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Strange magic

Started by Jamie D, August 28, 2013, 06:56:20 AM

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Devlyn

I'm not going to understand the answer, but what happens to pi as a circle approaches a hypothetical zero radius?
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Jamie D

It doesn't matter whether you use the radius or the diameter (as D = 2r).

When the radius is equal to 0, then you no longer have a circle, but rather, a point.
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Devlyn

That wasn't as bad as I expected. Thanks, mathemagician!
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Sandy

And in all of this mathematical mayhem, let us not forget our dear Benoit Mandelbrot, the father of fractals and fractal math who brought us the iconic mandelbrot set:



Fractal math brought us realistic clouds in CGI among many other things.  I wrote my first mandelbrot explorer program in the late 80's and sucked up an entire mainframe computer for hours.  I've run them on commodore 64's and they would run for days.

And each point on a mandelbrot points to an associated Julia set.  Each one unique and beautiful:



And each set is self similar.  I have run literally billions of calculations through my various computers and I am still in love with them to this day.

-Sandy
Out of the darkness, into the light.
Following my bliss.
I am complete...
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Jamie D

Thanks Sandy!  They are lovely, aren't they?
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Danielle Emmalee

Quote from: Jamie D on September 07, 2013, 08:07:32 AM
In fact, as it must be the case, almost all real numbers must be irrational!  Which, when you consider infinity, is irrational in and of itself.

Is there a proof for this?  I don't see how most, let alone almost all, real numbers must be irrational.  That's obviously setting aside the fact that there are infinite number of irrational numbers and and infinite number of rational numbers, which I'm not sure you can really do.
Discord, I'm howlin' at the moon
And sleepin' in the middle of a summer afternoon
Discord, whatever did we do
To make you take our world away?

Discord, are we your prey alone,
Or are we just a stepping stone for taking back the throne?
Discord, we won't take it anymore
So take your tyranny away!
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Devlyn

Duel! Chalkboards at twenty paces.
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Jamie D

Quote from: Alice Danielle on September 07, 2013, 02:45:58 PM
Is there a proof for this?  I don't see how most, let alone almost all, real numbers must be irrational.  That's obviously setting aside the fact that there are infinite number of irrational numbers and an infinite number of rational numbers, which I'm not sure you can really do.

Cantor's Proof

http://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument

Quote from: Devlyn Marie on September 07, 2013, 03:16:22 PM
Duel! Chalkboards at twenty paces.

I want the chalk that squeaks.  >:-)
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Danielle Emmalee

Quote from: Jamie D on September 07, 2013, 04:23:07 PM
Cantor's Proof

http://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument

I want the chalk that squeaks.  >:-)


cool.  makes sense now. Its really odd to think of the different levels of infinity.  Infinity is such a weird concept.
Discord, I'm howlin' at the moon
And sleepin' in the middle of a summer afternoon
Discord, whatever did we do
To make you take our world away?

Discord, are we your prey alone,
Or are we just a stepping stone for taking back the throne?
Discord, we won't take it anymore
So take your tyranny away!
  •  

Devlyn

Quote from: Alice Danielle on September 07, 2013, 04:37:09 PM

cool.  makes sense now. Its really odd to think of the different levels of infinity.  Infinity is such a weird concept.

Yes, I think about it ALL the time!
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Devlyn

Quote from: Jamie D on September 07, 2013, 04:23:07 PM


I want the chalk that squeaks.  >:-)

You ARE the chalk that squeaks!
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Danielle Emmalee

I always found it interesting that in school you are told that a negative number multiplied by a negative number equals a positive number.  You are never told why.  The teacher never proves this notion to you or even tries to explain how it makes sense.  You try it out on a calculator and it works, therefor you never question it again for the rest of your life, or at least I think most people don't.  I got interested in math at about 3 years old and by the time I came across negative numbers and multiplication from my older brother's math books that he brought home (I think I was about 6), I didn't have a calculator and I didn't have a math teacher, so I was really confused.  That and I'm sure most elementary teachers and most parents probably don't even know how to show proof of this concept if they were asked. So I was stuck at, I think, a grade 4 or 5 math level because I figured if kids in grade 4 understood this, surely I had reached my limit until I could have someone explain it to me.  That day never came.  Not sure why I never asked either, I probably just forgot about it and like the other kids, when I typed it in the calculator it worked. 

Not sure what the point of my rambling there was, I just think its interesting how many "simple" mathematical concepts are just taken for granted because of calculators.
Discord, I'm howlin' at the moon
And sleepin' in the middle of a summer afternoon
Discord, whatever did we do
To make you take our world away?

Discord, are we your prey alone,
Or are we just a stepping stone for taking back the throne?
Discord, we won't take it anymore
So take your tyranny away!
  •  

Danielle Emmalee

Oh and in case you were interested

Let x = ab + (-a)(b) + (-a)(-b)
case 1                                         case 2
=ab + -a(b+-b)                           = b(a+-a) + (-a)(-b)
=ab + -a(0)                                = b(0) + (-a)(-b)
=ab                                            =(-a)(-b)       

Therefore ab = (-a)(-b) 
Discord, I'm howlin' at the moon
And sleepin' in the middle of a summer afternoon
Discord, whatever did we do
To make you take our world away?

Discord, are we your prey alone,
Or are we just a stepping stone for taking back the throne?
Discord, we won't take it anymore
So take your tyranny away!
  •  

Jamie D

Oh yes, I remember using Reynold's number to determine whether fluid flow in a porous reservoir would act in a Darcian manner.

Darcy's Law, often used in the study of ground water aquifers, states:

Q = KA (h1-h2)/L

Where K = permeability
A = area
h1-h2= hydraulic head (or pressure drop)
L = length of the drop

Darcy's law describes the rate at which a fluid flows through a permeable medium. Darcy's law states that this rate is directly proportional to the drop in vertical elevation between two places in the medium and indirectly proportional to the distance between them. The law is used to describe the flow of water from one part of an aquifer to another and the flow of petroleum through sandstone and gravel.

But this only works when the Reynolds Number is relatively small.



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Jamie D

  111,111,111
x 111,111,111

12,345,678,987,654,321
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Late bloomer

The numbers of the Cosmos, particularly those pertaining to Galaxy Clusters, which draw me into their extreme distance.
Ooh, earthquake, nw California.  Lots of shaking going on.
We are never alone.  We're just temporarily having communications difficulties.
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Jamie D

Quote from: Late bloomer on March 10, 2014, 12:19:43 AM
The numbers of the Cosmos, particularly those pertaining to Galaxy Clusters, which draw me into their extreme distance.
Ooh, earthquake, nw California.  Lots of shaking going on.

http://earthquake.usgs.gov/earthquakes/eventpage/nc72182046#summary

Could be Blanco Fracture Zone, Gorda Plate, or related Gorda Ridge

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Jamie D

http://www.activetectonics.coas.oregonstate.edu/gorda.htm



The little goodies that look like beach balls are called "fault plane solutions" and show the relative motion on the fault - most on this map show strike-slip movement.
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Jamie D

Today's report from the USGS, regarding the 6.8 Ferndale Earthquake

Tectonic Summary

The March 10, 2014 Mw6.9 earthquake off the coast of northern California occurred as the result of the oblique strike slip motion on a fault approximately 80 km offshore of Eureka, California. The preliminary location places the earthquake within the Juan de Fuca plate (or Gorda subplate), which subducts beneath northern California, Oregon, and Washington at a rate of ~23 mm/yr. This location is outboard of the trench in the oceanic crust. The earthquake was widely felt along the coast of northern California and southern Oregon, particularly in the city of Eureka.

The general tectonics of this region are characterized by transitions between oceanic subduction of the Juan de Fuca plate beneath the Pacific northwest region and the continuation of the San Andreas Fault offshore. The intersection of the Juan de Fuca, North America, and Pacific plates forms the Mendocino Triple Junction off the west coast of California, with the subduction zone extending to the north and the San Andreas Fault diverging to the west offshore and continuing to the south. The offshore extension of the San Andreas Fault and southern extent of the Juan de Fuca plate are defined by the easternmost exposure of the Mendocino Fracture Zone. Several large earthquakes have occurred in this region since 1900 within 100 km of the March 2014 event, including events of M7.2 in 1922, M7.1 in 1923, M7.3 in 1980, M7.0 in 1994, M7.2 in 2005, as well as several events near the coast or inland of California, including the 1992 M7.2 Petrolia earthquake with its M6.6 and M6.4 aftershocks. Most recently, an earthquake of M6.5 in January 2010 with a similar faulting mechanism to the March 2014 event occurred.


The area discussed above is known as the "Mendocino triple junction."  Fascinating geology.
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