Big math/science results within easy reach

[jsilvela]

I hear you guys on not needing so many significant digits, engineer/physicist's approach etc. BUT (I like big buts)
without the reals, and infinite non-periodic decimals, the following statement would be false:

A continuous curve which takes both strictly negative values and strictly positive values MUST take the value 0 at some point.

Rational numbers are not enough to guarantee that.
That, I think, is another big result with an elementary proof.

 

[jsilvela]

Infinity, blah, I still have some papers and a book on aleph-null vs. aleph-one around someplace. I am definitely not going to look for it.

1 + 1 = 3 for very large values of 1.

Have you heard of the big point theorem?
Two parallel lines intersect at a point, if the point is sufficiently big.

 

[holla]

And then, as distinct from a mathematicians approach - you can have the engineers approach. Which looks somthing like:

How close does it have to be so it doesn't matter?

On the other side of things, there are plenty of statements in mathematics and computer science (complexity theory for example) which talk about facts that hold "for sufficiently large N".

Last semester I was teaching some complexity theory and ended up discussing Ackermann's function. This is a function on positive integers that grows so quickly that in order to write down the number of digits in Ack(4,4) you would need 2^65000 universes' worth of electrons. And then you can be left wondering: in what sense does a number like this "exist" and if "sufficiently large N" has to be this big, who actually cares what happens for that value of N?

Depending on your point of view this either renders those parts of mathematics useless or profound. I can't decide.

 

[jsilvela]

So to measure the size of the universe in units of the size of a helium atom you need less than 38 significant digits, and even in this ludicrous extreme example, 0.9bar with only 38 9's after the point is indistinguishable to 1.


Just from an engineers point of view, of course.

So for pretty much all mechanical engineering, you're not going to need more than 6 significant digits, and even that is probably massive overkill.
So 0.999999 is pretty much equivalent to 1 for practical purposes.

These are very good points.

But consider some practical issues that crop up for software engineering:

- for crypto / security, we need WAY more than 6 significant digits
- in computers, you can use the "floating point" representation of numbers. The venerable IEEE 754 standard uses a whopping 53 significant bits (or around 16 decimal digits). And, the sad reality is that they are unsuitable for *money* calculations, because 0.1 + 0.2 does not equal 0.3, and you can and will lose cents and dollars even in medium sized computations
- I think it was computed that the number of possible chess games are greater than the estimated number of atoms in the universe. We still want to be able to talk about those numbers.

Just saw @holla 's response, and agree...

 

[jsilvela]

Mathematicians just make everything too complicated.

 

[pjug]

By the way, the BDWoody proof that 0.999... = 1 is very nice from an intuitive point of view but takes a bit of work to make into precise mathematics. To define what 0.999... means, you need the notion of limit as I tried to describe above. Then to show that you are allowed to say

10 x 0.999... = 9.9999...

and other such operations -- which seem obvious from an intuitive point of view -- takes another pile of work.

I don't know if this is any cleaner for precise mathematics, but seems a simpler way to get there:
1 = 9*(1/9) = 9*(0.1bar) = 0.9bar

 

[IAtaman]

Proof of irrationality of square root of two is also a good one.
I always though Euler's identity was very easy to calculate and understand for its elegance once someone shows you how it works.
Special Relativity's math is Relatively simple for how Special the theory is.
Pythagorean Theorem has very interesting proofs that look convoluted at first but impressively simple when you get to end of it.
Riemann Hyphothesis is not very hard to understand and explain for how proof proof it has been.

 

[IAtaman]

I don't know if this is any cleaner for precise mathematics, but seems a simpler way to get there:
1 = 9*(1/9) = 9*(0.1bar) = 0.9bar

In my experience anything that resembles an infinite series should be left to professionals to handle. Things can go very wrong very fast

 

[Elitzur–Vaidman]

Infinity, blah, I still have some papers and a book on aleph-null vs. aleph-one around someplace. I am definitely not going to look for it.

1 + 1 = 3 for very large values of 1.

I've always wanted a reasonable proof showing that 2^aleph-null == aleph-one, but I think rigorously comparing the cardinality of infinite sets is about as far from "easily accessable" as mathematics gets

 

[antcollinet]

- for crypto / security, we need WAY more than 6 significant digits

Which is why I specified mechanical engineering for 6 significant digits.

 

[Philbo King]

I also just thought of when my father boggled my mind with this one:

1=.9bar

Edit for clarity.
.9bar = 0.999999999... I couldn't find a way to actually have a superscript, and ellipses I thought might be less clear, so, it is meant to represent 9s going on forever.

Nothing to do with pressure. Sorry 'bout that.
End edit.


The proof was simple enough for my young mind to grasp, but profound enough to leave quite the impression.

This one is conceptually simple.
1/9 = 0.11111...
Multiply by 9: 9/9 =1 (by definition)
and 9/9 = 0.9999... (by multiplication)

 

[DonH56]

I've always wanted a reasonable proof showing that 2^aleph-null == aleph-one, but I think rigorously comparing the cardinality of infinite sets is about as far from "easily accessable" as mathematics gets

I'm a simple hairy-knuckled country-boy engineer so the whole course was difficult for me to fit into the "reasonable" category..

I had a quantum physics prof who actually made it interesting and (somewhat) accessible, but the next course with a different prof was, uh, less so... About then I decided theoretical physics wasn't my cup of tea (ahem... Scotch). Although I attended a class or two by Feynman and taking one of his courses (as some friends did) might have been fun. Too far to drive.

 

[holla]

I've always wanted a reasonable proof showing that 2^aleph-null == aleph-one, but I think rigorously comparing the cardinality of infinite sets is about as far from "easily accessable" as mathematics gets

The question of whether 2^aleph-null = aleph-one is the continuum hypothesis and is independent of the axioms of set theory so no proof exists; and no counter-proof exists either. (This assumes we are working in Zermelo-Fraenkel set theory or something equivalent; other foundations of mathematics are available. We are clearly outside the realm of "readily graspable" here.)

 

[Elitzur–Vaidman]

The question of whether 2^aleph-null = aleph-one is the continuum hypothesis and is independent of the axioms of set theory so no proof exists; and no counter-proof exists either. (This assumes we are working in Zermelo-Fraenkel set theory or something equivalent; other foundations of mathematics are available. We are clearly outside the realm of "readily graspable" here.)

That would certainly explain why I never came across one while earning my mathematics degree.

 

[BDWoody]

Curious: how did your father explain it to you?

After reading the further comments, I have to say I don't believe I've ever had to 'show the class your work' in front of a class quite like this one.

It's not your typical audio forum...

 

[mhardy6647]

After reading the further comments, I have to say I don't believe I've ever had to 'show the class your work' in front of a class quite like this one.

It's not your typical audio forum...

It happens to the best of 'em...

 

[Ron Texas]

It reminds me of a friends home brew beta glasses or whatever they where called. He's a electronic technologist and took a pair of sunglasses and put diodes inside pointing at the eyes. Then he had circuitry to adjust the flashing rate, duration and brightness etc. They made one feel veryyy strange. Lol. Your picture makes me feel strange when looking deep into it.

I have no idea about how anyone dreams these things up.

 

[jsilvela]

I have no idea about how anyone dreams these things up.

OK, on the non-periodic pattern you posted on page 1, and the further comments of seeing visions and hallucinating...
I saw just a tile, nothing more.
Was there some sort of hidden "Magic Eye" picture?

 

[Ron Texas]

OK, on the non-periodic pattern you posted on page 1, and the further comments of seeing visions and hallucinating...
I saw just a tile, nothing more.
Was there some sort of hidden "Magic Eye" picture?

Not that I know of.

 

[jsilvela]

It's a bit deeper than that article suggests. This one does a better job I think.
https://www.quantamagazine.org/hobbyist-finds-maths-elusive-einstein-tile-20230404/

Finally read this. Pretty interesting.
Like many other things, you only realize the importance when you see the specialists excited.
BTW calling it an "einstein" tile is kinda cheeky attention-grabbing.