Math stuff that hasn't been proven

[micromass]

In elementary school or high school, we often use stuff that has never actually been proven (in that class). For example

- Pythagoras' theorem.
- Addition of natural numbers is associative.
- Every number can be uniquely (up to order) decomposed in prime factors.

Accepting such a things really annoyed me, I would always ask why something is true. The answer that most teachers gave me was "can you find an example where it doesn't work," sigh. I had to wait until university to actually see a proof for such a things...

So, were you ever annoyed that something wasn't proven in school?? And what would have liked to see a proof/reason of??

 

[BloodyFrozen]

- Every number can be uniquely (up to order) decomposed in prime factors.

This one bothers me alot!

Also proof of quadratic formula (easy to derive though)

and...

e^x= lim_x->∞ (1+1/x)^x

The teacher would just say that's what e^x is (and give the calculus notation:grumpy:), but not tell us how to get it, even in the most basic terms

And one last thing,

The SA and Volume formulas in the back of the book. Never knew how they got it until I read about rotational volume and Archimedes' way of proving some of them

 

[chiro]

In elementary school or high school, we often use stuff that has never actually been proven (in that class). For example

- Pythagoras' theorem.
- Addition of natural numbers is associative.
- Every number can be uniquely (up to order) decomposed in prime factors.

Accepting such a things really annoyed me, I would always ask why something is true. The answer that most teachers gave me was "can you find an example where it doesn't work," sigh. I had to wait until university to actually see a proof for such a things...

So, were you ever annoyed that something wasn't proven in school?? And what would have liked to see a proof/reason of??

I found pretty much that to be case of most (if not all) math taught in primary/high school.

Also most of the students would constantly remark why we even need to do an integral and that it has "no use in society".

In some ways I can empathize with those students because had they taken a few uni courses, they might have changed there perspective and maybe even enjoyed or appreciated what they were learning.

 

[disregardthat]

In statistics it has bothered me why we could use the normal distribution in certain situations. Even at basic university level it is not proved (at least where I study).

 

[micromass]

In statistics it has bothered me why we could use the normal distribution in certain situations.

Exactly! Every time when encountering a statistics problem, they assume a certain distribution. It was never very clear to me how we could ever know the distribution of an event. This has always bothered me!

 

[HallsofIvy]

In applications of mathematics, you have to start with some model. What model you use depends upon the situation. I don't know about you but when I first learned probability distributions I also learned why they would be useful for modeling specific situations. For example, you can develop the Poisson distribution as a model that "expresses the probability of a given number of events occurring in a fixed interval of time and/or space if these events occur with a known average rate and independently of the time since the last event" (http://en.wikipedia.org/wiki/Poisson_distribution)

The normal distribution that you mention is especially important because of one of the most important theorems in statistics, the "Central Limit Theorem":

If you have a large sample from any probability distribution (with finite moments) with mean $\mu$ and standard deviation $\sigma$ then the average value of the sample is approximately normally distributed with mean $\mu$ and standard distribution $\sigma/\sqrt{n}$. And the larger the sample, the better the approximation.

In other words, no matter what the actual distribution is, the average of your sample will be at least normally distributed.

 

[I like Serena]

So, were you ever annoyed that something wasn't proven in school?? And what would have liked to see a proof/reason of??

Nope. Those things never annoyed me.
I always knew these things were outside the scope of the class material and that none of the other students would be interested in it.
That didn't stop me from finding out for myself, elsewhere.
The nice thing about the class materials was that it gave me an overview of what there was, and that it triggered my curiosity to want to learn more!

By now I have discovered that there is simply too much to learn (or want to learn). :grumpy:
So some things I take for granted, and some things that peak my interest, I delve into wide and deep. :!)

 

[Dr. Seafood]

In high school I found it so silly and at times upsetting how much we take for granted.

As a math tutor, I try to prove almost all the results I use. I'm teaching really elementary calculus right now, but I'm trying to be as rigorous as possible without being silly -- "Silly" meaning that I go ahead and demonstrate existence and uniqueness of "0" with respect to ℝ when I'm just trying to teach a first-year science major what "derivative" means. I feel a good analysis book or course would probably have that expectation, but it's unnecessary as far as the scope of elementary calculus goes.

That said, I prove as much as possible. I present limits in their rigorous epsilon-delta form, and prove the useful so-called "limit laws" (not all of them because there are many, I leave some of them as exercises (lol math)). I take no differentiation rule for granted. I prove the squeeze theorem for real functions, Fermat's theorem, mean value theorem. I do not usually prove the extreme value theorem; I think that proof is too difficult and unnecessary for our purposes, so I omit it and present an intuitive geometrical sketch. That said, with one of my students I proved the intermediate value theorem, which I regret now because it's a pretty confusing proof (it relies on completeness, and I had to define supremum). I try not to stray too far from our chosen topic, but I also try to take for granted as little as possible.

Some rigour that I feel is necessary to present is when developing derivatives for transcendental functions: particularly exponential and trigonometric. I was tired of being told "without proof or development, there exists a number "e" such that e^x is its own derivative with respect to x." But I do this in my lessons at first, however, and then proceed to define function "ln(x)" to be the inverse of e^x. From this definition I prove ln(xy) = ln(x) + ln(y), and ln(x^y) = yln(x). I implicitly differentiate x = e^y to find the derivative of ln(x) wrt x. I use the fact that b = e^ln(b) for all b (by definition of ln) to finally find the derivative of b^x. I use continuity of the logarithm to show that e = lim (x + 1/x)^x and use this to approximate the decimal expansion of e.

Anyways, that's an example of the level of rigour I provide when I teach. It keeps the lessons interesting; I feel it becomes to laborious to just say "guess what, the derivative of the exponential is the log times the exponential" and then start using chain rule a billion times. I didn't prove something like that the log is continuous for positive real arguments, and I only make an intuitive "stretching of base" argument to persuade that e exists. But I prefer this kind of lesson because it shows that the number which makes the exponential its own derivative is approximately 2.71828.

 

[HallsofIvy]

In high school I found it so silly and at times upsetting how much we take for granted.

As a math tutor, I try to prove almost all the results I use. I'm teaching really elementary calculus right now, but I'm trying to be as rigorous as possible without being silly -- "Silly" meaning that I go ahead and demonstrate existence and uniqueness of "0" with respect to ℝ when I'm just trying to teach a first-year science major what "derivative" means. I feel a good analysis book or course would probably have that expectation, but it's unnecessary as far as the scope of elementary calculus goes.

That said, I prove as much as possible. I present limits in their rigorous epsilon-delta form, and prove the useful so-called "limit laws" (not all of them because there are many, I leave some of them as exercises (lol math)). I take no differentiation rule for granted. I prove the squeeze theorem for real functions, Fermat's theorem, mean value theorem. I do not usually prove the extreme value theorem; I think that proof is too difficult and unnecessary for our purposes, so I omit it and present an intuitive geometrical sketch. That said, with one of my students I proved the intermediate value theorem, which I regret now because it's a pretty confusing proof (it relies on completeness, and I had to define supremum). I try not to stray too far from our chosen topic, but I also try to take for granted as little as possible.

Some rigour that I feel is necessary to present is when developing derivatives for transcendental functions: particularly exponential and trigonometric. I was tired of being told "without proof or development, there exists a number "e" such that e^x is its own derivative with respect to x." But I do this in my lessons at first, however, and then proceed to define function "ln(x)" to be the inverse of e^x. From this definition I prove ln(xy) = ln(x) + ln(y), and ln(x^y) = yln(x). I implicitly differentiate x = e^y to find the derivative of ln(x) wrt x.

How do you prove that if you do not assume that the derivatrive of $e^x$ is $e^x$?

I use the fact that b = e^ln(b) for all b (by definition of ln) to finally find the derivative of b^x. I use continuity of the logarithm to show that e = lim (x + 1/x)^x and use this to approximate the decimal expansion of e.

Anyways, that's an example of the level of rigour I provide when I teach. It keeps the lessons interesting; I feel it becomes to laborious to just say "guess what, the derivative of the exponential is the log times the exponential" and then start using chain rule a billion times. I didn't prove something like that the log is continuous for positive real arguments, and I only make an intuitive "stretching of base" argument to persuade that e exists. But I prefer this kind of lesson because it shows that the number which makes the exponential its own derivative is approximately 2.71828.

 

[Dr. Seafood]

Well, I'm not assuming that e = 2.71828... . The thing that is assumed is that there exists a number e with the property that e^x is its own derivative. Developing the decimal approximation of e from that assumption works out pretty well, as I described before. Of course, side-stepping existence like that is not rigorous, but in defense of my teaching, it's the best I can do when presenting this material...
A limit of importance when developing the derivative of the exponential is L = (a^x - 1)/x, with x tending to 0. The specific assumption made is that there exists a number e such that, setting a = e, we have L → 1 as x → 0. This assumes that the limit exists.

I think it takes a whole different kind of rigour to show the existence of e. The most I can do at the elementary calculus level is make a "stretching" argument, i.e. consider a^kx = (a^k)^x, and show that we can change the base of an exponential (by "stretching") to fit the data points of another exponential. We want the derivative at x = 0 to be 1, and it seems you can choose the stretch factor k = 1/L (L defined as in the last paragraph) so that this is possible. This will "stretch" the base of the exponential to be the required number e. This is not rigorous at all; in fact, it's the exact thing micromass referred to in the OP. I just shrug and say "Oh, it exists, okay you better believe me." But I don't feel that the level of precision required here is necessary to teach this topic. I actually don't even know how to delve into rigor with this kind of argument, but the geometry usually makes this seem plausible enough for a student to believe me.

The point I'm trying to make is that at least from this perspective, you can see the motivation for the development of such a number e = 2.71828... . I think that's really important in a teaching setting. Omitting/ignoring the rigorous proof of the existence of e is much less annoying than just presenting the irrational number without providing motivation.

 

[Bogrune]

I've got one: How is π equal to the ratio of a circle's diameter to its circumference?

 

[dalcde]

We define pi to be the ratio. It's not a magic number that somehow is the ratio.

 

[micromass]

We define pi to be the ratio. It's not a magic number that somehow is the ratio.

Yes, but why is the ratio a constant?? That seems nontrivial to me...

 

[Dr. Seafood]

^ Actually, that's not surprising to me really: all circles are similar to one another, so the ratio between circumference and diameter shouldn't change when we change the diameter. That the ratio happens to be close to 3.14 is something I'm interested in finding ...

 

[Bogrune]

We define pi to be the ratio. It's not a magic number that somehow is the ratio.

Well visually it looks as if the ratio to a circle's diameter to it circumference is a bit less than 3.141592653...

 

[Dr. Seafood]

^ How did you draw that??

 

[Bogrune]

I simply used a compass, and I gave it a radius of 0.5. I then drew a line through it to make its diameter, and then I cut a few pieces of string its approximate size, and I "wrapped" them around the circle. Although I think I trimmed them innacurately...

 

[romsofia]

Why ln(0) is undefined, without looking at the graph. Never understood it.

 

[Dr. Seafood]

ln(x) is number y such that e^y = x, so ln(0) asks for y such that e^y = 0. But e^y > 0 for all y, so this is not possible in the real domain; i.e. ln(0) asks for a nonsense evaluation.

 

[daveb]

For quite some time I've been of the opinion that a rudimentary discussion of rings, fields and groups would be of great benefit to high school algebra students, so that then they understand why they are learning what they are larning.

 

[micromass]

For quite some time I've been of the opinion that a rudimentary discussion of rings, fields and groups would be of great benefit to high school algebra students, so that then they understand why they are learning what they are larning.

I agree! I did see some groups and fields when I was in high school (in Belgium), but it wasn't that much.

 

[I like Serena]

I agree! I did see some groups and fields when I was in high school (in Belgium), but it wasn't that much.

I don't get it.

Most high school algebra students have great difficulty with abstract thinking.
How would it help to make it even more abstract?
As I see it, you need to show how it applies in real life and apply it to real problems.

Explaining groups and rings and stuff would only be useful in an advanced class group.

 

[micromass]

I don't get it.

Most high school algebra students have great difficulty with abstract thinking.
How would it help to make it even more abstract?
As I see it, you need to show how it applies in real life and apply it to real problems.

Explaining groups and rings and stuff would only be useful in an advanced class group.

Well, it were only the advanced classes that saw these stuff We don't want to bother people who are struggling with basic algebra.
But I believe that there should be much more abstraction in high school (for advanced students). This can only help them later on, whether they go into science or not.

As I see it, set theory was deemed far too advanced to teach 40 years ago. But now it is introduced in elementary school! I don't think that this is a bad thing.

And of course we must show how it applies to real life and real problems. But a little bit of abstraction can only be benificial, I think.

 

[Char. Limit]

How do you prove that if you do not assume that the derivatrive of $e^x$ is $e^x$?

First you prove that $d(e^x) = e^x dx$ (I rearranged it so it would fit on one line).

3nQejB-XPoY[/youtube] EDIT: Fixed ...I got my order wrong too. My memory sucks...

 

[pwsnafu]

First you prove that $d(e^x) = e^x dx$ (I rearranged it so it would fit on one line).

That one uses the limit definition of e. If you don't want to use that:

1. By Taylor series there exists a solution to the IVP y'=y and y(0)=1.
2. Call this solution $\phi(x)$.
3. Show that $\psi(x) = \frac{\phi(x+y)}{\phi(y)}$ is also a solution.
4. Therefore by uniqueness $\phi(x+y)=\phi(x)\phi(y)$. Further, because phi exists via a convergent Taylor series, it is continuous, and so satisfies the http://en.wikipedia.org/wiki/Characterizations_of_the_exponential_function#Characterizations" and scroll down for a proof.

In this situation we define e = phi(1) but we haven't calculated it.

 

[Dr. Seafood]

First you prove that $d(e^x) = e^x dx$ (I rearranged it so it would fit on one line).

3nQejB-XPoY[/youtube] EDIT: Fixed ...The "mathematical discovery" is side-stepped.

 

[Anonymous217]

e^x= lim_x->∞ (1+1/x)^x

The teacher would just say that's what e^x is (and give the calculus notation:grumpy:), but not tell us how to get it, even in the most basic terms

But that IS the definition of e. (the standard one that is) And by the way, you mean e, not e^x.

 

[BloodyFrozen]

My bad

 

[HallsofIvy]

Many recent Calculus texts start by defining
$$ln(x)= \int_1^x \frac{1}{x}dx$$
so that $(ln(x))'= 1/x$ follows immediately from the definition,
then defining $exp(x)$ to be the inverse function to ln(x). That greatly simplifies finding the derivative of $exp(x)$. Of course you would still need to show that $exp(x)= e^x$- that is, that "exp(x)" really is some number to the x-power.

But that's easy. If y= exp(x), then x= ln(y). If $x\ne 0$, $1= ln(y)/x= ln(y^{1/x})$. Going back to the exponential form, $y^{1/x}= exp(1)$ so that $y= exp(x)= (exp(1))^x$. Define e= exp(1) and we have $exp(x)= e^x$.

 

[dalcde]

I personally don't like that definition since it doesn't show that the natural log is actually a logarithm and it is weird to define a logarithm like that (people will just be confused).

 

[Char. Limit]

Many recent Calculus texts start by defining
$$ln(x)= \int_1^x \frac{1}{t}dt$$
so that $(ln(x))'= 1/x$ follows immediately from the definition,
then defining $exp(x)$ to be the inverse function to ln(x). That greatly simplifies finding the derivative of $exp(x)$. Of course you would still need to show that $exp(x)= e^x$- that is, that "exp(x)" really is some number to the x-power.

But that's easy. If y= exp(x), then x= ln(y). If $x\ne 0$, $1= ln(y)/x= ln(y^{1/x})$. Going back to the exponential form, $y^{1/x}= exp(1)$ so that $y= exp(x)= (exp(1))^x$. Define e= exp(1) and we have $exp(x)= e^x$.

Just wanted to fix something there, as I've been told numerous times never to have the bounds contain the same variable as the integrand.

 

[disregardthat]

It's actually not wrong to use the same variable as bound and as a variable in the integrand. The bad thing about it is the confusion it might generate.

 

[Hurkyl]

I personally don't like that definition since it doesn't show that the natural log is actually a logarithm and it is weird to define a logarithm like that (people will just be confused).

Then they miss the point.

You aren't supposed to learn about logarithms and exponentials from this definition. You were supposed to have already learned in your previous classes the basic algebraic properties of these operations.

The point of this definition is for working out the technical details -- and of the four hooks I know into this set of operations, this one is by far the simplest.

The other two, incidentally, are:

• Define exp(x) via Taylor series
• Define x^y incrementally, first for positive integers y via repeated multiplication, then for positive rational numbers by demonstrating that you can take roots, then for positive real numbers by continuous extension
• Define exp(x) as the solution to a differential equation

 

[dalcde]

You aren't supposed to learn about logarithms and exponentials from this definition. You were supposed to have already learned in your previous classes the basic algebraic properties of these operations.

I'm not saying that you can't learn about logarithms. Looking at that definition, you won't even know that it is a logarithm. Why not define it as a logarithm with base e?

 

[kramer733]

In elementary school or high school, we often use stuff that has never actually been proven (in that class). For example

- Pythagoras' theorem.
- Addition of natural numbers is associative.
- Every number can be uniquely (up to order) decomposed in prime factors.

Accepting such a things really annoyed me, I would always ask why something is true. The answer that most teachers gave me was "can you find an example where it doesn't work," sigh. I had to wait until university to actually see a proof for such a things...

So, were you ever annoyed that something wasn't proven in school?? And what would have liked to see a proof/reason of??

You should've said "Can you prove that it works for EVERYTHING?!"

And yes I hated this with a passion too. Well I only started hating it 2 years ago. Before those 2 years ago, I didn't give a crap about math or school.