Let's discuss Mathematics

[Samson]

Why is a straight line y=mx+c?

A quadratic is y=ax^2+bx+c, a cubic is y=ax^3+bx^2+cx+d, why is a straight line not y=ax+b? Why c, and particularly why m?

 

[Kyriakos]

Why is a straight line y=mx+c?

A quadratic is y=ax^2+bx+c, a cubic is y=ax^3+bx^2+cx+d, why is a straight line not y=ax+b? Why c, and particularly why m?

While it's not written like that usually in Greek mathbooks (I most often see the coefficient of x in a linear function be "b" or even "a"), I would imagine that the "m" is there to help with calculations using that line as tangent to a curve? Since it would be unique.
Of course it's counterintuitive to start with that (in middle school), when it will be years before you use it for the tangent to a non-linear function. Then again all sorts of secondary school math are presented heterochronically (I am sure a simpler term exists in English, but apparently this one is also there in English).

 

[q1y11]

While it's not written like that usually in Greek mathbooks (I most often see the coefficient of x in a linear function be "b" or even "a"), I would imagine that the "m" is there to help with calculations using that line as tangent to a curve? Since it would be unique.
Of course it's counterintuitive to start with that (in middle school), when it will be years before you use it for the tangent to a non-linear function. Then again all sorts of secondary school math are presented heterochronically (I am sure a simpler term exists in English, but apparently this one is also there in English).

I like how you put meaning to those equations .
I found that to be the hardest thing to do coming out of high school and entering university with applied math as a degree. People tend to not explain every single thing in the equations and you could not find the meaning anywhere... That got me stunned for a long time.

 

[Kyriakos]

I like how you put meaning to those equations .
I found that to be the hardest thing to do coming out of high school and entering university with applied math as a degree. People tend to not explain every single thing in the equations and you could not find the meaning anywhere... That got me stunned for a long time.

I didn't see meaning in them in highschool either - back then I was only looking for personal 'tricks' to do things automatically. And sometimes the "meaning" is not even discovered - but for many part of it has been. What I like is that obvious (but abstract) tautologies (or even not so obvious ones) in algebra can also acquire very non-abstract meaning elsewhere, such as in geometry. Eg the inequality ab<=[(a+b)/2]^2 implies (I suppose among other things) that the square is the orthogonal parallelogram with the smallest perimeter for a set surface area => the algebraic phrase can be written also as functions with their min/max.

 

[q1y11]

I didn't see meaning in them in highschool either - back then I was only looking for personal 'tricks' to do things automatically. And sometimes the "meaning" is not even discovered - but for many part of it has been. What I like is that obvious (but abstract) tautologies (or even not so obvious ones) in algebra can also acquire very non-abstract meaning elsewhere, such as in geometry. Eg the inequality ab<=[(a+b)/2]^2 implies (I suppose among other things) that the square is the orthogonal parallelogram with the smallest perimeter for a set surface area => the algebraic phrase can be written also as functions with their min/max.

Yes , yes I like your point. I spend 12 years in and out of the university and I finally manage to get the lowest possible grade in order to finish it haha but I was really happy that I could explain to myself one thing. That functions and interpolation can be used to find different "things" in time. So just if we have a random sets of points in a dimension let's say a normal coordinate system we can find a representation of those points as a drawn out function, a representation of time and that Pi and the golden ratio can be used as a perfect process in time.... Don't know if I'm making sense haha to you at least I was happy when I though of that got the lowest point and never gave it much though after...

 

[Samson]

What is the definition of simplest form when it comes to Algebraic Fractions?

I have googled, and most definitions seem circular to me (eg. "In mathematics, the simplest form refers to the most reduced or simplified representation of a fraction.") This site is better than most, and says "In mathematical algebra, the simplest form is the least attainable fraction of a number or a linear equation." but I do no know the strict definition of attainable.

They give an example, but it just leaves me less sure. They say simplify this:


And the answer is:

But I do not understand how the simplest form can have an expression that can be further factorised on the top and an expression with brackets on the bottom. Depending on the definition of simple, surely the answer has to be one of these:

 

[Kyriakos]

@Samson Yes, the form (x^2+2x)/2(x+1) is inconsistent (as you have shown with the two counter-examples where the same way was applied. But I can imagine why it was the preferred way:
If you try to establish, through algebra (ie not through visualizing the graph) where this function is positive and where it is negative, you would never leave it as three functions because then you would need to establish combinations of +.- in three things instead of 2. Of course you can do the numerator as two first, but then keep the result as one polynomial.
This may still be a couple of years away from what the kid is asked to do - or at least one.

(edit, hm, now I will check if actually this is even true - namely if you can leave them as two without missing part of the range; will get back to you on this shortly)
(see two posts below: it wasn't true)

 

[Samson]

Yes, the form (x^2+2x)/2(x+1) is inconsistent (as you have shown with the two counter-examples where the same way was applied. But I can imagine why it was the preferred way:
If you try to establish, through algebra (ie not through visualizing the graph) where this function is positive and where it is negative, you would never leave it as three functions because then you would need to establish combinations of +.- in three things instead of 2. Of course you can do the numerator as two first, but then keep the result as one polynomial.

I see what you are getting at, but I am asking this from the point of view of tutoring a school child. If the question in the exam is "give this in the simplest form" what should the answer be to get all the marks? There must be a rule.

 

[Kyriakos]

I see what you are getting at, but I am asking this from the point of view of tutoring a school child. If the question in the exam is "give this in the simplest form" what should the answer be to get all the marks? There must be a rule.

From discussions in Facebook by math teachers for those grades, it sadly seems to be inconsistent. Better tell the kid to write both/all in the exam

 

[Kyriakos]

@Samson I was wrong to assume you could have the numerator be dealt with as one polynomial in the first place. Because indeed if you do that you will miss part of the range for +-. In this case I missed the added range for the full function being positive in also (-2,-1).

Desmos creates the graph -when I treated the numerator as one, I couldn't account for the function being positive in (-2,-1).

 

[AdrienIer]

I see what you are getting at, but I am asking this from the point of view of tutoring a school child. If the question in the exam is "give this in the simplest form" what should the answer be to get all the marks? There must be a rule.

As a high school math teacher my answer is :
-The question is badly worded. "In a simple form" is better because it doesn't imply that there is only one good answer
-Therefore I'd accept all 4 versions : (x²+2x)/(2x+2) (IMO the best one), (x²+2)/(2(x+1)), x(x+2)/(2x+2) and x(x+2)/(2(x+1))

 

[CKS]

I would say that simplest form means that the numerator and denominator have no common factors besides 1. I agree with Adrienler that all 4 versions should be acceptable, but I suspect that some teachers will have the belief that one of them is "best" and mark others wrong. I think the "best" variety will vary from teacher to teacher, and so you will have to ask the child's teacher what they want. If the teacher just goes by the answer in the book, you will be out of luck, since I expect the book answers will not be consistent.

Personally I prefer things factored out, since you have to factor everything to cancel terms, and re-distributing things at the end is an extra step, but I'm a lazy physicist, not a mathematician.

 

[Kyriakos]

If we take into account the class the pupil is in (middle school?), maybe the point of writing the function like that is strictly polynomial division? (which should be right around the corner)

For example:

It would be one rather direct reason to maintain the apparent inconsistency in how the numerator and denominator are left.

 

[Kyriakos]

Why does Desmos have no solutions for absolute variable functions when they equal 0?

For example:

It doesn't have a solution for |x|=0 either.
This does go against the definition of a function of an absolute variable, afaik, since it is defined as the distance of x, starting at a, from the value (in this case distance of x, starting from a=0, from 0), so you'd expect the line x=0 as solution to |x|=0 and x=+-3 for ||x|-3|=0.

edit: I asked on reddit, and perhaps it has to do with the program's method of establishing values of the function by comparing negative to analogous positive values.