biggest number

T@P

The question: You have to write the biggest number you can with a limited space, in other words each digit (including mathematical symbols) has an area of 1 unit. What is the greatest number that can be written on an area of x units? (note the area can be rearanged, so 2^5 would be two units, and your "paper" can go in any direction)

note i havent done the problem...

NoTime

[tex]\infty [/tex]

xxxxxxxxxxxxxxxx

Pyrrhus

infinity is not a number.

Noticibly F.A.T

9^9^9 maybe.

T@P

possibly, and actually 9^9 is what i thought too, but there are other ways. for example taking a eally smail number, and then 1 over that number would be quite big. But what i was actually lookng for was a proof that some way is the most efficient way... thanks for your posts

NateTG

Well, there's also:
9!!!!!!!!!!!!...
which is pretty huge, and arrow notation:
http://mathworld.wolfram.com/ArrowNotation.html

You really need to specify things a bit better to come up with a solid answer.

NoTime

I'm not so sure about that.
Mathematically any given line segment is composed of an infinite number of points.
Given the existence of a plank length, it would seem to resolve to a specific number.

wave

Infinity is an abstract concept. Please explain how you can "resolve" a specific number for infinity.

Rogerio

A line segment is not a real object! Plank doesn't apply!

It sounds the same as "there is a limited amount of real numbers in the [0,1] interval"...

Gokul43201

This makes no sense at all.

"Infinity" is simply not defined in the reals.

NoTime

Is a line segment that I draw any less real than say a square or a triangle?
In other words -> Why does Plank not apply?

Infinity is a concept that was developed long before knowledge of QM.
Ma Nature seems to be saying that this is indeed the case.
Personally, I'm inclined to take Ma's word for it rather than the imagination of man.

I will note that Zeno's arrow unerringly hits the target in real life.
Does this mean that there comes a place where you can no longer half the distance?

Hurkyl

Mathematical ideas and concepts are defined by axioms and definitions -- formal statements in the language of mathematics. In particular they are not defined by "reality".


There is no "Planck length" for a mathematical line segment because it is not a logical consequence of the axioms.



As an aside, there is no evidence (experimental or theoretical) that there is a smallest unit of length in reality either; "Planck length" doesn't mean what you think it means.

NoTime

I do know what an axiom is.
I also do not deny that they say exactly what you say they do.

Why should mathematics be exempt from reality?

To my knowledge the "Planck length" was determined from experimental data.
It is also the root of QM.
Is it a discontinuity or simply a region within which a determination cannot be made seems (at least to me) more problematical.
I connect with the idea that you do not like what I have said.
OTOH I have seen a number of arguments against the existence of infinities.

Why should I take one side or the other?

wave

Infinity is an abstract concept, so it does not have any concrete existence. Furthermore, infinity is not a particular number by definition. It's fine if you don't agree. Create a new word to label your concept, instead of arguing against an established definition.

Gokul43201

The reals are a well defined field and do not contain an element known as "infinity". That is how they are defined, and that is that.

However, "infinity" appears in Cantorian Set Theory, (surreal numbers) as the cardinality of certain sets.

Mathematics is simply a set of results obtained through an established logical method, based on a certain framework of axioms. There is no need for a mathematical structure to have any physical meaning.

And for your information, the Planck Length is not determined from experimental data. It is simply a length scale obtained by a suitable manipulation of fundamental physical constants ([itex]= \sqrt {hG/c^3} ~ or~about~10^{-35} m[/itex] ). Presently, that is too small to "measure".

Hurkyl

Because mathematics does not deal with reality; it deals with the formal consequences of axioms.

An axiom is simply a logical statement; it gets the name "axiom" because of the way we use the statement.

Here are a few of the axioms of the real numbers:

a + (b + c) = (a + b) + c
a * b = b * a
a * (b + c) = a * b + a * c

There are about 10 in all, of varying complexity.


Now, it is entirely possible to define "infinity" to mean "7". If you did so, then infinity really would be a real number. However, no standard definition of infinity yields a real number. In fact, no definition of infinity yields an object that is any sort of familiar number! In particular, Infinite numbers (such as those in the Surreals, or in the Cardinals) are not called "infinity".



Any sort of connection between mathematics and "reality" falls under the purview of science. It is science that says real numbers have some sort of connection to the real world. If science determined that there was a fundamental length, then that would mean that science would no longer attempt to say that real numbers are lengths; it does not mean that the mathematical meaning of a real number should change.

NoTime

Nice statement
It addresses the heart of the matter for me.
To the degree that attempts are being made to define "reality" using mathematics, then what is the requirement for mathematics to conform to "reality"?

The fundamental physical constants are called that because they can ONLY be determined by experiment.
Show me an equation that produces c. NOT a value for c, but c itself.

Take a simple well known case.
A bound electron has energy levels A and B.
What is the set of points in the interval AB?
I say that set is empty.
You can imagine that an electron can have any energy level between A and B.
It just is not true.