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

[Office_Shredder]

Ah, so they are both wrong!

Right. It should be $\frac{\sqrt{154}}{22}$

 

[symbolipoint]

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

 

[Orodruin]

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…

 

[Mark44]

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

Fixed

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

 

[Orodruin]

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

You’re welcome to rationalise it

 

[FactChecker]

Clebsch-Gordan not my areaMy 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]

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!

 

[PeroK]

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?

 

[martinbn]

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.

 

[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]

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.

 

[PeroK]

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.

 

[martinbn]

E.g. in the so-called singlet state:

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

There is a difference. Here it is normalizing a vector, so it is better to see it as $\frac{\vec{a}}{||a||}$.

 

[martinbn]

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.

 

[FactChecker]

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.

 

[PeroK]

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.

 

[berkeman]

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

 

[berkeman]

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.