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

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• NODARman
In summary: There is no good reason that I can see.-DanIn summary, the convention is to rationalize the denominator in fractional equations with roots in order to make the fraction easier to work with. There are cases where two solutions may look significantly different on the surface, but be equivalent.

#### 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?

NODARman said:
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

scottdave, chwala, FactChecker and 5 others
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.

WWGD and topsquark
NODARman said:
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.

scottdave and FactChecker
topsquark said:
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!

Last edited:
pasmith and topsquark
NODARman said:
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.

topsquark
PeroK said:
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.

scottdave, chwala, FactChecker and 1 other person
PeroK said:
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.

chwala, dextercioby and topsquark
Orodruin said:
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.

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

dextercioby and topsquark
PeroK said:
Who's going to revise all the Clebsch-Gordan tables? I nominate @symbolipoint.
Clebsch-Gordan not my area

My statement (post #7) was about Trigonometry instruction; not other topics.

scottdave, FactChecker and topsquark
PeroK said:
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

PeroK
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.

Mondayman said:
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.

Mondayman and PeroK
Orodruin said:
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.

Mondayman said:
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".

pasmith said:
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.

Orodruin said:
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
pasmith said:
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

symbolipoint
symbolipoint said:
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.

PeterDonis, dextercioby, topsquark and 2 others
Orodruin said:
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?

symbolipoint said:
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.

PeterDonis
topsquark said:
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
$$\frac 1 {1.41421356}$$

George Jones, dextercioby, epenguin and 6 others
symbolipoint said:
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.

chwala and topsquark
PeroK said:
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.

Orodruin said:
… 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.

topsquark
symbolipoint said:
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.

PeterDonis
Orodruin said:
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.

PeroK
symbolipoint said:
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.

chwala, PeterDonis and symbolipoint
symbolipoint said:
I was not referring to Statistics instruction or to practices there. I was referring to instruction in Trigonometry.
It's all mathematics.

topsquark
Orodruin said:
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 said:
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.

symbolipoint said:
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.

dextercioby and topsquark
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.

martinbn, FactChecker, PeroK and 2 others