Irrational numbers in real life.

[LogicX]

So I was thinking about numbers like pi. If you were to measure the area or circumference of a sphere in real life, you would get a never ending decimal. How can this exist in real life? How can an actual physical object have a circumference that is an irrational number?

 

[DaveC426913]

So I was thinking about numbers like pi. If you were to measure the area or circumference of a sphere in real life, you would get a never ending decimal. How can this exist in real life? How can an actual physical object have a circumference that is an irrational number?

Because the number is based on an artificial unit of measurement, perhaps inches. The circle is not obliged to fit into our (arbitrary) choice of units.

 

[Hurkyl]

So I was thinking about numbers like pi. If you were to measure the area or circumference of a sphere in real life, you would get a never ending decimal. How can this exist in real life? How can an actual physical object have a circumference that is an irrational number?

Forget pi. The decimal representation of one-third is never-ending. So is the decimal representation of zero.

 

[DaveC426913]

Forget pi. The decimal representation of one-third is never-ending. So is the decimal representation of zero.

Yeah. Simply choose another base and you're golden.

In ternary (base 3), one third is simply and precisely 0.1

 

[Antiphon]

Will some base allow the circumference and diameter to be simultaneously rational?

 

[I like Serena]

Afaik *any* non-discrete measurement we make yields an irrational number.
When we measure a length we take note of a number of decimals, but we know that the actual length would have infinitely many more decimals, which are unpredictable and therefore irrational.

 

[chiro]

So I was thinking about numbers like pi. If you were to measure the area or circumference of a sphere in real life, you would get a never ending decimal. How can this exist in real life? How can an actual physical object have a circumference that is an irrational number?

You could set up a triangle with sides of exact unit 1 and 1 and measure the hypotenuse. The tricky part of this debacle would be to find some measuring devices to show that it seems that there is no rational solution.

I'm sure at least one person has done it with either a triangle or a circle with sides or radius of unit lengths.

 

[micromass]

I don't believe that there are irrational numbers in real life, not even numbers like 1/3...
The point is that you can't make a perfect circle in real life, and you can't divide a cake exactly in three.

So, if irrational numbers don't enter in real life, why are they used in math then? Well, irrational numbers form a great approximation to real life. The area of something that looks like a circle can be very closely approximated by the area of a circle. It won't be correct, but it'll be everything we need...

 

[micromass]

Will some base allow the circumference and diameter to be simultaneously rational?

No, being rational is independent of the chosen base. So this won't happen.

 

[SteamKing]

If you have a square whose sides are length = 1, what is the length of the diagonal?

 

[micromass]

If you have a square whose sides are length = 1, what is the length of the diagonal?

Well, the real question is, does there exists a square with sides of length 1 in real life??

 

[I like Serena]

Well, the real question is, does there exists a square with sides of length 1 in real life??

Sure there is.
Take a bead necklass and form it into a square.
Suppose you have 10 beads on each side.
Now define your unit of length to be 10 beads.
Presto, a perfect square in real life!

 

[AlephZero]

Well, the real question is, does there exists a square with sides of length 1 in real life??

The answer is yes, provided you can invent a suffiently complicated metric for space-time which makes whatever you are looking at a square

 

[micromass]

Sure there is.
Take a bead necklass and form it into a square.
Suppose you have 10 beads on each side.
Now define your unit of length to be 10 beads.
Presto, a perfect square in real life!

Well, there are some obvious problems with that:
1) How do you know you bead necklace will really form a square? Sure, it may look like a square, but that doesn't mean it is one...
2) Is your unit of length well defined? That is, you have four sides full with 10 beads. But maybe some beads are bigger than other beads. So 10 beads on one side, may not have the same length as 10 beads on the other side...

 

[I like Serena]

Well, there are some obvious problems with that:
1) How do you know you bead necklace will really form a square? Sure, it may look like a square, but that doesn't mean it is one...

Aww, let me check that.
A square is defined as a figure with 4 sides with equal length, which it has.
But you are right I think, because the angles are supposed to be perpendicular, and I did not take care of that yet.
I'll have to look up what a square is exactly...
I'll get back to you on that :)

2) Is your unit of length well defined? That is, you have four sides full with 10 beads. But maybe some beads are bigger than other beads. So 10 beads on one side, may not have the same length as 10 beads on the other side...

You seem to be measuring the beads themselves, but I defined the unit in terms of beads.
Notice that I converted a length measurement to a discrete measurement.

 

[micromass]

You seem to be measuring the beads themselves, but I defined the unit in terms of beads.
Notice that I converted a length measurement to a discrete measurement.

Well, the problem I'm having is that some 10 beads are longer than other 10 beads. So you probably won't get a nice metric here...

But this is a philosophical argument really , that means that everybody can be correct...

 

[I like Serena]

Well, the problem I'm having is that some 10 beads are longer than other 10 beads. So you probably won't get a nice metric here...

But this is a philosophical argument really , that means that everybody can be correct...

I checked which subforum we're in and apparently it's "General Math".
I think a discussion about metrics is not out of place here.
I wouldn't want to post in the wrong forum. :)

A metric on a set X is a function (called the distance function or simply distance)
d : X × X → R
(where R is the set of real numbers).
For all x, y, z in X, this function is required to satisfy the following conditions:
1. d(x, y) ≥ 0 (non-negativity)
2. d(x, y) = 0 if and only if x = y (identity of indiscernibles. Note that condition 1 and 2 together produce positive definiteness)
3. d(x, y) = d(y, x) (symmetry)
4. d(x, z) ≤ d(x, y) + d(y, z) (subadditivity / triangle inequality).

I think these conditions are all satisfied.
And if some 10 beads are "longer" than some other 10 beads, we'll simply have to round it to an integer number of beads.

 

[micromass]

I checked which subforum we're in and apparently it's "General Math".
I think a discussion about metrics is not out of place here.
I wouldn't want to post in the wrong forum. :)

A metric on a set X is a function (called the distance function or simply distance)
d : X × X → R
(where R is the set of real numbers).
For all x, y, z in X, this function is required to satisfy the following conditions:
1. d(x, y) ≥ 0 (non-negativity)
2. d(x, y) = 0 if and only if x = y (identity of indiscernibles. Note that condition 1 and 2 together produce positive definiteness)
3. d(x, y) = d(y, x) (symmetry)
4. d(x, z) ≤ d(x, y) + d(y, z) (subadditivity / triangle inequality).

I think these conditions are all satisfied.
And if some 10 beads are "longer" than some other 10 beads, we'll simply have to round it to an integer number of beads.

Well, it's fun discussion...

Of course it will satisfy the properties of a metric. I should have realized that right away.
But the point is, if you want to talk about squares and other things, then you need geometry. So the question would be: does your definition of 'distance' satisfy the axioms of geometry. (I talk here about Hilbert's axioms, found in: www.gutenberg.org/files/17384/17384-pdf.pdf[/URL] )

Specifically, your notion of distance induces a notion of "congruence", which has the meaning "has the same distance". And I doubt very much that Hilbert's axioms of congruence are satisfied here...

 

[I like Serena]

But the point is, if you want to talk about squares and other things, then you need geometry.

I'm not so sure you need a geometry.

I think you only need enough to be able to properly define a square.
For instance, I just found in wikipedia under square, that the angles do not have to be 90 degrees (in non-euclidean geometries). It's just that all angles need to be the same.

So I think that all I need is a metric and a definition of an angle.

So the question would be: does your definition of 'distance' satisfy the axioms of geometry. (I talk here about Hilbert's axioms, found in: www.gutenberg.org/files/17384/17384-pdf.pdf[/URL] )[/QUOTE]

Well, my geometry axioms are a bit rusty.
Thanks for the link, I'll read it in more detail later. :)

[quote="micromass, post: 3278004"]Specifically, your notion of distance induces a notion of "congruence", which has the meaning "has the same distance". And I doubt very much that Hilbert's axioms of congruence are satisfied here...[/QUOTE]

Since right now, I'm not completely up to speed on geometry axioms.
Could you outline which congruence axiom is not satisfied and why?

 

[micromass]

Well, it says that, given a line segment AB and a line x, the there is a unique line segement CD on x congruent to AB.

But in this case it isn't fulfilled since "10 beads" can have thesame length. So there can be multiple line segements congruent to AB.

Of course, this can be solved by taking a "standard" 10 beads and call that a unit (instead of calling every 10 beads a unit). But then comes the trouble to defining a square: how to make sure the sides are exactly equal to the standard 10 beads...

 

[I like Serena]

Well, it says that, given a line segment AB and a line x, the there is a unique line segement CD on x congruent to AB.

But in this case it isn't fulfilled since "10 beads" can have thesame length. So there can be multiple line segements congruent to AB.

Of course, this can be solved by taking a "standard" 10 beads and call that a unit (instead of calling every 10 beads a unit). But then comes the trouble to defining a square: how to make sure the sides are exactly equal to the standard 10 beads...

Sorry, I don't get it.

I'm assuming you refer to:

IV, I. If A, B are two points on a straight line a, and if A' is a point upon the
same or another straight line a , then, upon a given side of A' on the straight line
a , we can always find one and only one point B' so that the segment AB (or BA)
is congruent to the segment A'B'. We indicate this relation by writing
AB ≡ A'B'.
Every segment is congruent to itself; that is, we always have
AB ≡ AB.

So if we take AB to be say "n" beads, and we pick C on line x, then we can count "n" beads to one side of C, and find D there. CD will be congruent to AB having the same number of beads...