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

• I
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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.

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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.

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

0±0+4∗92∗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.
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.)

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(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

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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 $$\left(0.002 + e^{i\pi} + \sum_{i = 1}^\infty \frac 1 {2^n}\right) \text{ dollars}$$

Mondayman, topsquark and berkeman
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Just copying or quoting a section of text that contains LaTeX doesn't work
Yet in post #40, you did make it work. How?

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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.

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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.

dextercioby
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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.

George Jones, symbolipoint and PeroK
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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, SammyS and berkeman
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.

PeroK
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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\, }##

dextercioby
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Either way, neither of these denominators is rationalized.

##\dfrac{\sqrt \pi\,}{\pi}## , ##\dfrac 1 {\sqrt \pi\, }##
Ah, so they are both wrong!

SammyS
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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
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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.

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Ah, so they are both wrong!
Right. It should be ##\frac{\sqrt{154}}{22}##

topsquark
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Here is maybe another answer:
Because pages cost money and ink to print different answers and different parts of answers cost money.

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Either way, neither of these denominators is rationalized.

##\dfrac{\sqrt \pi\,}{\pi}## , ##\dfrac 1 {\sqrt \pi\, }##
##\dfrac{1/\sqrt\pi}{1}##

Fixed

Anyway, I’d expect teachers to be rational…

vela, SammyS and PeroK
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##\dfrac{1/\sqrt\pi}{1}##

Fixed
But you still have a denominator (in the numerator fraction) that is irrational.

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But you still have a denominator (in the numerator fraction) that is irrational.
You’re welcome to rationalise it

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Clebsch-Gordan not my area

My statement (post #7) was about Trigonometry instruction; not other topics.
This is the way it was presented in the precalculus books I taught from in the '70s. I liked it because it made the pattern: ##sin(0)=\sqrt{0}/2; sin(30)=\sqrt{1}/2; sin(45)=\sqrt{2}/2; sin(60)=\sqrt{3}/2; sin(90)=\sqrt{4}/2##. The students seemed to like that.
But teaching conventions like that may have changed a lot since the '70s.

symbolipoint
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This is the way it was presented in the precalculus books I taught from in the '70s. I liked it because it made the pattern: ##sin(0)=\sqrt{0}/2; sin(30)=\sqrt{1}/2; sin(45)=\sqrt{2}/2; sin(60)=\sqrt{3}/2; sin(90)=\sqrt{4}/2##. The students seemed to like that.
But teaching conventions like that may have changed a lot since the '70s.
Interesting representation in keeping with denominator of 2. I really never saw it done like that.

romsofia
I would always use this following explanation to high schoolers when I tutored for why they "have" to rationalize their denominators. I personally didn't care, but a lot of teachers would take off points if they don't.

In human history, people first started counting things, like how many cows they own: 1,2,3... These are called the "natural numbers" because you naturally start at one. We next realized that hey, "no cow" should also have a number, which we call 0 and we have a new set of numbers called "whole numbers". Naturally, debt came into the number systems because you owe me a cow (-1) or two cows (-2), and now we have a whole new set of numbers called the integers (negative and positive whole numbers). Well, next humans started discussing parts of the whole. If we have 5 cows, and I take 2 of them, I now have 2 of the 5 cows, or to make life simple, I have ##\frac{2}{5}## of the cows. Number wise, we now have numbers between numbers, and we call these the "rational numbers" because they are just ratios, or parts of wholes. Notice how square roots haven't been "invented" yet? So, your teachers are just following history, and because it's their class, we will too. I don't actually know the history of numbers that well, but the story worked for most of them!

Now, the math reason (Which I do for students in pre-calc/algebra 2)... rational numbers are defined to be ##\frac{a}{b}## where ##a,b \in \mathbb{Z}, b \neq 0## so, yes, your teacher is correct to take points off your test because you're technically wrong to keep it as ##\frac{2}{\sqrt{5}}##. But then I also tell my students that, technically, you can't even write ##\frac{2\sqrt{5}}{5}## because ##2\sqrt{5} \notin \mathbb{Z}## and you should have to write it as ##\sqrt{5} \times \frac{2}{5}## so if you ever want to be petty, feel free to bring that up in class.

So, to OP, that's why the answer key in a math book won't have square roots in a denominator, because by definition of rational numbers, that number doesn't "exist". Although, as you can see, most of us won't really care (which I also tell the students I use to tutor if I helped them with these concepts), but when in school grades matter, so best not to let points get away!

topsquark and SammyS
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so, yes, your teacher is correct to take points off your test because you're technically wrong to keep it as ##\frac{2}{\sqrt{5}}##.
It's a convention, nothing more. Real numbers have reciprocals too!

What about calculators and computer language syntax? If you program ##1/0## you get a error. Are you seriously saying that if you program ##1/\sqrt 2## a computer should return a computational error?

How many students are turned off mathematics by this sort of pointless pedantry?

PeterDonis, topsquark and Orodruin
How many students are turned off mathematics by this sort of pointless pedantry?
I would guess none. If that is all it takes to turn someone off mathematics, he wasn't into it in the first place.

topsquark, FactChecker and PeroK
romsofia
Are you seriously saying that if you program ##1/\sqrt 2## a computer should return a computational error?
Of course not, but would I write down ##\frac{1}{\sqrt{2}}## on a math exam? Nope. In a similar vein, would you stop at ##\sqrt{80}##, or would you write ##4\sqrt{5}##? ##\frac{7\sqrt{10}}{\sqrt{2}}## or ##7\sqrt{5}##? Let's make it more algebraic, ##\frac{1}{i-\sqrt{3}}## or ##-\frac{i+\sqrt{3}}{4}##? ##\ln(x^3)## or ##3\ln(x)##? ##\ln(-1)## or ##i\pi##? None of these matter to a computer! If you're against reducing as a whole, I get the point.

Now, would I take points off an exam? No, I'm a physicist, I personally don't care. Numbers are numbers. But, if you write something down in the form of a ratio, you follow the definition of rational numbers.

Do I think students get tired of it? Of course.

PeroK
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I would guess none. If that is all it takes to turn someone off mathematics, he wasn't into it in the first place.
I would profoundly disagree with that attitude on a number of levels. First, that could be a reason that women are discouraged from studying mathematics. You would need some educational research to settle the matter, but it may be that women generally think more flexibly and are put off by that sort of hidebound thinking. In fact, your post in this day and age is quite explicitly sexist, as you assume the student is a "he".

More generally, that sort of attitude prevents fresh blood from entering institutions or professions. There is a certain type that inhabits the profession and they are at pains to ensure those entering the profession are of the same type.

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Of course not, but would I write down ##\frac{1}{\sqrt{2}}## on a math exam? Nope. In a similar vein, would you stop at ##\sqrt{80}##, or would you write ##4\sqrt{5}##?
There is nothing to choose between ##\sqrt{80}## and ##4\sqrt{5}##. They are both equally simple.
##\frac{7\sqrt{10}}{\sqrt{2}}## or ##7\sqrt{5}##?
##7\sqrt{5}## is clearly simpler. Whereas, there is no sense in which ##\frac {\sqrt 2} 2## is simpler than ##\frac 1 {\sqrt 2}##. This is why, outside of your dictatorship (e.g. in the realm of QM), ##\frac 1 {\sqrt 2}## is preferred. E.g. in the so-called singlet state:

https://en.wikipedia.org/wiki/Singlet_state#Singlets_and_entangled_states

I don't believe for one minute that quantum physicists fail to grasp the basics of elementary mathematics. Just because they don't obey your hidebound conventions.

No, I'm a physicist, I personally don't care.
I'm struggling to understand you now, as there is definitely no such convention in physics as the one you claim. Pick up any QM book if you don't believe me.

I would profoundly disagree with that attitude on a number of levels. First, that could be a reason that women are discouraged from studying mathematics. You would need some educational research to settle the matter, but it may be that women generally think more flexibly and are put off by that sort of hidebound thinking. In fact, your post in this day and age is quite explicitly sexist, as you assume the student is a "he".

More generally, that sort of attitude prevents fresh blood from entering institutions or professions. There is a certain type that inhabits the profession and they are at pains to ensure those entering the profession are of the same type.
I really don't know how to respond to this. May be you need to calm down a bit.

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Could it be because it is easier to do the calculation by hand? If calculating the long division by hand, I would prefer ##\sqrt{2}/2 = 1.414213.../2 = 0.707106...## over ##1/\sqrt{2}=1/1.414213... = ##? (just seems harder)
So maybe it's just a left-over from the days before calculators.

martinbn, George Jones and PeroK
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Could it be because it is easier to do the calculation by hand? If calculating the long division by hand, I would prefer ##\sqrt{2}/2 = 1.414213.../2 = 0.707106...## over ##1/\sqrt{2}=1/1.414213... = ##? (just seems harder)
So maybe it's just a left-over from the days before calculators.
Exactly. Which is why the attempts to ridicule the alternative have, quite frankly, made me angry.

George Jones
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Pausing the thread for a bit, and some possible Moderation...

SammyS and PeroK
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Just to tie off this thread, there may be times when one form or another of a solution is preferred or required, such as:

** The denominator needs to be rationalized -- this is required on some university problems and exam questions

** The denominator does not need to be rationalized, since the fraction is expressing the sides of a triangle so the fraction is intuitive the way it is

** The final numerical answer needs to be computed without the aid of a calculator, and one form of the fraction is more amenable to that hand calculation

Thread will remain closed.

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topsquark, FactChecker and PeroK