Is the Ratio of Circumference to Radius of a Circle Always Irrational?

In summary: An infinite number."In summary, the string has a length and that length is an integer. The string also has a ratio that is irrational.
  • #1
Trying2Learn
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TL;DR Summary
rationality of pi
I take a string. It has a length, and that length is a real, rational number (let's call it an integer)

I wrap it into a circle.

Is it safe to say the radius is represented by an irrational number, too?

Or is the "magic" (I would prefer not to use that word, but I go with it) of that number: that BOTH the radius AND the circumference are rational, while their ratio is irrational.
 
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  • #2
If you have a string of final length and call this length 1, then you need irrational portions of such strings to make a circumference.

If you have a string that makes a circumference and you call its length 1, then the part of it that represents the diameter is irrational.

You cannot have both at the same time.
 
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  • #3
let's suppose that you have a rational radius, say ##r=\frac{k}{l}## and a rational circumference, say ##c=\frac{m}{n}##, where ##k, l, m, n## are positive integers. Then the definition of ##\pi = \frac{c}{r} = \frac{ml}{nk}## must also be rational. So there's a bit of a problem there. You can not make an irrational number by adding, subtracting, multiplying or dividing rational numbers, that's built into their definition, I think.
 
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  • #4
Trying2Learn said:
TL;DR Summary: rationality of pi

I take a string. It has a length, and that length is a real, rational number (let's call it an integer)
Certainly the string length is a real number. How do you conclude that it is also rational?
 
  • #5
If both the radius and circumference are rational then pi is also rational.
 
  • #6
Trying2Learn said:
TL;DR Summary: rationality of pi
## \pi ## is irrational.
Trying2Learn said:
BOTH the radius AND the circumference are rational, while their ratio is irrational.
The ratio of two rational numbers is rational: that is what rational means.
 
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  • #7
but without trying to be pedantic, a circle is an abstract idea - you cannot actually physically make one, only an approximation. Same with irrational numbers
 
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  • #8
BWV said:
but without trying to be pedantic, a circle is an abstract idea - you cannot actually physically make one, only an approximation. Same with irrational numbers
As a practicing pedant, I would point out that all numbers are abstractions. In addition to not being able to create a piece of string with length ##\pi##, it is also impossible to create a piece of string with length 1.

Given a piece of string, that piece does not even have a length except as an approximation. It is fuzzy at the ends.
 
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  • #9
jbriggs444 said:
It is also impossible to create a piece of string with length 1.
But if you have a piece of string (and if it is meaningful to say that it has a length) then you can define its length as 1.

Also, if you have one piece of string (and a pair of scissors) you can create two pieces of string: God created the integers!
 
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  • #10
I could also declare its length is ##\pi##.
 
  • #11
pbuk said:
But if you have a piece of string (and if it is meaningful to say that it has a length) then you can define its length as 1.
Then your "1" is very ill defined since strings (and other physical objects) are not an exact length, as @jbriggs444 pointed out.
 
  • #12
dupe

(EDIT: when did I lose the ability to delete posts?)
 
  • #13
DaveC426913 said:
dupe

(EDIT: when did I lose the ability to delete posts?)
I think the delete capability only lasts for a minute or two (or maybe 5?)

EDIT: Hm ... I said that 'cause I see that my post in this thread can't be deleted but I checked an old post and it CAN be deleted. Weird.
 
  • #14
Anyway, I think this is wrong:

"strings (and other physical objects) are not an exact length"

The fact that we can't represent the length of a string with a finite number doesn't mean it doesn't have a length. But as always, see footnote *
 
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  • #15
DaveC426913 said:
The fact that we can't represent the length of a string with a finite number doesn't mean it doesn't have one. But as always, see footnote *
How do you measure it? From the farthest electron on one end to the farthest electron on the other end? Do you think electrons stay put to let you do that?
 
  • #16
phinds said:
How do you measure it?
Nobody said you could. Exact measurement is not the litmus test of a thing's existence.(Sincere question to all: is this discussion still in the realm of valid math? Or are we into "the sound of one hand clapping" philosophy now?)
 
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  • #17
DaveC426913 said:
Nobody said you could. Exact measurement is not the litmus test of a thing's existence.
No, but it IS a measure of its length. If you can't measure the length, how to you specify it?
 
  • #18
DaveC426913 said:
Nobody said you could. Exact measurement is not the litmus test of a thing's existence.
If you cannot measure an exact length, even in principle, then the existence of an exact length is dubious at best.

One can even go so far as to call the [physical] existence of integers into question. "How many apples are in that tree"? That question will not always have a definite answer.
 
  • #19
phinds said:
No, but it IS a measure of its length. If you can't measure the length, how to you specify it?
That's not what you said. you said "strings are not an exact length". That's different from the measurement.

I think you are conflating the map with the territory.
 
  • #20
Dave we are clearly talking past each other here. Let's just agree to disagree.
 
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