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glittergal96

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Being continuous is a property that says that a function is in some sense "nice" near a point.

Being differentiable is also a "niceness" property at a point. In fact it is a much stronger property, in the sense that differentiability at a point implies continuity at a point.

It is clearly possible to find functions that are continuous but not everywhere differentiable (absolute value function fails to be differentiable at 0).

What is surprising (well, I find it surprising, and the discovery went against the beliefs at the time) is that there exist a function that is continuous everywhere and differentiable nowhere! : https://en.wikipedia.org/wiki/Weierstrass_function

In fact, in a certain sense, MOST continuous functions are differentiable nowhere!
 

Paradoxica

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Being continuous is a property that says that a function is in some sense "nice" near a point.

Being differentiable is also a "niceness" property at a point. In fact it is a much stronger property, in the sense that differentiability at a point implies continuity at a point.

It is clearly possible to find functions that are continuous but not everywhere differentiable (absolute value function fails to be differentiable at 0).

What is surprising (well, I find it surprising, and the discovery went against the beliefs at the time) is that there exist a function that is continuous everywhere and differentiable nowhere! : https://en.wikipedia.org/wiki/Weierstrass_function

In fact, in a certain sense, MOST continuous functions are differentiable nowhere!
The concepts presented by uncountable nouns break down when applied to infinite sets. One must tread carefully and avoid fallacious conclusions when dealing with infinity. Remember to discern between uncountable and countable infinities.

Speaking of which, there is a one-to-one correspondence between the integers and the rationals. There are more transcendental numbers than algebraic numbers. And the set of all real numbers is exactly as big as the set of complex numbers.
 

glittergal96

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The concepts presented by uncountable nouns break down when applied to infinite sets. One must tread carefully and avoid fallacious conclusions when dealing with infinity. Remember to discern between uncountable and countable infinities.

Speaking of which, there is a one-to-one correspondence between the integers and the rationals. There are more transcendental numbers than algebraic numbers. And the set of all real numbers is exactly as big as the set of complex numbers.
I know lol, why are you reminding me?

In any case, none of the objects I mentioned is countably infinite.
 

InteGrand

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I know lol, why are you reminding me?

In any case, none of the objects I mentioned is countably infinite.
I don't think he was reminding you of those facts (countability of the rationals etc.); rather, I think he was adding them in as more 'interesting statements' for this thread. Or maybe you were referring to his first paragraph haha. Not sure in that case why (if at all that was his purpose) he was reminding you, maybe again just general statements for this thread.
 

glittergal96

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I more meant that countability is kind of an irrelevant notion to the thing I mentioned so it seemed weird to bring it up as a reply to my post.

(And yeah, meant first para. Second para is good for this thread :). Countability is such a nice and low-tech example of higher maths to show HS students.)
 

InteGrand

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I more meant that countability is kind of an irrelevant notion to the thing I mentioned so it seemed weird to bring it up as a reply to my post.

(And yeah, meant first para. Second para is good for this thread :). Countability is such a nice and low-tech example of higher maths to show HS students.)
My guess is his countability statements came from when you said "most" functions.
 

glittergal96

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My guess is his countability statements came from when you said "most" functions.
It's still unrelated to countability though. "Most" meant almost all with respect to a certain measure. Both the set and it's complement are still uncountable.
 
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Proof using base arithmetic that Christmas is a pagan festival:

OCT 31 = DEC 25
 

InteGrand

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It's still unrelated to countability though. "Most" meant almost all with respect to a certain measure. Both the set and it's complement are still uncountable.
Yeah lol I meant that I thought that he started talking about uncountability because of this, and once he started talking about uncountability, he mentioned countability too.
 

Paradoxica

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Yeah lol I meant that I thought that he started talking about uncountability because of this, and once he started talking about uncountability, he mentioned countability too.
My first statement was a comment on linguistic technicality. After that I went on a rant.
You can't apply the concept of "most", "some", etc. to infinite sets. Linguistically nonsensical.
 

glittergal96

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My first statement was a comment on linguistic technicality. After that I went on a rant.
You can't apply the concept of "most", "some", etc. to infinite sets. Linguistically nonsensical.
Ah okay cool. Yeah, of course using words like "most" is vague and imprecise. Talking about measure theory will go over most high schoolers heads though, so waving hands and being a bit colloquial in this thread seems more appropriate.
 

InteGrand

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My first statement was a comment on linguistic technicality. After that I went on a rant.
You can't apply the concept of "most", "some", etc. to infinite sets. Linguistically nonsensical.
Yeah this is what I thought you meant when I searched up 'uncountable noun' (the use of 'uncountable' here is unrelated to uncountability of sets so it confused me initially since I was thinking of the mathematical 'uncountable').
 
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Paradoxica

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:). Riemann showed that any conditionally convergent series can be rearranged so that the sum comes out to be any chosen real number, or diverge to $-\infty$ or $+\infty$. This is described in the link below for anyone interested.$"/>

https://en.wikipedia.org/wiki/Riemann_series_theorem
Another noteworthy point is that x=2 is at the edge of the radius of convergence for ln(1+x). This gives some insight as to why it is conditionally convergent, as it is the boundary between the zone of absolute convergence and the sea of absolute divergence.
 

glittergal96

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I have always found fixed point theorems quite pretty.

Probably the most well-known one is the Brouwer fixed point theorem. One version of this states that if you have a continuous function f from a closed n-ball (eg the set of points of distance =< 1 from the origin in n-dimensional Euclidean space) to itself, then this function must have a fixed point, which is an x such that f(x)=x.

So in one dimension, this says that a continuous function f defined on the interval [0,1] that takes values in [0,1] must have a fixed point. (The 1-d version can be proved at high school level, try it!)

Fixed point theorems can have some pretty whack consequences. Eg, if I am in Russia and I put a map of Russia on a table, there will be a point on this map that lies directly above the actual physical spot in Russia that it represents.

They are also quite useful in abstract mathematics, for things like showing non-constructively that a certain equation/system of equations has a solution.
 
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