An accurate representation of Irrational and rational numbers

[Dialog]

Is there an accurate way to write the value of an Irrational number?

If there is no an accurate way to write the value of an Irrational number, then can we conclude that no irrational number has an exact place on the real line?

And if there is an exact place to an irrational number on the real line, then how can we prove it?

What base do we need to write the value of 2/3 in finitely many digits?

 

[Integral]

Is there an accurate way to write the value of an Irrational number?

The current system provides for any precision you need.

If there is no an accurate way to write the value of an Irrational number, then can we conclude that no irrational number has an exact place on the real line?

No, the point is fixed, there can be no doubt about that, it is the representation which requires an infinite number of digits, this is a result of the real line being dense.

And if there is an exact place to an irrational number on the real line, then how can we prove it?

This can be done by defining a set of nested intervals each of which contain the number of interest.

What base do we need to write the value of 2/3 in finitely many digits?

Base 3 works great

$$\frac 2 3 = .2_3$$

 

[Hurkyl]

Is there something wrong with the representations 1/3 or √2?

And if there is an exact place to an irrational number on the real line, then how can we prove it?

It's more or less a tautology; real numbers are the "places" on the real line.

 

[TALewis]

There are different ways to get numerical values for various irrational numbers to any arbitrary degree of precision. For example, pi can be found from the following infinite series:

$$1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\dots=\frac{\pi}{4}$$

 

[quartodeciman]

Dialog asks about something called "the value of an Irrational number". Is that supposed to be something different from just "an irrational number" itself (not furthur specified in particular)?

 

[Dialog]

No, the point is fixed, there can be no doubt about that,...

Why there is no doubt about that?

Is there something wrong with the representations 1/3 or √2?

1/3 and √2 are arithmetical operations, and I am talking about an accurate representation of the right hand of the equations: 1/3 = ? or √2 = ?

Is that supposed to be something different from just "an irrational number" itself?

What is an irrational number itself, and how you define it if you have no way to represent it in an accurate way, which is not some arithmetical operation?

Also I have another question:

How can we be sure that different represetations of some irrational or rational number do not define new information which has an interesting Mathematical data, which is not cardinality or ordinality information?

For example: The different information that we get when 1/3 is represented by base 3 or by base 10.

 

[Integral]

Why there is no doubt about that?

Because this is the way the Real line in constructed. Read my post above, I told you how irrational numbers are "trapped".
Why are you so infatuated with the representation of a point? Each point on the line can be represented in many different ways. The basic construction of the real line does not use digital representations. They have their uses, but are not required or needed to define real numbers.

 

[Dialog]

This can be done by defining a set of nested intervals each of which contain the number of interest.

Can you explain in a non-formal way how you define pi for example?

Why are you so infatuated with the representation of a point?

Please prove that pi is no more than a "point" on the real-line.

An example:

If circle's diameter = 1 then circle's perimeter = pi.

Can we prove that there is an exact way to put pi on the real-line?

Am I right if I say that we cannot represent pi because it is like a prime number that can be divided only by 1 or by itself (which means that there is no common sub-part between pi and any other number)?

And if so then what kind of a number we get by the ratio between two irrational numbers like e/pi?

 

[Zurtex]

1/3 and √2 are arithmetical operations, and I am talking about an accurate representation of the right hand of the equations: 1/3 = ? or √2 = ?

How do you mean? Do you mean a polynomial expression with just integers, e.g:

For x = 1/3:

$$3x -1 = 0$$

For x = √2

$$x^2 - 2 = 0$$

 

[Dialog]

Hi Zurtex,

I mean can you represent an Irrational number in an accurate way by not using arithmetical operations at all?

It's more or less a tautology; real numbers are the "places" on the real line.

Can you prove it?

 

[matt grime]

Dialog, how do you propose to represent any number in a way "without using any arithmetic operations"? after all what does 35 mean but 3*10+5*1?

This is mathematics, not aesthetics. Any representation of a real number is just a shorthand for stating *what the number does* which is its definition: sqrt(2) is the unique positive real number that squares to give 2, which is itself the result of adding 1 to 1, where 1 is the multiplicative identity of the unique totally ordered complete field of the real numbers. Got it? Real numbers are not decimals, that we can only write real numbers as finite length decimal expansions in certain cases is just a fact of the system of representation. It is not a deficiecny with the real numbers.

You need to explain what you consider a 'reasonable' representation of a number, and then state which numbers satisfy the criteria. If you wish you may then make *subjective* statements about them, but the ones you've made so far just demonstrate your lack of mathematical sophistication.

Do you even know what the real numbers are? I mean as a mathematical object?

 

[Dialog]

Hi Matt,

I am talking about Irrational numbers (cannot be expressed by the ratio of two integers).

If circle's diameter = 1 then circle's perimeter = pi.

Am I right if I say that we cannot represent pi (by a ratio of two integers) because it is like a prime number that can be divided only by 1 or by itself (which means that there is no common sub-part between pi and any other number)?

And if so then what kind of a number we get by the ratio between two irrational numbers like e/pi?

 

[matt grime]

I am talking about Irrational numbers (cannot be expressed by the ratio of two integers).

If circle's diameter = 1 then circle's perimeter = pi.

Am I right if I say that we cannot represent pi (by a ratio of two intgers) because it is like a prime number that can be divided only by 1 or by itself (which means that there is no common sub-part between pi and any other number)?

And if so then what kind of a number we get by the ratio between two irrational numbers like e/pi?

Why does it matter what kind of numbers you're talking about when you've not justified what you mean by having a 'proper' kind of representation? So, what's a proper representation? You've already discounted 1/3 as improper, and sqrt(2).

THe stuff about pi is just plain wrong in any interpretation, not least because of the undefined subjective terms, and the observation that surely pi and 2pi must have a common subpart, whatever that might mean.

 

[Zurtex]

Well as stated before you can represent irrational number by infinite sum etc..

But the whole point of irrational numbers is you can not represent them with integers and your 4 basic mathematical operators.

 

[Dialog]

Why does it matter what kind of numbers you're talking about ?

Can you represent pi in an accurate (detailed) way by using finitely many symbols?

 

[matt grime]

yes, pi. now define what you mean by accurate, detail and symbol...

 

[Dialog]

and the observation that surely pi and 2pi must have a common subpart, whatever that might mean.

My mistake, I mean:

Am I right if I say that we cannot represent pi (by a ratio of two integers) because it is like a prime number that can be divided only by 1 or by itself (which means that there is no common sub-part between pi and any rational number)?

 

[matt grime]

No, you are not because you have not stated what you mean by divisbility for numbes not in the ring of integers - there must be some ring in which you are asking for the divisibility to be examined, what is it? 2/5 is represented as the ratio of two integers, yet it is like a prime surely since it cannot be divided by any integer.

We cannot represent pi as a ratio of two integers because it is irrational, that is the exact reason. It is moreover transcendental (and thus irrational), but so what? The real numbers aren't actually a line in any physical sense, this need to think of them as points, as dots on some infinite line, which you can't draw anyway, with some artificial and unjustified and unexplained meaning of 'accuracy' is slightly pointless if you don't even know what the real numbers are, and how mathematics works.

 

[Dialog]

And another qustion: Pi (when diameter = 1) is the value of a circle's perimeter .

Can you prove that any value which is related to a circle also holds for a straight line?

 

[matt grime]

And another qustion: Pi (when diameter = 1) is the value of a circle's perimeter .

Can you prove that any value which is related to a circle also holds for a straight line?

Why, when asked to explain yourself more clearly, do you always introduce more unexplained subjective ideas? That question just does not make any sense.

 

[Dialog]

Why not?

If our diameter is 1 on the real line, then our perimeter is pi on the real line.

Therefore i am asking if you can prove that any value which is related to a straight line also holds for a circle?

 

[matt grime]

you are asking me if a 'value', that is a noun, can 'hold' which would require an adjective, a property, for me to make any sense from it.
In short you question makes no sense in the English language, thus making it impossible to begin to work out its mathematical content.
You're still making the mistake of thinking that the real numbers *are* actually a line.

 

[Dialog]

Ok, i'll ask it again.

I am talking about the mathematical meaning of a circle and a stright line.

We know that if the quantitative value of the diameter (in the mathematical meaning) is 1, then the quantitative value of the perimeter (in the mathematical meaning) is pi.

Know we have two "points" on the real-line and we ask:

Can we prove that any number which is related to a mathematical straight line also holds for a any number which is related mathmatical circle, and vise versa?

 

[matt grime]

you can make the explanation as detailed as you want but you're still not asking a valid question that makes sense in English. What does it mean for a 'number' to 'hold'? It's nonsensical, and it doesn't matter how much you repeat yourself, it won't become meaningful.

 

[Zurtex]

I'm very confused by the way you have worded things, it doesn't seem to make any sense. However if you mean, can the length of the circumference of a circle be any number then yes as long as it is real and positive.

 

[Dialog]

You know what Matt?

Please define pi in a mathematical professional way and I'll ask my questions according to your definition.

 

[matt grime]

it is nothing to do with what you think pi is, it is entirely to do with the question that omitting unnecessary words reads: does a number hold? that makes no sense at all unless you explain what you mean for a 'number' to 'hold'

 

[Dialog]

it is nothing to do with what you think pi is

I am here to learn from you, so please define pi in a proffesional way and let me understand by your definition where i miss the point about irrational numbers.

 

[matt grime]

Here's one then:

pi is the ratio of a circle's diameter to its circumference.

if you prefer it is one hlaf the integral of path length about the contour C:={(x,y) in R^2 | x^2+y^2=1}

or it is 2 arcsin{1}

or the integral over R of some exponential function such as exp{-x^2} give or take some trivial arithemetic.

Now, none of that has anything to do with your question about numbers holding for circles if they are related to straight lines, so what were you asking?

 

[Dialog]

pi is the ratio of a circle's diameter to its circumference.

According this can I understand that pi can be any "place" on the real line and all we care about is the ratio of a circle's diameter to its circumference?

 

[matt grime]

pi can be any 'place' on the real line? of course not, if we must think of it as some physical object, going left to right, positive to negative, then pi is always to the right of all of the following numbers:

3, 3.1, 3.14, 3.141,...

and is in fact the sup of that set, which would be one definition of pi as a real number (dedekind cut)

why do you think pi can be any place on the real line? (ie be any element of the real numbers?)

 

[Dialog]

Because if the diameter can be any number, then pi (as you wrote) is the ratio of a circle's diameter to its circumference.

Therefore pi can be found in any "place" on the real line, which means, we only care for the ratio and not for any particular value.

 

[Hurkyl]

The diameter can be any number. The circumference can be any number. The ratio of the two is a constant; it can only be one number.

 

[matt grime]

Because if the diameter can be any number, then pi (as you wrote) is the ratio of a circle's diameter to its circumference.

Therefore pi can be found in any "place" on the real line, which means, we only care for the ratio and not for any particular value.

usually in a 'therefore' the consequent is implied by the antecedent. I don't see anything that can remotely justify that assertion mathematically. pi is fixed, the ratio between a circle's circumference and diameter is independent of the circle, in the sense that it is the same for any two circles.

 

[Dialog]

The diameter can be any number. The circumference can be any number. The ratio of the two is a constant; it can only be one number.

So, is this constant is the "place" of pi on the real line?

the ratio between a circle's circumference and diameter is independent of the circle, in the sense that it is the same for any two circles.

This is exactly what i mean, the pair(circumference,diameter) can be in any "place" on the real line as long as they stick together by their constant ratio.

In more general view, from this point of view there is no such a thing like a one real-line made up of fixed numbers, but infinitely many real-lines on top of each other where each real-line has its own 1 and we get a fractal, which its depth is infinitely many real-lines.