Am I wrong? Is this correct thinking? Am I weird? Am I crazy?

[1MileCrash]

http://www.teacherschoice.com.au/maths_library/area and sa/area_2.htm

"NOTE: The value of p can never be known exactly, so surface areas of spheres cannot be calculated exactly."

p = pi here.

It always irks when people say "(insert irrational constant here) doesn't have an exact value" or even "we don't know pi's exact value."

I don't understand. I can't express it completely as a decimal, so what?

I know the exact value of pi through many other ways. Series, the square root of gamma function evaluated at 1/2, and at the risk of being yelled at for circular reasoning, the ratio of a perfect circle's circumference to diameter, are all pi. I can only approximate it in decimal form, but saying that means "we don't know it's exact value" or even worse that it "has no exact value" makes no more sense than saying I don't know exactly what my dog looks like because I can't draw an exact picture of him on an etchasketch.

A simpler way to show the exact value of pi is as follows:

[itex]\pi[/itex]

I mean, I just don't understand this. It's a number, it has a place on the number line, it is clearly defined.

e is exactly 1/1! + 1/2! + 1/3! ... so yes, I do know it's exact value. It is a clearly defined number!

 

[nucl34rgg]

You are absolutely correct. We know pi, e, etc exactly. We cannot represent them nicely as a terminating or repeating decimals in any rational base, but that doesn't change the fact that pi is an exact number.

 

[Office_Shredder]

Besides, what if [itex] r=1/\sqrt{\pi} [/itex]

 

[micromass]

1MileCrash, you think like a mathematician

Here's a little joke on the topic:
A physicist, an engineer and a mathematician are asked what pi is. The physicist says that it is [itex]3.14 \pm 1[/itex]. The engineer says that it is approximately 3.1415. The mathematician thinks long and hard. After several hours he proudly exclaims: it is equal to pi!

Sorry, your post reminded me of this.

 

[nitsuj]

We cannot represent them nicely as a terminating or repeating decimals in any rational base, but that doesn't change the fact that pi is an exact number.

So I can draw a perfect circle and measure the length of the line? Neato!

 

[DaveC426913]

But, 1MileCrash:

... that doesn't change the fact that pi is an exact number.

But it does change the fact that "we can't calculate the surface of a sphere" with a given radius -which is the reference you mentioned in your OP.

You'd be able to express it algebraically, but could not give it a unit.

The measurement of the surface of your sphere will either be 4*pi*r^2 or it will be some value whose error is proportional the square of our imprecision in knowing pi.

 

[Integral]

But, 1MileCrash:

But it does change the fact that "we can't calculate the surface of a sphere" with a given radius -which is the reference you mentioned in your OP.

You'd be able to express it algebraically, but could not give it a unit.

The measurement of the surface of your sphere will either be 4*pi*r^2 or it will be some value whose error is proportional the square of our imprecision in knowing pi.

You cannot measure circumference or radius to a greater precision then our knowledge of the digits of Pi. So you can indeed calculate the circumference or area. The precision of the result will be determined by your ability to measure, not the value of Pi.

 

[DaveC426913]

You cannot measure circumference or radius to a greater precision then our knowledge of the digits of Pi. So you can indeed calculate the circumference or area. The precision of the result will be determined by your ability to measure, not the value of Pi.

If I define a hypothetical sphere as precisely 1 inch in diameter, it's circumference and other measurements will only be as accurate as our measurement of pi.

 

[nitsuj]

But, 1MileCrash:
You'd be able to express it algebraically, but could not give it a unit.

That good to see. in my first post I deleted having noticed by looking at my geometry set's circular compass that units defined the "length", so maybe say dividing circumference by diameter does something wonky with the units.

 

[DaveC426913]

That good to see. in my first post I deleted having noticed by looking at my circular geometry set compass that units defined the "length", so maybe say dividing circumference by diameter does something wonky with the units.

I am not really sure what you are trying to say.

To divide circumference by diameter you must take two measurements. Both will be only as accurate as your measuring device.

But even if we use ideal, hypothetical objects, wherein we define the measurement as an exact unit, there is still an element of error in what value we use for pi.

 

[1MileCrash]

If I define a hypothetical sphere as precisely 1 inch in diameter, it's circumference and other measurements will only be as accurate as our measurement of pi.

You're cheating if you say "the sphere is exactly 1 inch diameter" and then in the very next breath limit pi by our measurement accuracy. I don't see how saying something is "exactly 1 inch" and "exactly pi" are any different. "Exactly one inch" and "exactly pi" require the same exact measurement accuracy... infinite.

Furthermore, if your hypothetical sphere is exactly one inch in diameter not by measurement but by definition, then of course the circumference is exactly pi by definition as well, not an approximation!

You'd be able to express it algebraically, but could not give it a unit.

Why couldn't I give a unit? 4pi inches. There is absolutely nothing wrong with that. 4pi is a number. Just like 5, 6, or 2.

Expressing numbers algebraically is just as valid as writing them out, we express numbers in the most convenient way. I would choose to express a large number algebraically, 6.9 x 10^20 for example, and I could have that many inches. 6.9x10^10 inches, 4pi inches, 8 inches, I don't understand what you're saying.

The measurement of the surface of your sphere will either be 4*pi*r^2 or it will be some value whose error is proportional the square of our imprecision in knowing pi.

If we define the radius to be 1 cm exactly, then the surface area of the sphere is exactly 4pi cm^2 by definition.

If we measure the radius to be 1 to k amount of significant figures, there is nothing stopping you from measuring just as many significant figures of pi. 1.000 meters means "we stopped measuring after millimeters" and 3.141 meters means "we stopped measuring after millimeters." Exactly 1 meters has just as many decimal places as exactly pi meters.

Therefore that calculation based on measurement works exactly the same as calculating the area of a rectangle or any other calculation based on measurement that didn't necessarily involve any irrational constant. Take two measurements, and obey significant figure rules.

If you allow yourself to say "exactly one by definition" then you can say "exactly pi by definition" without a second thought.


Does that make sense?

 

[nitsuj]

I am not really sure what you are trying to say.

To divide circumference by diameter you must take two measurements. Both will be only as accurate as your measuring device.

But even if we use ideal, hypothetical objects, wherein we define the measurement as an exact unit, there is still an element of error in what value we use for pi.

Ah ha, either do I really.

I guess I am thinking of it from the perspective of mutiply two lines of equal length at a right angle make a square. The two dimensional area of the square is calculated exactly. This can't be done with a circle. Said differently units in the x direction are equal to units in the y direction when calculating the area of a square.

 

[MrsLorinda]

Dave is completely correct. You must always consider the element of error with Pi.

 

[DaveC426913]

You're cheating if you say "the sphere is exactly 1 inch diameter" and then in the very next breath limit pi by our measurement accuracy. I don't see how saying something is "exactly 1 inch" and "exactly pi" are any different. "Exactly one inch" and "exactly pi" require the same exact measurement accuracy... infinite.

[strike]Units can be exact. One foot is precisely 12 inches to an infinite number of decimal places.

A square that is one inch on a side is precisely one square inch in area. Zero error.

What precisely, in square inches, is the area of a circle whose diameter is precisely one inch?[/strike]

Furthermore, if your hypothetical sphere is exactly one inch in diameter not by measurement but by definition, then of course the circumference is exactly pi by definition as well, not an approximation!

[strike]Yes. But it is unitless. You have only expressed it algebraically.[/strike]

Why couldn't I give a unit? 4pi inches. There is absolutely nothing wrong with that. 4pi is a number. Just like 5, 6, or 2.

Does that make sense?

Hm. Yes it does.

I was thinking of pi as an algebraic value. It's not.

"pi" is no different from "one third". They represent precise points on the number line, even if they can't easily be located with a straight pin on that line or be succinctly be represented in decimal form. No one would claim "one third" (or even "1/3") is imprecise or "can never be known exactly".

Well reasoned 1MileCrash, well reasoned. You are correct and point is conceded.

 

[1MileCrash]

I've learned a lot from your posts, happy to finally return the favor.

 

[romsofia]

Irrational constant means you can't represent it using a fraction of real numbers. An infinite series isn't a ratio of a/b (where a and b are real positive numbers, and b != 0), and the gamma function is just an integral which converges absolutely for x>=1*, so that means we can switch the summation and integration signs, which then leads us back to an infinite series, which once again ISN'T a ratio of a/b!

So yes, you're wrong because you're not sticking true to the definition of an irrational constant.

*Assuming we're talking about: [tex]{\Gamma (x) = \int^\infty_0 \frac {t^n}{te^t} dt}[/tex]

 

[1MileCrash]

I've never claimed that anything was a ratio of a/b. The true definition of an irrational number is that is *can't* be expressed as a/b where and and b are integers. When have I ever implied otherwise?

 

[pwsnafu]

Irrational constant means you can't represent it using a fraction of real numbers.

Nonsense. Every irrational number can be written as a ratio of two reals, namely itself and 1.

On top of that, the fact that pi doesn't have a periodic decimal representation has no bearing about whether there is a finite representation of pi. Of course it does have one: π. Or you can write everything in base-π. Our choice of decimals is arbitrary anyway.

Dave is completely correct. You must always consider the element of error with Pi.

MrsLorinda, any error in calculating pi is always dwarfed by the error in your apparatus, for the simple reason we can increase the precision of pi without bound. And if you are not discussing a physical problem, just a hypothetical, then the error of π is always zero. You can argue that that was what you meant in your post, but then you are contradicting Dave (who already retracted his argument).

Edit: And another thing. One thing that annoys me about the Teacher's Choice article is that they take the square root 39.81 and don't discuss error of that operation. But they needed to point out that pi can "never be known exactly". Huh.

 

[Deveno]

what most people mean by "cannot be determined exactly" (whether they know it or not) is simply that a number is irrational.

one can view the real numbers as a "perfect measurement system". we, on the other hand, are not-so-perfect. there is a recursion problem with measurement, which is: what do you take as a standard? more precision in measurement requires a very small unit of measurement, and we can only reliably go so small. so in this sense, rational numbers represent an optimal compromise: we can get "close enough" without needing to be "exact".

so even though we can produce, at will (in theory, our computers ARE limited in power), rational numbers to any desired (finite) degree of accuracy near pi, the "exact" value eludes us.

the teacher could be seen as wrong, or right, depending on your point of view. even though there is a 1-1 correspondence between positive reals and their magnitudes, the real numbers are just points, sitting there on the number line, whether we know about them or not, but their magnitudes are something we measure, which requires choosing a "unit", which limits us to rational approximations, which are always "precisely wrong" to any desired degree of inaccuracy.

 

[nucl34rgg]

So I can draw a perfect circle and measure the length of the line? Neato!

You can't draw a perfect circle, but that's a physical limitation. Pi is still a number whose exact value is known. It can be represented in many different ways. It's not a rational number, so we can't represent it as a finite or repeating decimal number in a rational base. Still, pi is exactly...pi or [insert anyone of the many series for pi]. I don't understand why we can only "calculate" rational numbers. Surely we know what sqrt(2) is. It is the number whose square is 2. What does it mean to "exactly calculate" something, anyway? The area of a circle is exactly calculated to be pi*r^2. Whether someone wants to round it off and express a number within some arbitrary distance to this number in decimal form is purely up to the person, but we had an exact calculated value before. The teacher's explanation on that site is technically wrong and will probably confuse some of his/her more careful students.

what most people mean by "cannot be determined exactly" (whether they know it or not) is simply that a number is irrational.

one can view the real numbers as a "perfect measurement system". we, on the other hand, are not-so-perfect. there is a recursion problem with measurement, which is: what do you take as a standard? more precision in measurement requires a very small unit of measurement, and we can only reliably go so small. so in this sense, rational numbers represent an optimal compromise: we can get "close enough" without needing to be "exact".

so even though we can produce, at will (in theory, our computers ARE limited in power), rational numbers to any desired (finite) degree of accuracy near pi, the "exact" value eludes us.

the teacher could be seen as wrong, or right, depending on your point of view. even though there is a 1-1 correspondence between positive reals and their magnitudes, the real numbers are just points, sitting there on the number line, whether we know about them or not, but their magnitudes are something we measure, which requires choosing a "unit", which limits us to rational approximations, which are always "precisely wrong" to any desired degree of inaccuracy.

Right, but even if the teacher meant by "cannot be calculated exactly" as "cannot be expressed as a rational number," that teacher is again wrong because, as office shredder said earlier, if r= 1/sqrt(pi), then the surface area is exactly 4, an "exactly calculated" number as defined by him/her earlier. Thus, his/her statement,"The value of pi can never be known exactly, so surface areas of spheres cannot be calculated exactly." is blatantly false. The algebraic relationship between numbers still holds.

 

[nitsuj]

You can't draw a perfect circle, but that's a physical limitation. Pi is still a number whose exact value is known. It can be represented in many different ways. It's not a rational number, so we can't represent it as a finite or repeating decimal number in a rational base. Still, pi is exactly...pi or [insert anyone of the many series for pi]. I don't understand why we can only "calculate" rational numbers. Surely we know what sqrt(2) is. It is the number whose square is 2. What does it mean to "exactly calculate" something, anyway? The area of a circle is exactly calculated to be pi*r^2. Whether someone wants to round it off and express a number within some arbitrary distance to this number in decimal form is purely up to the person, but we had an exact calculated value before. The teacher's explanation on that site is technically wrong and will probably confuse some of his/her more careful students.

Can I draw an acceptably perfect circle and work from there?

If I drew my perfect circle with a pencil, is it not "perfect" because my pencil line doesn't have a perfectly smooth edge to it? Is that the physical limitation to drawing a circle? If so, I can't see how it would be different for any shape including a square. Or is it because the curve cannot be perfectly "smooth"? Oh that kinda makes sense with distance measurments being "straight". Ah and that's why pi keeps going forever (more defined) because there are no straight lines in a perfect circle.

 

[Deveno]

Right, but even if the teacher meant by "cannot be calculated exactly" as "cannot be expressed as a rational number," that teacher is again wrong because, as office shredder said earlier, if r= 1/sqrt(pi), then the surface area is exactly 4, an "exactly calculated" number as defined by him/her earlier. Thus, his/her statement,"The value of pi can never be known exactly, so surface areas of spheres cannot be calculated exactly." is blatantly false. The algebraic relationship between numbers still holds.

that is an excellent point. perhaps a better way to express what the teacher should have said, is that pi and 1 are not commensurable. to draw an analogy with the circle, we might declare the radius OR the circumference to be our "unit" standard, in which case we cannot accurately ("exactly") measure one in terms of the other (it doesn't really matter which one we choose. i suppose it is a sign of a general lack of imagination that most people intuitively pick the linear, rather than the radial, measure as "more natural").

of course, if we devise a rule(r) with markings only at a radius length, and a circumference length, all sorts of "round" things lend themselves well to measurement. and such a thing is easy to imagine, if not actually possible to manufacture (intuitively, one uses something like a string to measure the 2 markings we have - this is only approximate, but higher precision devices could be devised). finding a perfect circle to measure, well...just buy those at the same hardware store that sells "massless springs".

 

[Zula110100100]

I would have to agree that based on the base comment that we know where pi is just as well as we know where 1 is. Consider if we did use base pi, the digit 1 would equal pi, but the number in decimal we know as 1 should be irrational

 

[1MileCrash]

that is an excellent point. perhaps a better way to express what the teacher should have said, is that pi and 1 are not commensurable. to draw an analogy with the circle, we might declare the radius OR the circumference to be our "unit" standard, in which case we cannot accurately ("exactly") measure one in terms of the other (it doesn't really matter which one we choose. i suppose it is a sign of a general lack of imagination that most people intuitively pick the linear, rather than the radial, measure as "more natural").

I disagree on that as well. Pi still behaves as any other number in your example, and here's why I think that:

If we define the radius to be one, I agree that we cannot accurately measure the circumference, if we are talking about a mundane form of measurement. I also agree that if we define the circumference to be 1, we cannot mundanely measure the radius of that circle accurately. By mundanely, I simply mean with an apparatus alone.

Do you agree that if I define the edge of a square to be 3, we cannot accurately, mundanely measure its area? Apparatus alone, could we get its exact area to an infinite number of decimal places? No.

It is only by a mathematical relationship that we know the area of that square is exactly 9.
And I also know by a mathematical relationship that the circumference of your circle is exactly 2pi when the radius is our unit standard.

 

[Zula110100100]

I was counting sheep last night and I change my vote. I can give you 1 sheep, you cannot give me pi sheep, eve if you cut a 4th sheep in half, there are a finite number of atoms in a sheep, so they cannot divide in (pi - 3)|(4-pi) You could then split the atoms but there is still a finite number of particles that make them up. You could argue this comes down to measuring, or you could argue this comes down to "Is matter infinitely divisibe?" Which I would think it is not.

 

[micromass]

I was counting sheep last night and I change my vote. I can give you 1 sheep, you cannot give me pi sheep, eve if you cut a 4th sheep in half, there are a finite number of atoms in a sheep, so they cannot divide in (pi - 3)|(4-pi) You could then split the atoms but there is still a finite number of particles that make them up. You could argue this comes down to measuring, or you could argue this comes down to "Is matter infinitely divisibe?" Which I would think it is not.

That way, I also can't give you 1/3 sheep...

 

[Zula110100100]

So I would say repeating numbers are equally as irrational as pi, but less "real" than 1, or other finite decimal numbers.

 

[micromass]

"Irrational" is a bad word. It has a definite meaning in mathematics.

But finite decimal numbers have the same problem. I can't give you 1/2 or 2/10 sheep as well...

 

[Zula110100100]

Ah I see, unless it is an even-number of atoms in the sheep..which is not guaranteed.

 

[nitsuj]

I disagree on that as well. Pi still behaves as any other number in your example, and here's why I think that:

If we define the radius to be one, I agree that we cannot accurately measure the circumference, if we are talking about a mundane form of measurement. I also agree that if we define the circumference to be 1, we cannot mundanely measure the radius of that circle accurately. By mundanely, I simply mean with an apparatus alone.

Do you agree that if I define the edge of a square to be 3, we cannot accurately, mundanely measure its area? Apparatus alone, could we get its exact area to an infinite number of decimal places? No.

It is only by a mathematical relationship that we know the area of that square is exactly 9.
And I also know by a mathematical relationship that the circumference of your circle is exactly 2pi when the radius is our unit standard.

I think I see it simulary to you. Mathamaticaly the length units "work well" with length units.

I think curves complicate this, perfect circles infinitely so. (i.e. circumference / diameter) Lengths don't divide well into curves. Circle gets the square!

 

[pergradus]

I certainly don't think Pi can be thought of as a physical unit.

Consider the way we measure the length of string of length 10 cm:

You can start by placing your ruler down, and starting at 0 cm, count by 1 cm until you get to the end of the string, and at at 10 cm - which is equivalent to adding 1 + 1 + 1... + 1 until you get 10.

Suppose it was now 10 and 1/3 cm:

You add 10 cm together, and then add an extra 1/3 cm. It doesn't matter if you can't write down what 1/3cm is in terms of mm or any other decimal, physically you can divide 1 cm into exactly three pieces - there's nothing stopping you from doing that.

Now try to get to 10 + pi cm - you start with adding 10 cm up, then you can add three more, then you have to add 1 mm, and then a little more, and a little more, and just a little bit more... and so on forever. You have to add an infinite number of smaller and smaller measurements to ever achieve that extra pi cm. You can never actually measure out an irrational physical quantity - and hence I don't think it's possible for physical quantities to obtain irrational values.

The fact that the billiard ball doesn't have an irrational surface area is because it's not a perfect sphere, and never will be. Just like you can't ever draw a perfect triangle with a hypotenuse of Sqrt(2).

 

[1MileCrash]

I certainly don't think Pi can be thought of as a physical unit.

Consider the way we measure the length of string of length 10 cm:

You can start by placing your ruler down, and starting at 0 cm, count by 1 cm until you get to the end of the string, and at at 10 cm - which is equivalent to adding 1 + 1 + 1... + 1 until you get 10.

Suppose it was now 10 and 1/3 cm:

You add 10 cm together, and then add an extra 1/3 cm. It doesn't matter if you can't write down what 1/3cm is in terms of mm or any other decimal, physically you can divide 1 cm into exactly three pieces - there's nothing stopping you from doing that.

Now try to get to 10 + pi cm - you start with adding 10 cm up, then you can add three more, then you have to add 1 mm, and then a little more, and a little more, and just a little bit more... and so on forever. You have to add an infinite number of smaller and smaller measurements to ever achieve that extra pi cm. You can never actually measure out an irrational physical quantity - and hence I don't think it's possible for physical quantities to obtain irrational values.

The fact that the billiard ball doesn't have an irrational surface area is because it's not a perfect sphere, and never will be. Just like you can't ever draw a perfect triangle with a hypotenuse of Sqrt(2).

Again, counting exactly 1 cm and counting exactly pi cm are equally daunting tasks. You can measure 1.00000000000 cm perhaps, but that's no easier than measuring 3.141592653589 cm. Both are approximations to 1 and pi respectively when we are talking about measurement. 1.0 cm is not equal to exactly 1 cm, it means we stopped measuring after millimeters. You can't just ignore that in the rational number's case.

 

[Number Nine]

The issue here that computations with pi (and irrational numbers) can only work with rational approximations. This is a technological limitation, it has nothing to do with "not knowing pi exactly"; we do know pi exactly, we just can't work with it exactly. It's just a number.

 

[Deveno]

assuming we know how to define it, any real number is "knowable" in the sense that we can compute it to any desired accuracy.

i mean what is a mathematical relationship?

when we say a = f(b), for some formula involving b, there is a tacit appeal to a and b being "the same kind of something". if our formula is:

A = [itex]\pi[/itex]r², than A is not a rational number. so we have to have some kind of notion that allows us to say what equality of two real numbers IS.

analytically, we say that A converges to [itex]\pi[/itex]. some people have trouble resolving this notion of convergence to our everyday notion of "is-ness". probably because it relies on the notion of a "completed infinity" ([itex]\pi[/itex] is not finitely expressible as a rational, it is an infinite sequence of such rationals).

my point being, saying a = b, when a and b are two real numbers, sweeps a lot of complicated machinery "under the rug", the truth of such a statement (and how complicated it actually is) is hidden behind a formalism which makes it appear as if it is just as simple as a statement like 2+2 = 4, or 1/3 + 2/3 = 1.

to go a bit further, how does one actually prove that the formula for an area of a circle actually holds? if you can show me a derivation that does not, in fact, rely on some limiting process (and thus a "completed infinity"), i will withdraw my statements.

 

[pwsnafu]

to go a bit further, how does one actually prove that the formula for an area of a circle actually holds?

Before one can do that you need to define the problem of finding "area of a circle" without analysis.