math

[Vraith]

Where Pythagorean Theorem is a 3rd order equation.
Has anybody tried to formulate a 4th order equation?

I'm sure they have...those crazy mathematicians will try anything at least once.
Except for the ones who will only try one thing...but in every possible way.

[Vraith]

How often do you get math, crayon-like-work, and commentary all in one?
[[[well...fairly often, if you look...but this was a good one, and no deep math necessary]]]

mathwithbaddrawings.com/2015/09/23/what-does-probability-mean-in-your-profession/

[Avatar]

I like it.

--A

[Vraith]

I like it.

--A

Then you'll probably like these, too. This time it's dry-erase on whiteboard drawing.
The third or 4th one down relates to the recent conversation that involved Orlion and peer-review.

Many chuckles...this is one of my favorites [because I hate the constant panic-press, and wish people would quit falling for it]

Our World: One Dead in Shark Attack; See Tips for Shark Safety Inside
Mathematically Literate World: One Dead in Tragic, Highly Unlikely Event; See Tips for Something Useful Inside

mathwithbaddrawings.com/2013/12/02/headlines-from-a-mathematically-literate-world/

[Avatar]

Our World: Firm’s Meteoric Rise Explained by Daring Strategy, Bold Leadership
Mathematically Literate World: Firm’s Meteoric Rise Explained by Good Luck, Selection Bias

--A

[peter]

Been reading a bit about this 'imaginary number' i, which if I get it right, is the square root of minus 1. Apparently this number is absolutely central to the development of quantum theory [in some way I haven't got my head around] and is what is known as a 'complex number'. All this is pretty much 'greek' to me; can anyone put some flesh on the bones of what a complex number is - apparently it's not one of the normal integers or fractions thereof that we use in everyday calculatione, but is something else alltogether.

The book I'm reading is by Neil Turok, a physicist who has done work alongside Hawking, and is now a major player in the development of the 'cyclic model' for the origin of the Universe, which is an alternative to the 'inflationary model' and proposes that the Universe came into being as a result of a collision between two of the 'brane-worlds' of M-Theory. [Hashi has mentioned his support of this idea on a number of occasions IIRC].

How bizzare that these deep [and complex in their derivation] quantities such as i and e should be found to be intrinsic to the very funamental nature of 'being' itself; as someone said "If God was a mathmatician - he was a damn good one!"

[Hashi Lebwohl]

To start, recall that multiplying two positive numbers gives you a positive number, multiplying two negative numbers gives you a positive number, and that multiplying a positive and a negative numbers is the only way to get a negative number. Now recall that the "square root" of a real number k is some other real number h where two things are known to be true: h < k and h^2 = k (presuming k <> 1, in which case h = k = 1). This means that for any given number k it has two square roots, h and -h.

Imagine, then, what happens if we try to find the square root of -1. There must be some number that multiplies by itself to give a negative number, but this contradicts the multiplication statements we made earlier. Since the number could not be real it must, literally, be imaginary. Mathematicians rarely use the term "imaginary" these days, instead using the term "complex".

Here is the Wolfram MathWorld article on complex numbers.

Most of modern physics would be impossible without complex numbers.

Interestingly, if you work with matrices you will encounter quaternions. It turns out that the matrix we can identify as "=1", which has the form [{-1, 0}, {0, -1}] has *three* different "square roots" denoted i, j, and k (where i^2 = j^2 = k^2 = -1).

i itself has more than one square root; (1 + i)/sqrt(2) is one of them:

(1 + i)/sqrt(2) * (1 + i)/sqrt(2) = [(1 + i)(1 + i)]/[sqrt(2)*sqrt(2)] =

(1 + i + i + i^2)/2 = (1 + 2i - 1)/2 = 2i/2 = i

*************

I support the brane hypothesis because it allows for a reasonable and rational explanation of how the universe could exist before our known laws of physics and it offers an explanation for things to occur *before* the event colloquially known as "The Big Bang". Incidentally, our "laws" of physics are merely side-effects of the interactions of the branes as they collided, forming a "pocket" which, for now, is stable, but which could--will?--collapse again someday. It also accounts for the expanding universe we can verify via testing--the branes are still in the process of colliding, creating more "space" into which our intersection sits.

[JIkj fjds j]

My understanding of imaginary, or complex numbers, is simply that they are one of the scientific wonders of the world. Like pulling rabbits from a hat.
I've seen this method of calculation -j in electronic circuits and capacitor reactance, or complex impedance.

[peter]

Can i actually be given a numerical value then, or when it is used in equations is it always in such a way as to 'cancel itself out' (or by some other such trick as to avoid it's value having to be made explicit)?

[Hashi Lebwohl]

It doesn't have a numerical value; rather, it is merely a concept that allows for working with or solving certain equations. As Rune notes, i shows up in physical constructs like circuits but it isn't really there because when you work the equations the complex parts will cancel out or collapse into only real components.

[Avatar]

Imaginary I tell you.

--A

[peter]

Yesterday I put together a question about how the square root of minus one had [surely?] to equal one, and that surely it had to be the number that gave the result minus one as it's square root that was imaginary [coz this is impossible [isn't it?] ........ but got 404'd so I must have been talking crap!

[Hashi Lebwohl]

In math, "imaginary" and "impossible" are two totally different things. It is impossible to find the exact value of e because that number is transcendental--it never repeats and it never terminates.

[Avatar]

So it can be anything, which is really useful for proving an equation...

--A

[Hashi Lebwohl]

No, it has an exact value but the problem is we can never know what that exact value really is. All we can do is find more accurate approximations of that value but because of time and/or memory limitations we must stop calculating. When we stop calculating the decimal approximation stops, converting our answer into a rational number...but e is not a rational number.

That link in the thread in the Close, the one about "non-computable" numbers, was cute. What the author there called "non-computable" numbers mathematicians know as "irrational" numbers (which includes the subset of "transcendental numbers") and is the uncountable portion that makes the real numbers an uncountable set. The most sophisticated and powerful supercomputer we can ever build will not able to give us the *exact* value of sqrt(2) because that decimal representation never terminates.

The concept of a set being uncountable can be hair-pulling. Consider the Cantor set--start with the closed interval [0, 1], now throw out the middle third, leaving you with [0, 1/3] U [2/3, 1], then throw out the middle thirds of those sets, and so on and so forth an infinite number of times. The set you are left with is still uncountable even though you threw out most of the original set.

[peter]

That sounds a bit like some of the crazy stuff I've read about infinity (hotel's and buses full of people and bags of disappearing rubbish etc.) That math is a pretty weird thing at it's fringes!

[Hashi Lebwohl]

After a few years the weirdness wears off and then you find that things like uncountability or Fourier transforms become second nature.

[Avatar]

You just make it all up.

--A

[Vraith]

That sounds a bit like some of the crazy stuff I've read about infinity (hotel's and buses full of people and bags of disappearing rubbish etc.) That math is a pretty weird thing at it's fringes!

Math is pretty weird even in its middle [if you can find the middle, if it even exists].

For most of two millennia, there was potential and actual infinity. Infinite--- numbers and such in math---was fine. Infinite real things, like an infinite number of stars or anything else, were simply impossible. Smart people believed in the first, only crazy or dumb people the second. Then things got funny...

And then these peeps started building bridges:
https://www.quantamagazine.org/20160524 ... te-divide/

I don't think I believe in actual, material infinities. However, it is extremely important to human knowledge that math that includes infinities is applicable...and why/how it is so.

[Hashi Lebwohl]

The term "infinite" is really subjective because you have at least two types on inifinity, starting with countable (like the set of rational numbers) and uncountable (like the set of irrational numbers).

Infinity never really takes effect in the real world, at least no always. Consider black holes--the formula which originally predicted their existence has only the term r in its denominator, so as r approaches zero the force of gravity approaches infinity. Since the universe didn't get sucked into the first black hole which ever formed we know that it doesn't have infinite gravity and therefore doesn't have a zero radius but the math which says "it becomes inifinite" is still valid. Perhaps a side-effect of being non-quantum in nature prevents matter from attaining any sort of "infinite" status.