Cardinality stuff is cool.

Like the proof there are the same number of fractions as integers.

And the proof there aren't the same number of real numbers.

It's amazing that the continuum hypothesis is still a thing, and may be insoluble...

I'll try my best tomorrow when not watching footy and tapping on my phone.

Sure. guess it shows how boring this game is that i'm in the maths thread while watching. I normally hate maths

equinox_code wrote:

What is meant by imaginary numbers, for example?

Imaginary numbers are square roots of negative numbers. Since -1 x -1 = +1, the square root of +1 can be plus or minus 1.

But the square root of -1 can't be +1 or -1, and it is denoted commonly as either i or j.

A complex number is something that has a real (ie normal numbers between plus and minus infinity) and an imaginary component.

For example
 1.896725 is a real number.
i is an imaginary number.
1.864586396 + 4i is a complex number.

acemuzzy wrote:

SpaceGazelle wrote:

On phone so no links. Not sure if there are links tbh. There are no particles and there are no waves, there is only equations and constants

That doesn't mean it's not useful to talk about particles and waves though

Absolutely. If you're going to teach QM well you're going to need particles as a philosophical foothold to start with. You don't need waves at all actually, and they're fairly redundant once you're aware of the madness.

equinox_code wrote:

What is meant by imaginary numbers, for example?

The same as negative numbers with a square root thrown in. Philosophically, the root is merely decoration. The negative is the important bit, so you can see it's not as complex as it first seems

That is illogical

Horseshit.

@spock

Spock wrote:

I don't like maths. It is an ultimately soulless pursuit that teaches us nothing about humanity.

The only real depressing thing about humanity is it's self importance.

@ Spock - you wouldnt have a train, or a Manchester were it not for the advances in maths that underpins everything we do. But beyond that, it is also incredibly effective at describing the universe itself.

I'd say mathematics has taught us one important thing in particular. Back when theologically inclined mathematicians insisted Pi was exactly 3, the proof to the contrary was an important reminder that reality does not bend to our whims or desires.

Spock wrote:

Maths is a tool to facilitate the mundane e.g.  If space gazelle has x amount of horse shit, when will vela's train arrive at Manchester at y?

You equate maths to quantity and not relationship. It's a common trait tbf.

OK.

acemuzzy wrote:

Cardinality stuff is cool. Like the proof there are the same number of fractions as integers. And the proof there aren't the same number of real numbers. It's amazing that the continuum hypothesis is still a thing, and may be insoluble...

It is cool. Integers are countably infinite. That means you can go 1, 2, 3, 4....and so on up to infinity.

Reals are uncountably infinite. What comes after 1? 1.1? 1.01? 1.0000000.........01? No. there's an infinite number of numbers between 1 and 1.1 for example. They are uncountable because you can't get past 1 when trying to count.

It means integers are a subset of reals - an infinitely a much smaller set of numbers compared to reals - and are a level of infinity below the reals.

Integers are designated aleph0 , reals are aleph1. 

Now think about functions represented by code in computer RAM. RAM is a sequence of binary bits that  ultimately represents an integer. So every computer program in existence, and will ever exist in current architecture, will equate to an integer. 

Therefore the number of functions that a computer can deal with is an infinitely small a minute subset of the number of computable functions that map onto reals. It's a miracle computers can do anything.

I like to think the functions not representable by integers are the great imponderables - the real human problems. Since we haven't really come up against the limits of computers (at least in theory - we have a theoretically infinitely powerful computer called a Turning machine with infinite RAM which can still only process functions mapping to integers - even if they are infinitely long and can deal with an infinite number of problems), then we aren't asking the right questions about life.

I thought we'd finally got a Rachel Riley thread!
I guess I'll have to read it instead when I'm less tired.

She m..m...m....m....makes my @prick happy.