[Answered] who can prove that 1 > 0 ?

[tart666]

it's been awhile, i forgot how...

(edit Note: this question was for people that think "math" is a branch of knowledge rather than part of accounting)

using the following:


Addition is an operation on the field of real numbers, taking in two, returning one, with properties:
commutative, associative, adding 0 returns same number, for any a there always exists number b such that a+b=0

Multiplication is an operation on the field of real numbers, taking in two, returning one, with properties:
commutative, associative, multiplying by 1 returns same number, distributive ie a(b+c)=ab+ac

using these definitions of 1 and 0, how do you prove that 1>0 ?

 

[uart]

I think you could just take it as an axiom that the natural numbers 0, 1, 2, 3, .. are ordered couldn't you?


If however you want to prove it "strictly" from the rules that you have posted above then there is one slight problem, you haven't defined ">" anywhere in those rules.

If you allow me to define ">" as : a>b if and only if a=b+c for some positive c, then it follows that

1 = 0 + 1 therefore 1 > 0


Is this supposed to be a silly question?

 

[Peter]

The symbols, including the glyphs we use to represent numbers, are the basic prerequisites of math altogether. They're defined to mean what they mean.

Nothing to prove here. Please move on.

 

[f95toli]

Try "Principia Mathematica" , a (famous) book by Bertrand Russel. It probably contains the proof you are looking for (as far as I remember there is also a rather long proof of 1+1=2).

 

[drag]

Howabout you get a scale. On one side put 1 apple. On the other side put 0 apples. See which weighs more. Now put anything, a car, a brick, a leaf, anything that's not lighter then air.

 

[Peter]

Even if you do something like that, you still need a definition of what it means when you say "0" and "1".

That's not something that is to be proven - the numerical glyphs and the amount of (whatever) each represents is the most basic of all basics of the math language.

 

[f95toli]

I disagree, I am almost sure you can prove that 1>0 (I was not kidding when I wrote about the proof of 1+1=2, I have it in a book somewhere). I don't think you need to use it as an axiom.

 

[Krakerjak]

common sense

 

[ZeroNine8]

By definition of how the number system is arranged, 1 > 0, there is no proof of this as it is a fundamental idea on which our number system is founded. The very nature of how 0 and 1 are defined places 0 less than 1, there is nothing to be proven.

 

[wacki]

who can prove that 1 > 0 ?

I will never understand the impracticality of acadamia.

The University I got my undergrade at, for a long time, didn't teach C, which constists of 90% of all the real world programming. Instead everything was taught using Scheme because of it's "mathematical beauty". It was great as a learning tool, but sucked if you were a CS major trying to get a job.

 

[TerryMathews]

Originally posted by: wacki
The University I got my undergrade at, for a long time, didn't teach C, which constists of 90% of all the real world programming. Instead everything was taught using Scheme because of it's "mathematical beauty".

Translation: Your univ. didn't have enough professors on hand that understood C/C++ well enough to construct a cirruculum around it.

 

[Rainsford]

Originally posted by: TerryMathews

Originally posted by: wacki
The University I got my undergrade at, for a long time, didn't teach C, which constists of 90% of all the real world programming. Instead everything was taught using Scheme because of it's "mathematical beauty".

Translation: Your univ. didn't have enough professors on hand that understood C/C++ well enough to construct a cirruculum around it.

How is that possible. I don't think there is a programmer of even moderate talent who doesn't know C or C++ fairly well. Unless they all went to wacki's university I guess.

 

[Shalmanese]

I know someone who works as a Uni professor and earns $150,000 a year and doesn't know a jot of C. He works in AI and is involved in reasonably specialised applications for which there are better suited languages. C isn't the end of the world.

 

[f95toli]

Zeronine8: But that is still not a proof. The trick is to use as few axioms and rules as possible and then deduce everything from them, "common sense" is a valid method to "prove" something in engineering and maybe even in physics but not in math.

The whole point of "principa mathematics" was to show that math was self-consistent (that you do not need really axioms, only a few basic rules of logic), of course they failed but no one really knew why until Gödel came along.

 

[tart666]

Originally posted by: uart


If you allow me to define ">" as : a>b if and only if a=b+c for some positive c, then it follows that

1 = 0 + 1 therefore 1 > 0


Is this supposed to be a silly question?

The problem is, you use "positive" in your definition of ">", and "positive" is usually defined as "greater than 0". This way both definitions are circular and therefore invalid.

The bigger problem is, I don't have any number theory books anymore, does anyone have a definition of either ">" or "positive" on the field of reals?

 

[Peter]

Listen. Once again. 0,1,2,3,4,5,6,7,8,9. In that order. Representing increasing amounts. That's one of the foundation of all things mathematic. So are all the expressions, like + = >.


Can YOU prove that the greek letter Epsilon sums up stuff? You're asking the same thing. Get over it.
Ain't no proving that, yust as well as there is no proving 1>0. It is the way it is.

 

[tart666]

Originally posted by: Peter
Listen. Once again. 0,1,2,3,4,5,6,7,8,9. In that order. Representing increasing amounts. That's one of the foundation of all things mathematic. So are all the expressions, like + = >.


Can YOU prove that the greek letter Epsilon sums up stuff? You're asking the same thing. Get over it.
Ain't no proving that, yust as well as there is no proving 1>0. It is the way it is.

jebus, no need to crap the thread if you can't comprehend the question...

 

[Peter]

Once again.

9>8>7>6>5>4>3>2>1>0, the order of the numbers we use to express quantities, is one of the basic conventions math is founded on altogether. You can't prove that using math, because math is rooted on it.

What you need to comprehend is that your ORIGINAL question in itself is pointless. Even after you edited it to look more smartass.

 

[RossGr]

this is proven as a proposition in "Principles of Analysis" by Rudin.

The basic statement is

if x<>0 then x^2>0, in particular, 1>0

The key definition is

An ordered field is a field F which is also an ordered set, such that

(i) x+y< x + z if x,y,z are in F and y<z ,
(ii) xy>0 if x,y in F and x>0, y>0
We also need
(d) (-x)(-y)=xy
and from the Axioms of multiplication
M4 F contains an element 1<>0 such that 1x=x for every x in F

Proof
If x>0 (ii) gives x^2>0 . IF x<0, then -x>0 hence (-x)^2>0. But x^2= (-x)^2 by (d) Since 1=1^2, 1>0


Edit:

There that should read a bit better. I has the WYSISYG editor selected. Looks like that causes some troubles.

Edit, it seems like I do not see the same thing in edit mode as what gets posted. Makes it hard of fix typos! sorry for the confusion after (i) I tried to fix it.

 

[tart666]

thx, RossGr, this is what I was looking for!

edit: and to Peter: I did mention in the original question I was talking about "reals" and not naturals, no?

 

[LurchFrinky]

First of all, thank you RossGr for showing us what tart666 meant to say.

tart,

If you notice from Ross's post, he had to use definitions for ordered fields and sets, along with definitions for >,<, and -(x). None of these were present in your original post. His proof also contains the statement "if x>0", which is a pretty important assumption.

The arguments by Peter and others weren't that it was impossible, it's that you were leaving out some pretty important bits.

 

[tart666]

i did mention above that a definition of "order" ( ">" and "positive") was needed, no? (edit: and the definition of "-(x)" is included in the original post...)

 

[uart]

After reading the entire thread I vote Drag's proof to be the best.

Howabout you get a scale. On one side put 1 apple. On the other side put 0 apples. See which weighs more. Now put anything, a car, a brick, a leaf, anything that's not lighter then air.

 

[rjain]

the point is that we need to prove these things to make sure that our abstract number system is consistent with the concrete reality that we based the system on.

 

[drag]

The point is pointless.

The reason that 1 is greater then zero is because we say so. Mathmatics is just a language, not some sort of freakish law of nature.

What does a apple care if it made up of 1000000000000000000000000000000000 atoms or 3 atoms. It doesn't it just exists.

If you feel like it you can make 1 more then zero. Who cares? So what we then count 1 0 2 3 4 5 6 7 8 9 10 etc etc

then 1 + 0 = 0

Can you prove a circle is round, or a square has pointy edges?

A circle is round because we all agree that round things are sometimes circles. We could easily say that circles are now the ones with pointy edges and then that would be true.

Can you PROVE IT that a square has 4 sides?

The point is ABSTRACTION. A way to seperate our perceptions from reality. We use numbers such as 1 and zero to FIND fundamental truths, not that they are fundamental truths in themselves.

It's like a game. In poker certian hands represent certian scores. Some hands equal more then other hands. Because we say so. Otherwise cards are just peices of plastic covered paper. Meaningless without the abstract concepts we inflict on them.

Just like numbers are REALY just marks on pieces of paper. Our minds are what give them value.

Thus by following a strict langauge such as math we can quantify objects and put them thru filters (mathmatic equations). We could do the same thing with a natural language. That's what people been doing for ages, but mathmatics eliminate variables thus allowing scientific methods to be applied to information.