Looking for internet famous math prob on dist law.

[houlahound]

A few years ago it was a big thing where high school basic math definition stumped a lot of pro mathematicians.

The prob from memory involved order of operations. Could be wrong but I think it was getting the correct answer to;

a(b+c) for specific values of a,b,c. All integers and no tricks.

Some maths profs argued for their answer some changed but the only answer was it was not a well defined question. Few conceded the other guy was right, or how they were wrong.

Sorry I can't define the actual problem but it started a math educator war. Hope my vague definition triggers someone's memory.

 

[houlahound]

OK pretty sure this is the problem although the numbers are irrelevant.

6÷2(2+1)

What is the solution to 6÷2(1+2)=?: Professor of …:

 

[micromass]

I'm going to doubt that this stumped any pro mathematician

 

[houlahound]

May well be urban myth but apparently different calculator brands give different answers although I have not tested that assertion.

Interesting break down of the issues;

http://www.slate.com/articles/healt...emdas_doesn_t_always_give_a_clear_answer.html

And a more general background why its even a problem;

http://knowyourmeme.com/memes/48293

 

[micromass]

May well be urban myth but apparently different calculator brands give different answers although I have not tested that assertion.

That is correct. That means that the calculators were programmed incorrectly. They were programmed incorrectly because when typical students type in 6÷2(1+2) they often mean the incorrect thing.

Interesting break down of the issues;

http://www.slate.com/articles/healt...emdas_doesn_t_always_give_a_clear_answer.html

The PEMDAs do work and they do give a clear answer.

 

[houlahound]

 

[micromass]

Position 2 is crap. Find me some mathbooks that define "implied multiplication". You won't find it. I have never even heard of "implied multiplication" before this problem came around. Go ahead, search in Rudin, Bloch, Landau, or any other math book that rigorously defines numbers and their operations. Nowhere will you see that "juxtaposition" in any way behaves as position 2 tells us.

 

[houlahound]

Still a good discussion topic for students tho.

 

[pwsnafu]

The problem stems from some people assume that

1. $a/bc$ is equal to $\frac{a}{bc}$ instead of $\frac{ac}{b}$, or
   
2. people who think "implicit multiplication" (i.e. mathematical dot) and "explicit multiplication" (i.e. times symbol) are different. This is nonsense, there is only one multiplication of real numbers.

 

[micromass]

Still a good discussion topic for students tho.

Not really. It's completely irrelevant to mathematics. It's a stupid convention. Besides, nobody uses ÷ to denote division anymore.

 

[micromass]

If you want a good discussion topic for students, then talk to them about the value of $0^0$. For this, both sides actually do have a good point.

 

[pwsnafu]

If you want a good discussion topic for students, then talk to them about the value of $0^0$. For this, both sides actually do have a good point.

There is also "Is zero a natural number?"

 

[micromass]

There is also "Is zero a natural number?"

Or $0\cdot \infty$, where the convention that it equals 0 sometimes makes a lot of things easier in some parts of math (measure theory).

 

[houlahound]

Great discussion topics, will do some work. Thanks.
What about this one;2^2^2^2^2^2^2^2^2...etc

Sorry can't make it nested.

 

[pwsnafu]

See the Wikipedia article on Lambert W, example 3.

 

[micromass]

Great discussion topics, will do some work. Thanks.
What about this one;2^2^2^2^2^2^2^2^2...etc

Sorry can't make it nested.

Or $\sqrt{2\sqrt{2\sqrt{2\sqrt{2\sqrt{...}}}}}$.

Or the truly amazing

$$\pi = \frac{4}{1+ \frac{1^2}{3 + \frac{2^2}{5+\frac{3^2}{...}}}}$$

 

[micromass]

And if you really want to upset the class:

 

[houlahound]

See the Wikipedia article on Lambert W, example 3.

Interesting until I got to this and my heart sank;

"The Lambert W relation cannot be expressed in terms of elementary functions"

 

[micromass]

Interesting until I got to this and my heart sank;

"The Lambert W relation cannot be expressed in terms of elementary functions"

That doesn't mean anything. Why do you think the sine function is any more natural than the Lambert W function? We only know a limited number of exact values for the sine function too.

 

[houlahound]

On a simple note the correct answer to this finite problem?

2^2^2^2^2^2^2^2^2

 

[houlahound]

I think the sine function is more natural cos it is more natural;

tide comes in, tide goes out therefore sine.

That doesn't mean anything. Why do you think the sine function is any more natural than the Lambert W function? We only know a limited number of exact values for the sine function too.

 

[micromass]

On a simple note the correct answer to this finite problem?

2^2^2^2^2^2^2^2^2

Like expected, my computer gave an overflow error.

 

[micromass]

I think the sine function is more natural cos it is more natural;

tide comes in, tide goes out therefore sine.

So it's only natural because you can find an application of it in nature?

 

[houlahound]

By definition of the word natural your comment is trivial.

 

[micromass]

OK. But then (all from wiki)

The Lambert W function has been recently (2013) shown to be the optimal solution for the required magnetic field of a Zeeman slower

The Lambert W function was employed in the field of Neuroimaging for linking cerebral blood flow and oxygen consumption changes within a brain voxel, to the corresponding Blood Oxygenation Level Dependent (BOLD) signal.

The Lambert W function was employed in the field of Chemical Engineering for modelling the porous electrode film thickness in a glassy carbon based supercapacitor for electrochemical energy storage. The Lambert "W" function turned out to be the exact solution for a gas phase thermal activation process where growth of carbon film and combustion of the same film compete with each other.[14][15]

The Lambert W function was employed in the field of epitaxial film growth for the determination of the critical dislocation onset film thickness. This is the calculated thickness of an epitaxial film, where due to thermodynamic principles the film will develop crystallographic dislocations in order to minimise the elastic energy stored in the films. Prior to application of Lambert "W" for this problem, the critical thickness had to be determined via solving an implicit equation. Lambert "W" turns it in an explicit equation for analytical handling with ease

The Lambert W function has been employed in the field of fluid flow in porous media to model the tilt of an interface separating two gravitationally segregated fluids in a homogeneus tilted porous bed of constant dip and thickness where the heavier fluid, injected at the bottom end, displaces the lighter fluid that is produced at the same rate from the top end. The principal branch of the solution corresponds to stable displacements while the -1 branch applies if the displacement is unstable with the heavier fluid running underneath the ligther fluid.[17]

So the Lambert W function is natural too.

 

[houlahound]

I would argue as a laymen (which .makes completely unqualified to argue) that those examples (without reading how they are are applied) are just tools not different to how Laplace transforms are used to find real measurable voltages/currents in in real electric circuits but the tool itself is not a real thing in the physics sense.

 

[micromass]

I would argue as a laymen (which .makes completely unqualified to argue) that those examples (without reading how they are are applied) are just tools not different to how Laplace transforms are used to find real measurable voltages/currents in in real electric circuits but the tool itself is not a real thing in the physics sense.

Agreed. But is the sine not just a tool in order to study waves too? In reality, there are no such things as circles and perfect waves. Why don't you consider the sine a tool then?

 

[houlahound]

Have to look closer at the W function to answer that which will take some time but my my off the hip answer is that the sine function has a 1:1 mapping onto the phenomena it describes and requires no invoking of other functions inside other functions inside some nested nightmare of algorithms pointing to yet more functions hence in my view sine as natural and elementary.

In a boxing match sine would beat down W within the first round... that's how natural it is.

 

[pwsnafu]

Have to look closer at the W function to answer that which will take some time but my my off the hip answer is that the sine function has a 1:1 mapping onto the phenomena it describes and requires no invoking of other functions inside other functions inside some nested nightmare of algorithms pointing to yet more functions hence in my view sine as natural and elementary.

In a boxing match sine would beat down W within the first round... that's how natural it is.

Actually, real life waves are rarely pure sine waves. This is because you have seabeds/coastlines causing interference, eddys etc, not to mention the effect of the wind. Notice how waves "break" under certain conditions.

 

[micromass]

Have to look closer at the W function to answer that which will take some time but my my off the hip answer is that the sine function has a 1:1 mapping onto the phenomena it describes and requires no invoking of other functions inside other functions inside some nested nightmare of algorithms pointing to yet more functions hence in my view sine as natural and elementary.

In a boxing match sine would beat down W within the first round... that's how natural it is.

The sine wave is a perfect solution to a mathematical model. A model never occures in practice, it is an idealized situation.
In the same way, the W function is a perfect solution to a mathematical model.

The W function is very easy to define, $W(z) = a$ if and only if $ae^a = z$.
Compare this to the definition of the logarithm: $log(z)=a$ if and only if $e^a = z$.
Looks rather analogous to me. If you accept the log as natural, then you must accept the W function too.

 

[houlahound]

Can a table of values for W be computed?

ie

z in cloum a, W(z) in colum b.

 

[micromass]

Can a table of values for W be computed?

ie

z in cloum a, W(z) in colum b.

Yes, in exactly the same way that one can be computed for the log and the sin function.

 

[houlahound]

K, I will give the link a proper read, I stopped when it defined W as inverse branches of something else, what does that even mean.

 

[micromass]

It means that the $W$ function is an inverse function.
For example, $log(z) = w$ if and only if $e^w = z$. This means that the $\log$ is the inverse function of the exponential.

But situations like the log and the exponential aren't always this simple. Take $w =\sqrt{z}$. It is certainly true that $w^2 = z$, but $w$ isn't the only value for which $w^2 = z$, also $(-w)^2 = z$. So there are two solutions to $w^2 = z$. We choose one solution (namely the positive one) and call that $\sqrt{z}$. But we have made a quite arbitrary choice here, we could also take the negative. We say that the square root function has two branches (one for each choice). So there is a negative branch of numbers whose square is $z$ and a positive branch.

In the same way, there is no unique $w$ such that $we^w = z$. A choice must be made (in some case, in others there is a unique solution). These two choices define two "branches" of the W function.

 

[houlahound]

OK this I can do, expansion of W

This also

ETA