ratio of chord to its arc

[enhandle]

is there any chord of a circle that is commensurable with its arc?
or is there a chord that has an algebraic relation to its arc?
I think not but am not sure.

 

[stapel]

is there any chord of a circle that is commensurable with its arc?
or is there a chord that has an algebraic relation to its arc?

Please reply with the full and exact text of the exercise, the complete instructions, a scan of (or detailed description of) any graphical or other supplementary information, and a clear listing of your thoughts and efforts so far. Thank you!

 

[enhandle]

Please reply with the full and exact text of the exercise, the complete instructions, a scan of (or detailed description of) any graphical or other supplementary information, and a clear listing of your thoughts and efforts so far. Thank you!

I am not a math student and this is not an exercise, just a question of mine. and not for any practical purpose.

Let me elaborate a bit. the ratio of circumference to diameter is called pi and is a transcendental number. but can some some smaller chord be a rational fraction of its arc? if not, can some chord have at least an algebraic relation (roots and powers) to its arc?
I suspect the answer is no but would like to know for sure.

 

[pka]

is there any chord of a circle that is commensurable with its arc?
or is there a chord that has an algebraic relation to its arc?
I think not but am not sure.

Have a look at this webpage.
That answers most of your questions.

But for the very first: What is the shortest distance between two points?

 

[enhandle]

Have a look at this webpage.
That answers most of your questions.

But for the very first: What is the shortest distance between two points?

uh, no the web page does not answer my questions nor do I have any idea what you meant by asking about distance between two points.

I should have posted this in the geometry/trig forum.
anyway, surely most chords are not in simple ratio to their arcs. the wolfram page shows how to calculate dimensions of a sector in a general way.
it does not help with my query at all.

 

[enhandle]

mods, can we move this to geometry/trig forum?
thanks.

 

[stapel]

mods, can we move this to geometry/trig forum?
thanks.

Done.

 

[pka]

uh, no the web page does not answer my questions nor do I have any idea what you meant by asking about distance between two points.

You seemingly lack the vocabulary to under the answer to your own question.
That webpage does indeed answer any question one could ask about an arc.
The measure of an arc is the measure of the central angle subtending the arc.
Therefore, the chord determined by the arc must be less in length than the arc itself because the shortest distance between two points is the length of the line segment they determine (the chord).

 

[enhandle]

You seemingly lack the vocabulary to under the answer to your own question.
That webpage does indeed answer any question one could ask about an arc.
The measure of an arc is the measure of the central angle subtending the arc.
Therefore, the chord determined by the arc must be less in length than the arc itself because the shortest distance between two points is the length of the line segment they determine (the chord).

Uh you seem to lack the mathematical logic to even understand the question.
I didn't ask, "hey, which is longer, the arc or its chord".

I asked if the ratio is always irrational or even always transcendental. (turning the question inside out). I think so, but would like someone who really knows to tell me.

The questions require a yes or no answer. a proof or reasoning of the answer would also be helpful.

Responding to me that an arc is obviously longer than its chord is non-responsive unnecessary and ridiculous besides.

 

[pka]

is there any chord of a circle that is commensurable with its arc?
or is there a chord that has an algebraic relation to its arc?

That is exactly as you posted the question. Trained mathematicians are not accustomed to dealing with such sloppy use of language. See how we use commensurable. That concept occurs in many different ways.
As I said before the webpage does help you answer the above question.
$\displaystyle s=R\cdot\theta$ is the arc length.
$\displaystyle a=2\cdot R\cdot\sin\left(\frac{1}{2}\theta\right)$ is the cord length.

It up to you to use that information to answer the particular question you have.

Uh you seem to lack the mathematical logic to even understand the question.
I didn't ask, "hey, which is longer, the arc or its chord". Responding to me that an arc is obviously longer than its chord is non-responsive unnecessary and ridiculous besides.

I don't think much of your opinion. I did teach mathematical logic, axiomatic geometry, and topology at the graduate level for many years.

 

[enhandle]

An arc is longer than its chord.

I know that but thanks (snicker) for trying to be helpful.

please stop replying to this thread.
you are either a troll (my best guess)or someone who feels they must post without even bothering to understand the question.

for the benefit of anyone else
note the following analogous facts:
Most trig functions are irrational numbers (but not transcendental numbers). but some are rational such as sine 30 or tan 45.

now the ratio of a semi-circle (not a trig function) to its arc (the diameter) = pi/2, which is not only irrational, but transcendental.
I do not know if this is true of all arcs of a circle, perhaps some arcs are in simple ratio to the chord. perhaps some are irrational but not transcendental.
I doubt both of those possibilities but would like to know for sure.

 

[enhandle]

That is exactly as you posted the question. Trained mathematicians are not accustomed to dealing with such sloppy use of language. See how we use commensurable. That concept occurs in many different ways.
As I said before the webpage does help you answer the above question.
$\displaystyle s=R\cdot\theta$ is the arc length.
$\displaystyle a=2\cdot R\cdot\sin\left(\frac{1}{2}\theta\right)$ is the cord length.

It up to you to use that information to answer the particular question you have.


I don't think much of your opinion. I did teach mathematical logic, axiomatic geometry, and topology at the graduate level for many years.

What a pity.
--------------------------------------------------------------------------------------------------------------
Indeed that is exactly how I posted it. it is clear succinct and understandable. besides I explained it further several times, but you feel the overwhelming urge to post without even considering what I wrote.

1) I did not ask which is longer. And if you thought that I did even after my explaining, you simply did not care to pay attention.
2) I did not ask how to calculate the length of an arc given its chord or vise versa. And if you thought that I did even after my explaining, you simply did not care to pay attention

3) I did ask whether an arc and its chord meeting my condition can exist. Perhaps the answer should be obvious but I don't see it. That's is why I posted. You refuse even to see the question.

Saying that the answer is found on that page is like saying that starting with a few ideas one can reason out the rest of mathematics. perhaps. but then there would be no need for this website. or math teachers for that matter.

Now please go prove the Riemann hypothesis and leave me alone.

 

[enhandle]

If you can't recognize a joke, I pity you. Best of luck anyway.

"He with nary a smile
is but the skeleton of his inner child"

sorry, but it was only after I replied that I realized the your comment was not from troll pka being insistant.

that's when I realized you were kidding.

 

[enhandle]

is there any chord of a circle that is commensurable with its arc?
or is there a chord that has an algebraic relation to its arc?
I think not but am not sure.

ok I figured it out on my morning commute. it is almost trivial and perhaps I should have realized it.
if my reasoning is correct, there are infinitely many such chords.
one can find a chord and arc ratio for any rational number > 1 but < pi/2.
and for any other number in that range as well, for that matter.

to defend my use of commensurable
suppose I had asked:
does there exist a rectangle whose diagonal is commensurable with its sides?.
that question should be answerable by any high school geometry student,

but it is the case of the square where the term incommensurable shockingly
came from.

I am therefore justified in using the term asking about any 2 comparable lengths such as a chord and its arc.
I have no idea why he called it sloppy. it isn't.

anyway, thanks everybody it's time for me to move on.

 

[Bob Brown MSEE]

none

http://math.stackexchange.com/questions/299124/is-sinx-necessarily-irrational-where-x-is-rational

sin x is irrational at non-zero rational values of x. This result is Theorem 2.5 and Corollary 2.7 in Ivan Niven's Irrational Numbers.

From PKA's reference, you can use this fact to prove that...
The ratio of any chord length to the length of its sectioned arc is necessarily irrational.



Good question, prompted by observing that arc/diameter = pi/2. and wondering if it where generally true for all chords.

 

[enhandle]

http://math.stackexchange.com/questions/299124/is-sinx-necessarily-irrational-where-x-is-rational

sin x is irrational at non-zero rational values of x. This result is Theorem 2.5 and Corollary 2.7 in Ivan Niven's Irrational Numbers.

From PKA's reference, you can use this fact to prove that...
The ratio of any chord length to the length of its sectioned arc is necessarily irrational.




Good question, prompted by observing that arc/diameter = pi/2. and wondering if it where generally true for all chords.

but then what is wrong with the following reasoning?

as a central angle increases from 0 to 180 degrees (sorry, I think in degrees ), its chord and arc also increase. but the ratio between the arc and chord also smoothly increases from 1 to pi/2 and must pass through every value between 1 and pi /2. That includes rational irrational and transcendental numbers.

so where am I going wrong?

 

[enhandle]

http://math.stackexchange.com/questions/299124/is-sinx-necessarily-irrational-where-x-is-rational

sin x is irrational at non-zero rational values of x. This result is Theorem 2.5 and Corollary 2.7 in Ivan Niven's Irrational Numbers.

From PKA's reference, you can use this fact to prove that...
The ratio of any chord length to the length of its sectioned arc is necessarily irrational.



Good question, prompted by observing that arc/diameter = pi/2. and wondering if it where generally true for all chords.

and also
what do you mean all sin values are irrational? sin(30) = .5 !?

oh, I get it! when the theorem says all rational values of x, he means in radians. 30 degrees equals an irrational value of radians. about 0.5235987756


the theorem therefore does not contradict my reasoning

 

[pka]

and also
what do you mean all sin values are irrational? sin(30) = .5 !?

oh, I get it! when the theorem says all rational values of x, he means in radians. 30 degrees equals an irrational value of radians. about 0.5235987756
the theorem therefore does not contradict my reasoning

No one doing serious mathematics has used degrees in a hundred years.

 

[enhandle]

No one doing serious mathematics has used degrees in a hundred years.

another irrational response.

 

[enhandle]

Dunno....but I find all this kinda hilarious...
as if it will decrease the price of groceries...

Anyhoo, condensed formula for this poor innocent "ratio r":
r = pi * u / [360 * SIN(u / 2)]
where u = central angle. Amen!

very interesting.
you and your evil twin seem to be at odds, because the formula you quoted uses degrees.

anywho thanks
but the formula itself does not help much, anymore than knowing how to multiply will by itself reveal if
222233456789061 is prime.

as for groceries, who knows, sounds like a topic for another thread.