Proof that pi is rational - lol

kriminal99 said:
No the staircase does converge to the diagonol, and gives us the infinite summation which results in sqrt(2). Note that the start and endpoints of this staircase are not on the endpoints of this diagonol, but rather get closer and closer to the diagonol as the number of stairs increases.
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You keep adding to your limitingscenario. Well there's holes in this one aswell. Say the base and side of the triangle both have length 1. According to pythagoras theorem the hypotenuse have length sqrt(2). Ok, so in your limit the endpoints of the stairs goes closer and closer to the base and side. When I've taken the limit far enough the distance between the stairs endpoints and the base and side will be small enough that I can put it aside and we have the earlier scenario again where you add together the horizontal portion of the stairs and the vertical portion of the stairs and end up with the sum approximately 2 which is far from sqrt(2). This limit of stairs just doesn't converge to the length of the hypotenuse according to pythagoras and everyone after him.


kriminal99 said:
If you give me any definition of square root 2, I will eventually be able to reduce it to this one.
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Ok, here goes.

Definition of the square root of 2:
The number a equal to or larger than 0, that satisfies the equation a*a=2.

Do you want the proof that it is irrational aswell?
 
kriminal99 said:
I don't need to show that the expansion terminates or repeats, because this does not rule out trivial difference from a rational, nor do any of the proofs that sqrt(2) is not rational.
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And maybe you'd like to define "trivial difference" for us.
 
AndersHermansson said:
You keep adding to your limitingscenario. Well there's holes in this one aswell. Say the base and side of the triangle both have length 1. According to pythagoras theorem the hypotenuse have length sqrt(2). Ok, so in your limit the endpoints of the stairs goes closer and closer to the base and side. When I've taken the limit far enough the distance between the stairs endpoints and the base and side will be small enough that I can put it aside and we have the earlier scenario again where you add together the horizontal portion of the stairs and the vertical portion of the stairs and end up with the sum approximately 2 which is far from sqrt(2). This limit of stairs just doesn't converge to the length of the hypotenuse according to pythagoras and everyone after him.

Ok, here goes.

Definition of the square root of 2:
The number a equal to or larger than 0, that satisfies the equation a*a=2.

Do you want the proof that it is irrational aswell?
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Im not adding to it, Im just trying to remember the damn thing. Keep in mind that the holes you were referring to 2 seconds ago were in a straw man version of what I was describing. At some point you should realize that you cannot know that you are considering the same argument that I am until I admit that something you bring up is a hole in my argument... And therefore while presenting the problems with what I have been able to communicate to you so far making propaganda statements regarding me or my arguments is not mature or justified.

EDIT: BTW it is also possible (and it is what has happened here) that someone concieve of an argument, realize its logical value, and then a long time later forget the argument and try to recall it and make a mistake. This is what has happened here, I have made a mistake in reconstructing this argument. You are correct regarding the argument as I have provided it thus far... Thankyou...

The real argument involves doing everything similarly but I am leaving something out. I will come back with it when I have finished completely reconstructing it. No I don't expect you to believe me until I come back with it.

Differ trivially means the difference is like 1/infinity and therefore they are equal.

The definition of the square root hasn't given the number a value. It merely states that if such a number happens to exist that satisfies the equation A*A = 2 it would be the square root of two. What I am trying to show you is that the others which supposedly give it a numerical value are based on the same concept and therefore cannot give it a numerical value.
 
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kriminal99 said:
And therefore while presenting the problems with what I have been able to communicate to you so far making propaganda statements regarding me or my arguments is not mature or justified.
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I'm trying my best do decipher your statements!

kriminal99 said:
EDIT: BTW it is also possible (and it is what has happened here) that someone concieve of an argument, realize its logical value, and then a long time later forget the argument and try to recall it and make a mistake. This is what has happened here, I have made a mistake in reconstructing this argument. You are correct regarding the argument as I have provided it thus far... Thankyou...
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Ok that can absolutely happen! But you are making statements that need rigorous mathematical treatment.

kriminal99 said:
The real argument involves doing everything similarly but I am leaving something out. I will come back with it when I have finished completely reconstructing it. No I don't expect you to believe me until I come back with it.
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Very good. Then it can be dealt with properly!
 
Let me just forget about the vizualization for a moment and just sum up the point of this thread:

1. A rational number is a number that can be written as a ratio of two integers p/q
2. As an immediate consequence of definitions of rational numbers and addition and integers summations of rational numbers are also rational.
3. Infinite summations are sometimes defined as not rational sometimes, where infinite simply means never ending.

Once again in simple logic :

1. All summations of rationals result in other rationals
2. Infinite summations of rationals are not rational

The definitions of rational, integer, and addition dictate that ALL summations of rationals are rational. Not SOME, and not ALL BUT ones that never end.

This is an immediately obvious logical contradiction. Mathematics constantly redefines terms in its belief set for convenience at the moment and do not try and connect their actions to the real world in any way. Even if you ignore reality or what things are equivalent to in the real world, You can only do this once without having a large probability of having your belief set contradict each other. As long as defining a certain term a certain way has ramifications that are not immediately obvious this is the case.

And if all ramifications of defining a term a certain way WERE obvious, then we wouldn't be finite beings and wouldn't even need mathematics. We could just look at a pile of 4 billion coins and immediately recognize it as such. Complex equations would be as obvious to us from the definition of our terms as deducing bob is short from the statement "bob is bob is not tall or average height" is now. However we are finite beings and that is why we are using math to begin with.

The point regarding rationals is trivial. The REAL point is any number of contradictions could come up anywhere in mathematics as a direct result of defining terms for convienience.
 
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kriminal99 said:
2. Infinite summations of rationals are not rational
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If and only if the decimal representation is not cyclic.
kriminal99 said:
The definitions of rational, integer, and addition dictate that ALL summations of rationals are rational. Not SOME, and not ALL BUT ones that never end.
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No infinite summations can be irrational. Most of them are not cyclic, which means that they can't be written as p/q.
kriminal99 said:
This is an immediately obvious logical contradiction. Mathematics constantly redefines terms in its belief set for convenience at the moment
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I think it is you who are confused.
 
2. As an immediate consequence of definitions of rational numbers and addition and integers summations of rational numbers are also rational.
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Only finite summations (i.e. summations with a finite number of terms). It's a simple point.
 
kriminal99 said:
The definitions of rational, integer, and addition dictate that ALL summations of rationals are rational. Not SOME, and not ALL BUT ones that never end.
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If by 'summation' you are including infinite series then no. The usual axioms for addition (whether on the reals, integers, or rationals) state that it's an operation that takes two elements of your set to another element of your set and satisfies some other properties you'd expect 'addition' to have (asssociativity, commutivitiy, inverses, how it meshes with multiplication, etc), but none of these properties says anything at all about infinite series. You have assumed this property from I don't know where, it's not one that mathematicians assume when they talk about addition (eg. see the axioms for rings, fields, etc.)
 
Now you are simply fabricating definitions for infinity.

If anywhere in math that it is held that infinity is simply a never ending sequence then you cannot simply change that here for convienience with out it contradicting what was said before. This means that you can't say things like "could go on forever" or "doesn't end" or things of this nature.

In reality nothing changes regarding a summation of rationals just because it doesn't end. At what point do you think a summation suddenly becomes irrational? If you add a million terms its still necessarily rational. A billion etc... A summation being infinite simply means you don't stop adding more terms. It doesn't magically change the properties of the things being added or how.

James R said:
Only finite summations (i.e. summations with a finite number of terms). It's a simple point.
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This is just flat out wrong, as hallsofivy pointed out. 3/10 + 3/100 + 3/1000... is a counter example (1/3)

shmoe said:
If by 'summation' you are including infinite series then no. The usual axioms for addition (whether on the reals, integers, or rationals) state that it's an operation that takes two elements of your set to another element of your set and satisfies some other properties you'd expect 'addition' to have (asssociativity, commutivitiy, inverses, how it meshes with multiplication, etc), but none of these properties says anything at all about infinite series. You have assumed this property from I don't know where, it's not one that mathematicians assume when they talk about addition (eg. see the axioms for rings, fields, etc.)
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This is just logic. Mathematicians can arbitrarily define terms all day long, but they don't get to define what is logically dictated by something that you claim. To begin with you are giving infinite series a seperation from finite series that they are not afforded by the definition of infinite used often in mathematics.

Also it is clear that there is some confusion regarding differences in opinion in mathematics on these types of subjects. Just because one particuarly naive professor tells you that something "is" a certain way or even that "most people think this in the field" doesn't mean it is true. It is not like there is some kind of unanomous consensus among people capable of considering such ideas that things "are" a certain way. And even if there was some kind of consensus, if it was incorrect (in the sense that it causes problems when dealing with, and doesn't match, reality) then even if you want to go the naive "sociology" type reasoning you have to consider that a potentially infinite number of people in the future are going to agree with the correct viewpoint.

It doesn't matter what path you take, there are going to be large numbers of contradictions as long as you are arbitrarily defining terms. In the end there are only two paths to take: Define terms based on reality, or define terms without any connection to reality whatsoever but rather based on the fact that these arbitrary definitions do not contradict your earlier ones. In the latter case your construction has 0 value to the real world. Right now mathematics is a patchwork monster from created from both of these approaches, the result of which is something that is often self contradicting.
 
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Seems to me like you decided that standard mathematics was inconsistent, and then went to find where it was. You have made up your mind yet have not produced a rigorous proof of your claims.
 
K99:

Right now mathematics is a patchwork monster from created from both of these approaches, the result of which is something that is often self contradicting.
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It is? Just because you don't understand or agree with it? Are you a professional mathemetician who, upon a lifetime of long and deep thought has found deep, disturbing "inconsistencies" in modern mathematics? Or are you just another crank with a big ego who thinks that THEY, above all others, have seen the light? Note yet another hallmark of the true wackjob.
 
The proof of this is purely logical in nature, although it seems more and more that logic is something beyond most mathematiciains that come here...

There is reality, and there is your belief set. Most people think it is a goal to model their belief set BASED on reality. If you are modeling your belief set on reality than there are things which YOU DONT GET TO DEFINE ARBITRARILLY. These are the things that reality dictates. If you in combination with this arbitrarilly define things that you, AS OF YET, have not seen to be determined by reality, then there are very likely going to be contradictions.

On the other hand you could build a belief set without any regard for reality whatsoever, and THEN its ok to define things just so they won't contradict what you already have. Don't expect this belief set to be of any use however, and also expect people to look at you funny when you talk about how star wars is real and they found light sabers buried in the desert somewhere...

Additionally, our ability to choose the second path is limited. You will constantly find yourself using and applying things and reasoning you see has value in reality to your fabricated belief set. When this happens, you get a patchwork monster and will end up contradicting yourself.

Allow me to provide an example:

John is taller than sally.
Sally is taller than joey.
Joey is the same height as Billy.
Billy is the same height as John.

Lets say these are the axioms of mathematics. Obviously this example is drastically simplified. Now if you follow the logical consequences of each of these statements, statements 1 and 4 contradict each other. However the hairless monkey stating the axioms of mathematics is finite in nature and cannot always immediately see this contradiction. In fact the contradiction might be really hard to see without an extreme amount of effort. You might start making up your own definitions of "taller", "than" etc like mathematicians do, but then your just plain stupid.

Just so you aren't confused, I (and people that make questions like this on iq tests) have made these small enough such that I could still know the consequences and that the first three are consistent but the last is not. In a non artificial setting that is often beyond anyone's capabillity.
 
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Math is a language you kook. Hinduism is a belief system. English is a language. UFOlogy is a belief system.

Math is a made-up notational language to organize ideas on how things and ideas relate to other things and ideas. pi is not rational by definition. RATIOnal number can be expressed as the RATIO of integers. By definition.

Why don't you argue the merits of changing the definition of CAT to mean WINDOW or something just as stupid. CAT is a misleading concept and I don't like it.
 
I swear you get less intelligent every time you hit that post button.

And how exactly, do you differentiate math from any other belief set? Because you "like" math, or because you primitively reason that some of the ideas contained in math model reality in a certain context (like 1 + 1 = 2) You don't think hinduism started out with the intention of being a "made-up notational language to organize ideas on how things and ideas relate to other things and ideas"? It has its own words just like math, that are no different from the symbols. The people who hold hinduism believe their reasoning is solid. How is math somehow different than hinduism as far as being a belief set is concerned? Or have we suddenly exceeded your mental capacity?
 
Ugh. Intellectual child. Belief systems rely on faith. Math and the sciences rely on proof, experimental or logical. La dee da. Need a nap K99? Want your blankie?
 
Define faith and proof? Does not the theologist claim it to be proof when a major accident is averted? Does not the mathematician desire to interpret things a certain way that adheres to his views? Does he not desire to prove that a certain thing is the case or preserve the beliefs he has so far built up?

Do you really think you are so aggressively insulting me because you ACTUALLY believe that math is unfallable and totally based on unobjective proof?
 
Faith by definition requires no physical evidence or logic. Faith is based on the unquestiong BELIEF in the truth of a thing.

Proof by definition consists of concrete evidence or logical reasoning that is consistent with everything currently postulated about a particular subject.

Theologians are cranks.
 
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