/\sqrt 2$ is equivalent to $\sqrt 2 / 2$ should not be teaching math.

 

[symbolipoint]

Someone not able to recognise that $1/\sqrt 2$ is equivalent to $\sqrt 2 / 2$ should not be teaching math.

Topic drift? Student checking his answer in the book answer key should realize the book might or might not be using unrationalized form and think accordingly. Work assessor checking students' answers may apply standard instructions to accept final answers in only rationalized form, but depending on steps needed and shown, should be permitted to give partial credit.

 

[topsquark]

Possibly the marker could not recognise that the answers were equivalent, or the mark scheme was poorly drafted and said "\sqrt{2}/2" rather than "\sqrt{2}/2 or equivalent".

I obviously am not in the class but when I was in both High School and College my Math instructors always required that we rationalize the denominator. If that's the case here then we should expect that points will be lost. (My Physics instructors didn't care so much.)

-Dan

 

[Orodruin]

may apply standard instructions to accept final answers in only rationalized form

Such ”standard” instructions should be avoided. It is like deducting points in a history exam for writing that the father of Elizabeth II was George VI instead of writing that Elisabeth II was the daughter of George VI.

To recognize equivalent answers should be imperative to students, but even more so for teachers and people correcting standardized exams.

 

[symbolipoint]

Such ”standard” instructions should be avoided. It is like deducting points in a history exam for writing that the father of Elizabeth II was George VI instead of writing that Elisabeth II was the daughter of George VI.

To recognize equivalent answers should be imperative to students, but even more so for teachers and people correcting standardized exams.

Realize that students are still learning their subjects and students are supposed to learn to follow instructions.

 

[PeroK]

If you look up any page on the normal distribution you'll find the normalisation factor of $\frac 1 {\sqrt{2\pi}}$. What would these maths instructors say about that?

 

[Orodruin]

Realize that students are still learning their subjects and students are supposed to learn to follow instructions.

… which puts even more impetus on instructions being reasonable, which I would argue that they are not if they require rationalization of the denominator.

 

[Mark44]

I obviously am not in the class but when I was in both High School and College my Math instructors always required that we rationalize the denominator.

Most algebra or precalculus textbooks I've seen spend a fair amount of time on exercises related to simplifying rational expressions. I believe that the preference for, say $\frac {\sqrt 2} 2$ over $\frac 1 {\sqrt 2}$ goes back to the time before calculators and such.

It's a lot easier to divide an approximation to $\sqrt 2$ by 2, than to divide 1 by an approximation to $\sqrt 2$.

For example, try long division (on paper) of each of these:
$$\frac {1.41421356} 2$$
versus
$$\frac 1 {1.41421356}$$

 

[symbolipoint]

Realize that students are still learning their subjects and students are supposed to learn to follow instructions.

Yes. That is the truth, and students do lose credit for not following instructions. I am not lying.

 

[symbolipoint]

If you look up any page on the normal distribution you'll find the normalisation factor of $\frac 1 {\sqrt{2\pi}}$. What would these maths instructors say about that?

I was not referring to Statistics instruction or to practices there. I was referring to instruction in Trigonometry.

 

[symbolipoint]

… which puts even more impetus on instructions being reasonable, which I would argue that they are not if they require rationalization of the denominator.

Not within my control. The practices are the practices. If instructor expects final answer in a certain form, either follow instructions of receive less credit.

 

[Orodruin]

Not within my control. The practices are the practices. If instructor expects final answer in a certain form, either follow instructions of receive less credit.

I never said you can control particular instructors. That practices are practices is a cheap argument. If the practice is bad then one has to be able to say so. There is no particular reason to require rationalizing the denominator here. It does not add anything of value.

 

[symbolipoint]

I never said you can control particular instructors. That practices are practices is a cheap argument. If the practice is bad then one has to be able to say so. There is no particular reason to require rationalizing the denominator here. It does not add anything of value.

Still the practices as they may be conducted or expected is the truth. Try arguing about this with your professors and see how much progress you make.

 

[Orodruin]

Still the practices as they may be conducted or expected is the truth. Try arguing about this with your professors and see how much progress you make.

I have not called anyone ”my professor” in decades. As a university professor I’d happily argue with the people making such bad standards.

 

[PeroK]

I was not referring to Statistics instruction or to practices there. I was referring to instruction in Trigonometry.

It's all mathematics.

 

[symbolipoint]

I have not called anyone ”my professor” in decades. As a university professor I’d happily argue with the people making such bad standards.

So what is this? Cultural difference among the various universities' departments?

 

[symbolipoint]

Still the practices as they may be conducted or expected is the truth. Try arguing about this with your professors and see how much progress you make.

And I see your reaction, but as I say that is (or for sure was) the truth, not kidding. Professors were often enough, busy. Their graders were busy. If a certain form of an answer was specified, then that was what we needed to use. If not followed, then less credit for an exercise. If answer in the back of book was in a different form, then we knew what to do. If grader checks students homework and the work did not conform to a specified outline, then either less credit or no credit; if professor gave a test or quiz and said, "Do not do computations! Give answer in symbolic form only!", then students who did not follow received no credit. If prof. gave test or quiz and said, "round all answers to the nearest tenths place", then any answer not so reported received no credit. If computer science professor expected exercise assignments turned into include data table and flow-diagrams, he meant it. When students were given low scores on such assignments and tried to ask the professor about this, students were cut-off from receiving those discussions or explanations. Most of these professors were not lenient.

 

[Orodruin]

Professors were often enough, busy. Their graders were busy.

Generally I have little pity left over for this. Yes, I am generally quite busy, but quality in teaching is one of the university's main pillars and students should expect better.

 

[Office_Shredder]

I think requiring a rationalized denominator is part of setting a standard for students who don't have good taste in what answers can look like.For example if the problem is to evaluate $\cos(\pi/3)$, is $\sin(\pi/6)$ an acceptable answer? It's equivalent to the right answer, so why not?

Or let's say you have solve $x^2-9=0$. Your student carefully applies the quadratic formula and writes

$\frac{0 \pm \sqrt{0+4*9}}{2*1}$That's their final answer. You have to assign this answer a grade that reflects their understanding of the concepts of Algebra. What do you give them? An A? That's what people are kind of arguing for here.

Write all answers as rationalized fractions is strict, but at this level you need a policy to demonstrate if people understand what they've written or if they're just regurgitating symbols back at you.

 

[PeroK]

And I see your reaction, but as I say that is (or for sure was) the truth, not kidding. Professors were often enough, busy. Their graders were busy. If a certain form of an answer was specified, then that was what we needed to use. If not followed, then less credit for an exercise. If answer in the back of book was in a different form, then we knew what to do. If grader checks students homework and the work did not conform to a specified outline, then either less credit or no credit; if professor gave a test or quiz and said, "Do not do computations! Give answer in symbolic form only!", then students who did not follow received no credit. If prof. gave test or quiz and said, "round all answers to the nearest tenths place", then any answer not so reported received no credit. If computer science professor expected exercise assignments turned into include data table and flow-diagrams, he meant it. When students were given low scores on such assignments and tried to ask the professor about this, students were cut-off from receiving those discussions or explanations. Most of these professors were not lenient.

It seems to me that, if what you say is true, some of your professors were suffering from psychological disorders. Which may be a good reason not to be unduly influenced by them and not to promote their somewhat sadistic methods or excessive pedantry as a productive way to teach mathematics.

 

[symbolipoint]

It seems to me that, if what you say is true, some of your professors were suffering from psychological disorders. Which may be a good reason not to be unduly influenced by them and not to promote their somewhat sadistic methods or excessive pedantry as a productive way to teach mathematics.

Refering to the quote of message in post #36,
Most of those professors actually did teach well, but had some very strict instructional standards. They were tough for some of the students. Some of the reasons could be supportable. As you might imagine, some of the students did complain; to the professors during class time, and among themselves in and out of the classroom.

 

[symbolipoint]

Or let's say you have solve x2−9=0. Your student carefully applies the quadratic formula and writes

0±0+4∗92∗1That's their final answer. You have to assign this answer a grade that reflects their understanding of the concepts of Algebra. What do you give them? An A? That's what people are kind of arguing for here.

Write all answers as rationalized fractions is strict, but at this level you need a policy to demonstrate if people understand what they've written or if they're just regurgitating symbols back at you.

The example you gave would often be a factorization fact exercise for some Algebra 1 students who may have not yet been in Algebra 2; and most typically the students would learn about general solution for quadratic equation in Algebra 2. So that example could be a situation in which instructor expects the use of factorization and not use general formula solution.

(note: the formatting for part of the quoting did not work properly, to show that quadratic formula solution of your example.)

 

[Mark44]

(note: the formatting for part of the quoting did not work properly, to show that quadratic formula solution of your example.)

Just copying or quoting a section of text that contains LaTeX doesn't work

 

[Mark44]

Or let's say you have solve
$x^2-9=0$. Your student carefully applies the quadratic formula and writes

$\frac{0 \pm \sqrt{0+4*9}}{2*1}$

This discussion of whether equivalent answers are allowed or not puts me in mind of Randall Munroe's response after a lengthy and fruitless discussion with a Verizon customer service drone over a bill for $.002. In case the name is not familiar, he's the author of the XKCD blog/cartoons.

The amount on the check is $$\left(0.002 + e^{i\pi} + \sum_{i = 1}^\infty \frac 1 {2^n}\right) \text{ dollars}$$

 

[symbolipoint]

Just copying or quoting a section of text that contains LaTeX doesn't work

Yet in post #40, you did make it work. How?

 

[Mark44]

Yet in post #40, you did make it work. How?

I quoted the section of text up to the start of the LaTeX, and then opened the edit window to copy the unrendered LaTeX and paste it onto the end of what I had quoted. As a mentor I'm able to edit posts, but that capability probably isn't available to regular members.

If you use the Quote feature to quote text with LaTeX in it, you'll undoubtedly need to redo any LaTeX that was there. Since you didn't do this, that's why the quadratic equation solution that you quoted came out as it did.

 

[PeroK]

As a mentor I'm able to edit posts, but that capability probably isn't available to regular members.

Perhaps it's an idea for you to go round all the threads on PF and edit the absurd and inexplicable instances of $\dfrac 1 {\sqrt 2}$? Since it is such an abomination to those with heightened mathematical sensibilities.

 

[Mark44]

Perhaps it's an idea for you to go round all the threads on PF and edit the absurd and inexplicable instances of $\dfrac 1 {\sqrt 2}$? Since it is such an abomination to those with heightened mathematical sensibilities.

No, thanks. I don't have a problem with $\frac 1 {\sqrt 2}$ -- I was just offering a plausible explanation for why elementary algebra textbooks spend so much time on simplifying these types of expressions.

 

[WWGD]

high school algebra text are based of Euler's Elements of Algebra. Where Euler does not rationalize the denominator. Who are we to argue wit Euler.

No wonder I'm a fan of the Houston Eulers.

 

[Mondayman]

If you are learning about rationalizing denominators I can understand why. But as long as the answer is correct it shouldn't matter whether the radical is in the denominator or numerator. Or if it's written with exponents instead. I actually preferred doing so in my calculus and DE courses sometimes, I felt like I could read it more clearly. Instructor had no issue.

 

[SammyS]

To the best of my knowledge, no one writes $\dfrac{\sqrt \pi}{\pi}$ instead of $\dfrac 1 {\sqrt \pi}$.

Square roots appear in the denominator all over statistics, quantum mechanics, and the gamma factor in SR!

Either way, neither of these denominators is rationalized.

$\dfrac{\sqrt \pi\,}{\pi}$ , $\dfrac 1 {\sqrt \pi\, }$

 

[PeroK]

Either way, neither of these denominators is rationalized.

$\dfrac{\sqrt \pi\,}{\pi}$ , $\dfrac 1 {\sqrt \pi\, }$

Ah, so they are both wrong!

 

[WWGD]

Sorry to tell the OP that there are infiitely -many equivalent ways of writing a mathematical expression. You're lucky you're dealig with essentially numerical answers. If the answer was more complex, such as , e.g., a Fourier Series solution, there would be many more different (all correct) expressions for it.

 

[Mondayman]

The most recent Numberphile video is about that. You can integrate functions using different methods and end up with different answers. Shows why the constant of integration is important. I would recommend it to anyone learning calculus.