What is considered most simple

[GreenPrint]

What is considered most "simple"

My professor this year in multiariable calculus made a very argument for why you shouldn't leave something in "exact form". All previous years I was told to leave something in "exact form" because it's more exact.
For example on a exam this
squareroot(2)
is correct and not
about 1.4... (whatever it is I can't remember)
because 1.4... is only a approximation and is therefore incorrect and squareroot(2) is correct because it not a approximation but only the correct answer.

My professor argued that squareroot(2) is not simple enough because it's a operation on a number. I found this odd because like I said I was always told to leave it in "exact form" because it was more correct. He argued that you wouldn't leave something like this

integral[2,4] x^2 dx

on a test so then why leave squareroot(2) as a answer on a test? Both are operations on a argument. I found this rather persuasive but it contradicts what I've always been told. Even when I was in high school and taking AP Calculus I believe I would of lost points if I put numerical approximations and not the "exact form" of an answer.

I'm not exactly sure which one is correct or more simple and what I should put on a test next semester because I have a different professor.

 

[Infinitum]

It depends on how your professor thinks, because he's going to be awarding you your marks

As for what is correct, \sqrt{2} is an irrational number. Its not an operation on a number, the square root of 2 is \sqrt{2}, just like the square root of 4 is 2. 1.414... is just the decimal form(not a simpler form) of \sqrt{2}, which is approximate, and therefore inaccurate.

Edit : If I were you, I would prefer writing \sqrt{2} as the answer over 1.414.

 

[ehild]

Write √2≈1.414... as the answer. Sometimes it is said how many significant digits you need to use.

ehild

 

[Infinitum]

Write √2≈1.414... as the answer. Sometimes it is said how many significant digits you need to use.

ehild

Isn't this usually done in physics, rather than mathematics? I always have had the opinion that math is perfect.

 

[ehild]

Well, some Math problems require numerical results. Word problems, for example. How big is the area of a given land, what is the volume of a container, and so on.
Math is perfect, but the problems are not .

If somebody wants the solution in exact form, he/she should indicate it. There are the final test in my country just now, and I read the phrases either "give the exact solution" or write the result with 3 significant digits"

ehild

 

[HallsofIvy]

There is a differerence between a single direct operation like taking the square root and finding an integral which might require many different techniques. Your teacher is a bit whacky (unlike me, of course) and he and I would really go at it hammer and tongs!

 

[SteveL27]

There is a differerence between a single direct operation like taking the square root and finding an integral which might require many different techniques. Your teacher is a bit whacky (unlike me, of course) and he and I would really go at it hammer and tongs!

I second that. If the answer is sqrt(2) and someone writes "1.414..." or even worse, "1.414" without the dots, I think they don't understand that sqrt(2) is more precise and better notation than the numeric forms. And when they write 1.414, they show lack of understanding of the difference between an irrational number and a truncated decimal approximation.

You are definitely right on this, unless the question specifically asked for an answer in numeric form.

There is a convention for radical expressions, which is to put the radical on top. So for example 1/sqrt(2) is more properly written sqrt(2)/2. Once you get that far you're done.

However, you need to do what the prof says! Just go with the flow till the end of the semester.

 

[Curious3141]

If I had your teacher, the next answer I'd give would be: [1;2,2,2,…]. Continued fractions are "exact" as far as I'm concerned, and what can be "simpler" than addition and division?