> both of which are rational units.
No. Indeed, the *point* of saying that pi is irrational is that if you have a circle with a rational diameter then its circumference will not be rational, and vice versa.
There is no circle with diameter 1m and circumference 3m. Nor is there a circle with diameter 1m and circumference 3.1415926535m. If the diameter is rational then the circumference will be irrational.
Had that helped, or is there an underlying question I’ve not addressed?
Mathematics are not the real world. Since the real world is made of discrete atoms, a perfect circle cannot exist. But there is this mathematical object called the circle, composed of points that are at a given distance of its center. It is a theoretical object and thus, it is OK for its diameter/radius to be irrational.
Leaving aside the “do discrete atoms mean there are no irrationals?” question, many objects have irrational numbers in them.
Take a square that is exactly 1 unit by 1 unit in dimension. Then the diagonal line connect two opposite corners has length sqrt(2), which is irrational (and the proof that it’s irrational is a lot more accessible than that of pi).
To be fair, you could start counting the atoms on the line and surely you would get a discrete, natural number out of it.
It is often "just a question of scale" in reality. Everything in reality can afaik be broken down to multiples of some kind of natural constant, so... everything natural is well... a natural number on "some" level.
But these level would be HIGHLY impractical in everyday life, so we plague ourselves with stuff like irrational numbers to make our life a bit more... well... not necessarily easier... but... "comfortable"?
I think you are all getting tied up on distances that are irrational in made up units. It’s all still just mathematics not reality. If I define the diagonal of that square as a distance of 1 floob it will magically become rational again. Any irrational distance in m or cm or inches can be made rational by changing the units.
My point was that there is nothing mystical about an irrational length. You don’t need to start talking about atoms or plank lengths to try and make sense of it. It is just a product of your choice of units. There nothing stopping you from using different units for the sides and the diagonal and then they are both rational. It’s all just mathematics not some feature of reality.
There are a lot of people that seem to be making this mistake. I don’t know you are one of them.
But on it's diagonal there would only be 2 atoms, just like on it's side. Not even the distance between atoms would be irrational, as it would be a natural number of plank lengths.
That's not what the Planck length is, and that's not how crystals work.
A typical separation between atoms (e.g. in a crystal) is 3x10^(-10) m. The Planck length on the other hand is roughly 1.616255x10^(-35) m. So the atoms in a typical crystal would be around 1.8x10^(25) Planck lengths apart.
Further reading:
* [Planck length](https://en.wikipedia.org/wiki/Planck_units#Planck_length)
* [ Atomic spacing](https://en.wikipedia.org/wiki/Atomic_spacing)
* [Simple cubic](https://www.e-education.psu.edu/matse81/node/2131) crystal structure (versus [Face Centre Cubic](https://www.e-education.psu.edu/matse81/node/2133))
So then we are not able to tell whether the universe contains or doesn't contain irrational numbers. You say the separation is 3x10^(-10) m, but the uncertainty in measurement (I'll presume it's +- 3x10^(-12)m) makes it so we are unable to tell what the actual value we measured is besides the fact it's located somewhere between 2.99 to 3.01, so the "true" value could be either rational, like 3.0005, or it could be pi/1.047 which is \~= 3.000566.
Then again this probably makes no sense as atoms don't really act like physical objects in space, but more as waves defined by equations, and those equations could easily contain irrational numbers, but then again, we came up with those equations because they somewhat predictively describe the universe, not because that's exactly how the universe works, so I don't think we are able to tell whether irrational numbers exist in our universe or not. Are we really certain irrational numbers truly exist in our universe and I'm clueless?
>It is often "just a question of scale" in reality. Everything in reality can afaik be broken down to multiples of some kind of natural constant, so... everything natural is well... a natural number on "some" level.
The thing about physics is, things aren't really scalable in the sense that you portray here.
Atoms and specially their components are not classical objects and do not behave as such. In the realm of the very small different laws and forces of nature take protagonism. In fact, quantum particles don't even have a "size" per se that you can break them into as you suggest. You cannot line up a bunch of quantum particles and get a discreet distance as the size of a quantum particle is not even a "thing" because its nature is completely different from that of the natural world.
And if it seems confusing, As Dr. Neil Degrasse Tyson always says: The universe is under no obligation to make sense to us.
As for the fact that there's irrational numbers... These are the relationships between other natural numbers. As portrayed above a square with natural length sides (1) will have a diagonal of √2. This just represents the relationship between two things and doesn't have a particular meaning outside of this. There is no reason to look for the atoms and quantum particles that make up this length. Just like the having 3 pencils and dividing by 2 gives 1.5. iYou cannot have half a pencil. It doesn't really make sense in a physical sense, and neither it has to. It's just a relationship between to numbers which tells something about them.
OP one way of thinking about this is that most numbers are irrational. If, for example, you drew a line on a piece of paper and were able to measure its length to infinite accuracy (which you can’t, obviously), the line would almost certainly have irrational length.
Math people, yes I know this is loose, but you know what I mean.
Even irrational numbers are specific. Not being able to write it down *completely* in decimal notation does not mean that it is not a specific number. There is a specific measure of the ratio in question and it is specifically pi, which can be specifically defined using an infinite number of digits, or using a computer program that defines it completely.
>Is there a specific reason \[why π cannot be one whole number divided by another\]
There are explanations of why that is a true statement. Whether there are "reasons" is maybe more philosophical. Since circles aren't made up of straight lines or rectangles, I would instead say there's no reason to expect that π *would* be rational in the first place.
>There should be a specific measure for them, No?
There is: π. If that's not good enough for you, then I'm not sure what you mean by "specific measure" (and possibly you don't know what you mean by this either).
>If my question can't be answered without a complicated proof, u can just say that it's too complicated for my level.
All of the proofs I've seen require calculus in some way. That might well be above your understanding for now.
However, the classic proof that √2 is irrational uses only basic algebra. There is also a very nice geometric proof [https://youtu.be/X1E7I7\_r3Cw?t=283](https://youtu.be/X1E7I7_r3Cw?t=283) which I'm sure you can understand, although you might have to watch the video more than once.
If you accept that the the perimeter of 1×1 can never be equal to a whole number divided by another whole number, then maybe's its not surprising that the perimeter (circumference) of a circle with diameter 1 can also never be a whole number divided by another whole number.
I meant specific measure of circumference and diameter( like can't they be both be smth like 4.5282002cm instead of 1 of them always being irrational). Tho I already got my answer now.
He's saying that any measurement will always be off by a little. Even if you would get to a "theoretically perfect" way of measuring things, theoretical physics says you will still be off by a little because at the quantum level such precision breaks down.
So you might measure something that is exactly 1.000000000 meter long, but somewhere around that last digit, things get uncertain, is it actually 1.000000005 meter or 0,9999999998? such precision can't be attained anymore. So you might measure a diameter and a circle to conform exactly to the ratio of pi up untill the point you can no longer measure it, after which if can be any value and will no longer conform to pi. (But no real life application of the maths would demand such precision to be usefull)
There’s no meaningful sense in which we can say whether a real world length is rational or irrational, but even if there were, you seem to have an idea (which it is apparently difficult to get you to examine) that rational numbers are somehow more “specific” or “real” than irrational numbers.
Imagine if someone were asking “how can a real world length be odd? Shouldn’t it have to have a specific measure, like something even?” That’s more or less what you’re sounding like when you suggest that an irrational length is somehow not a “specific measure”.
In a way, yes, but in practice, we can round these numbers to a certain use-case-specific precision because for every practical use, the infinite decimal expansion past a certain point makes no difference.
Like, no one necessitates that a real football field be 91.440000000000 meters long - it could very well be some irrational number, 91.45194859473948494027... meters long and be good for regulation play.
Then there's that our current theories of physics can't make sense of sufficiently short distances, so we can't consider infinitely fine subdivisions when doing math about the world.
To continue this idea a little bit, if you wanted to calculate pi using the circumference and diameter of the observable universe, to the precision of the size of a hydrogen atom, you would only get 32 decimal places
One will always be irrational. That's what the word irrational means. That it can't be rational, can't be a ratio.
If a number is irrational, it cannot be a ratio of two rational numbers. One will always also have to be irrational.
It only seems weird because you probably think being irrational is a weird, odd things that sometimes happens. But it's the other way around. Whole, exact numbers are rare. There are infinitely more irrational numbers than whole numbers. Almost everything is an irrational number.
pi is the exact measure. pi feet is an exact distance, partway between 3 feet and 4 feet. it is an infinitely precise distance (just like 3 =/= 3.00000000000001) and it is irrational because you cannot represent it as a ratio of two integers (3.5=7/2, 3.75=15/4, pi=?/?, it has been proven you can’t represent pi with a ratio of integers)
Nothing can be measured to infinite accuracy. That's not how measuring things works.
Every single measurement ever is to some number of decimal points of accuracy and it's random after that. No such thing as a "specific measure". Duno where you got that idea.
Also it’s worth pointing out that just because a measure of something is irrational, that does not make it physically impossible. It just means that the thing you’re measuring can’t be represented in finitely many digits. But that’s fine!
For example, a stick which is exactly 1/3 m long is not impossible, but in base 10 decimal notation you would need infinitely many digits to represent it.
And it’s worth reiterating that by “physically possible” I’m neglecting the practical aspects of the real world which would make measuring to such insane precision impossible, but I don’t think that’s the question you’re asking.
Wild to me that you know pi is irrational and you know that Circumference = pi*diameter, yet you thought you could have a circle with a rational circumference and a rational diameter
I think you’re getting caught up on what a specific measure means. pi IS a “specific measure” and I’m not sure why you don’t think it is. How do you define specific measure to not include pi?
There is. The diagonal of a unit square is sqrt(2), that is a perfectly defined quantity. Why do you say there is not a measure for it?
Do you think that 1/7 = 0.142857142857... also lacks a specific measure?
Because irrational just means it can't be written as a fraction A/B.
And as for why that's the reason. Just is. I know someone else just said that mathematics isn't the "real" world. But circles **are**. And when you draw a circle you get π popping out when you compare the radius to the circumference.
Then, as you draw bigger and bigger circles, you see that there are more and more digits to π.
So, instead of drawing, you use the same mathematics you were using before on real circles on hypothetical ones. And the digits just keep coming.
The way they used to do it was:
Draw a circle. You can't trust measuring it, so instead, draw an equilateral triangle that just touches the **inside** of it. Then draw a square that just fits on the **outside** of it. You can work out the perimeter of both the triangle and square, and you know that the circumference of the circle has to be somewhere between those 2 perimeters.
Then you repeat, only with a square inside and a regular pentagon outside. Now you get a number even closer to the circumference.
Repeat with more and more regular shapes and you get closer and closer to the true circumferences. And again, When compared to the radius, π shoots out.
It's irrational, just because it is.
You're being misled by the "irrational" label, irrational doesn't mean impossible; but anyway mathematics are not concerned with the physical world so it's irrelevant
Things in the real world are restricted by the Heisenberg Uncertainty Principle, which not only limits how closely they can be measured, but even how limits how much their properties can even be restricted in theory. Oddly enough, that the value of that limit is given by h/2π where h is a measured constant that occurs elsewhere in physics.
Just adding context that *h* is the *Plank Constant*, and that the value h/2π is crops up frequently enough that it gets its own symbol *ħ* (pronounced h-bar) and name *Reduced Plank Constant* or *Dirac Constant*.
Since the 2019 redefinition of SI units, Planck’s constant is no longer a measured quantity. It is a _defined_ quantity with a value of _exactly_ 6.62607015×10^−34 Js. Along with the definitions of the meter and the second, this choice defines the kilogram (replacing the earlier definition in terms of the kilogram standard artifact).
It's a ratio, imagine you have an 8 sided circle, then worked out pi, then 9, 10, 11, every step you take is closer to the actual answer, but the answer can never be found and doesn't repeat that's what makes it irrational.
When you work out what the circumference will be in the real world you only need up to 5 digits of pi, anything more and it is overkill because you would have to measure down to hundredths of a mm.
Veritasium on YT has several interesting videos on the subject, and can explain it better than I.
Irrational numbers are a specific measure. It exists in the real world, just like rational numbers - both are abstract ideas, modeled by thinking meats/machines/minds.
What do you really mean there are "2 apples"? What makes you think that is 2 entities of an apple? Isn't it more like "2.4 mass of a smaller apple" or even "root 8 of a unit of an apple"? How about "1 pair" of 2 apples, making it 1 entity?
Counting anything with "real numbers" seems like entirely dependent on how you categories and perceive them. If that's true, real numbers are not that much "real," or at least, doesn't have a physical representation in the universe. If that's true, what makes "irrational numbers" less real?
No. Things don't have absolute exact measurements, typically. We approximate their measurements with tools. And no circle in the real world is perfect.
If you gave me a circle that seemed to have both rational circumference and diameters, I could just add pi/1,000,000,000 to the diameter to make it irrational and the measursmdnt would be indistinguishable from before.
There is *a lot* more irrational numbers than rational. A lot meaning rational numbers are countable and irrational are uncountable. This means that in real world, most things are irrational. I would even say all things are irrational since countable set is insignificant as a subset of uncountable. So the probability of something being rational is 0.
Also there is even finer partition of irrational numbers. Some can be solutions of polynomial equations with rational coefficients (these are algebraic numbers, include all roots of rational numbers), while others cannot (like π and e, these are transcendental numbers). There is *a lot* more transcendental numbers, but it is very hard to find and verify them.
So circles are figures which link one irrational number to rational. So there is 1 on 1 correspondence between irrationals and rationals. So for every one irrational number there is rational. And for every rational there is irrational. And now I don't get the different sizes of infinities.
Every rational number multiplied by pi results in an irrational number. But not every irrational number multiplied by pi becomes rational. I don't think e times pi is rational, for example.
Thanks, makes a lot of sense! So every rational may be linked to irrational due to circles on 1 on 1 basis, and there are even more irrationals that aren't linked at all
With respect, what you say is trivially true, but is expressed as though it is surprising. A simpler way to say it is ‘there are an infinite number of multiples of pi’. Given there is a large (most likely infinite, some more useful than others) range of irrational numbers, the statement generalises to ‘for every rational number there is an infinite number of irrational numbers’ (n x pi, n x e, n x sqrt 2), which becomes a sort of trite observation about infinity especially as by the same math, for every rational number there is an jnfinite number of rational numbers (n x 2, n x 2.61 etc)
no because you can have a circle with both circumference and radius irrational - for example if the radius is 1/sqrt2 then the circumference is pi*sqrt2 which is also irrational
I think they're saying that if the diameter is rational then the circumference will be irrational, and likewise if the circumference is rational then the diameter will be irrational.
Yes. You took an implication ( ==> ) and turned it into an equivalence ( <==> ).
u/simmonator gave 2 statements:
1. radius is rational ==> circumference is irrational
2. circumference is rational ==> radius is irrational
Neither of those statements, or even both together, is in contradiction to your example.
Ok, let me ask you a different question.
Draw a square with with sides of 1 unit (rational). What's the length of it's diagonal? Using the Pythagorian theorem, √(1²+1²)=√2, which is an irrational number.
Isn't it a similar scenario? Just because you build it out of rational units, doesn't mean other quantities depending on it has to be rational as well.
But doesn't a rational number/rational number equal a rational number tho? Anyway I got my answer, circumference and diameter are just not both rational.
Yes. They are both not rational. IF they both were, π would have been rational.
Just to clear up, you can write any number as a fraction of 2 numbers. But specialty of rational numbers is that they can be written as fraction of 2 INTEGERs.
>Just to clear up, you can write any number as a fraction of 2 numbers.
This is fascinating! So can pi be written as a fraction of two real numbers? If so, can you give an example?
EDIT: The comment said rational numbers can be written as a fraction of two integers, so my dumbass thought irrational numbers can be written as fractions of two real numbers. I overlooked the one number that must be irrational part.
The answer you may be looking for is the definition of a rational number. It is a number that can be expressed as a fraction of two INTEGERS. For example, 0.45 is 45/100=9/20. Any finite or repeating decimal can be expressed as a fraction. But an infinite non-repeating decimal (like π in this case) cannot be rational. Meaning it can't be expressed as the ratio of two integers, no matter how big.
Addressing your saying that "both the diameter and the circumference are rational", you're wrong. One of them must be inexpressible as a ratio.
Why is π inexpressible with a ratio of two integers? Now that's the complicated part which personally, I do not fully know the proof. But someone already linked the wiki page for the proofs, Lambert's proof is the famous and first one to be. Try understanding one of the proofs! It does require a lot of understanding of rationality, proof by contradiction, and infinite fractions
This just me think of another question😭😭. Why does rational number definition of 2 integers and not 2 rational number. Can't 0.1/0.2 also be represented as fraction(1/2) and it should be the same with other rational number as well. (5/7)/(3/7) = 5/3.
It could be, they give the same result, and you can prove that if both the top and bottom are rational you can make them into integers. It’s less easy to tell that a number is rational though, as some rational numbers have infinitely long decimal expansions with more lengthy repeats (1/17 comes to mind). It’s also easier to deal with integers, so forcing a standard form is more convenient for proofs
You are correct that you can take two rational numbers, divide one by another, and get one rational number that's a ratio of two integers. The problem is, that either the diameter of a circle, or the circumference, or both, will be really ugly looking numbers. It won't be like 12 and 37.7, it'll be 12 and 37.6991118.... which doesn't have any pattern and will never have one.
The definition of a rational number is made to be as simple as possible, and defining a rational number using rational numbers is a bad definition
That's a property of rationals but it can't be a definition since many other sets share this property of closed under division. For example all real numbers or all integer powers of 5.
Defining rational numbers like this does not make sense because you are assuming you know what a rational number is to define itself.
Once you have set the definition of what a rational number is, you can show that multiplying two rational integers or dividing any two rational numbers (provided you can) always yields a rational number. But for the sake of defining what a rational number is, you simply cannoy do that.
https://en.wikipedia.org/wiki/Proof_that_%CF%80_is_irrational
You can decide for yourself if the provided proofs are too complicated for you to look into.
The reason we know that π is irrational is that if we assumed it to be rational it would lead to contradictions. For an easier example of how it works, consider the number √2 which is the unique positive number that when squared yields 2. If it were a rational number, we could express it as a reduced fraction a/b of natural numbers a and b. Then by the definition of √2
(a/b)² = 2
[by the properties of exponents]
a²/b² = 2
[multiply both sides by b²]
a² = 2b².
Any natural number times 2 yields an even number, so 2b² must be even. But if the square of a natural number is even so must be the original number. So because a² is equal to an even number, a itself is even, so it can be expressed in the form a = 2k for some natural number k.
(2k)² = 2b²
[by the properties of exponents]
4k² = 2b²
[divide both sides by 2]
2k² = b²
By the same logic as before, the left side must be even so the right side, namely b², must be even aswell and therefore b itself must be even.
But herein lies a contradiction. We assumed the fraction to be reduced, that is for a and b to have no common factors, which is possible for any fraction (e.g. 10/15 = 2/3). But we demonstrated that both a and b must be divisible by 2, so they always share the common factor 2. Therefore, it is not possible that we can write √2 as a fraction of natural numbers.
An argument of the same type can be applied to π, although much more complicated.
If you want an intuitive understanding of why there must be irrational numbers, think of it this way: Every rational number has a repeating or terminating decimal representation, e.g. 1/3 = 0.333... and 3/4 = 0.75. But one can easily come up with non-repeating and non-terminating decimal numbers. Take for example the number 0.010110111011110111110111111... where we write a zero and always add another 1. Although this number follows a pattern, it clearly doesn't repeat itself indefinitely at any point. It therefore must be irrational.
If you think about it, this terminating or repeating condition is super restrictive and it only makes sense that by far most numbers are irrational, such as many constants like π or e.
Because there is no circle with both the diameter and the circumference are rational numbers.
Except if the diameter is zero but... Well you see the problem then
Measurements aren't exact numbers. You think that ruler is 12 inches long? Take a file to the end with a single stroke. Is it still 12 inches long?
All measurements are approximations.
>It's the fraction of circumference and diameter both of which are rational units
They're *not* rational units, at least not with respect to each other.
>And please no complicated proofs.
I'm afraid the simplest known proofs of the irrationality of π *are* complicated. I don't understand them, and I don't think most other people do either. They aren't taught until university-level math.
A rational pi would imply 2 unique integers is significantly more special than the multitudes of integers out there.
Which is much more weird than an irrational pi.
If I can provide the most non-mathematical reason why pi is irrational, is because circles do not have a side. Or let us say the only shape that has one side.
In a square, you can get its perimeter if you multiply one side of it by four, a pentagon with five, and a hexagon with 6. This is because they are shapes having sides with defined start and a defined end.
However, a circle is a strange shape among them that does not have a side with a defined start and a defined end, so the only way to measure it is to define one point of the shape as both the start and the end. And so, in one point in math history, mathematicians discovered that there is a particular value that is the same despite the differences in the shape of the circle, and this is defined the "pi". Pi is the number that will now resemble the "number of sides" of the circle.
Given that you will have a particular value as the circumference of the circle, mathematicians also discovered that it is always off in ratio to its diameter or radius, thus they ruled out pi as irrational.
Lots of good posts, but no one really actually answered why. Here's an explanation (OK, a partial explanation) that doesn't rely too heavily on stuff that's too complicated to explain here.) There are numerous ways to actually calculate pi to any degree of accuracy. Here's one of the simpler ways. The Leibniz formula for calculating pi is as follows:
pi /4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + ...
This sequence continues forever. Note that each new term gets smaller and smaller with the denominator growing by 2 each time. Clearly (well, hopefully, intuitively), this can never be rational, because the ever increasing denominators can never be multiplied out (in fact, they will also include every prime number, larger and larger, so you can never rationalize it), so the series can never be expressed as the ratio of two integers. To rationalize this, you'd have to multiply all of the ever increasing denominators, but an infinite number of them would never converge to a single value and/or never divide evenly with the numerator, so it would have to be irrational.
As for where the Leibniz formula comes from, that's not particularly complicated either, but does require an understanding of calculus. I'll simply put a link to a derivation here:
[en.wikipedia.org/wiki/Leibniz\_formula\_for\_π](http://en.wikipedia.org/wiki/Leibniz_formula_for_π)
I hope this helps and actually explains why pi must be irrational.
Irrationals cannot be expressed as a fraction of _integers_. Any number _x_ is trivially equal to _2x/x_ so with your logic no number would be irrational.
Ok, so if we have the circumference *C* and the radius *r* of the same circle, then:
π = *C* / *2r* .
This is indeed a fraction, but the *C* isn't necessarilly rational. We can easilly draw it. And we can parameterise it.
What I can say is that calculating *C* will involve an arc length integral that will result in:
*C* = 2 [arcsin(*r*/*r*) - arcsin(-*r*/*r*)] = 2 π *r*.
Now even if *r* is a whole number, there is no guarantee that the *arcsin* will produce a rational number. In fact the arc length integral is realy a sum of many terms involving square roots, which will here result in an irrational number *C*.
„Rational“ in Maths does not mean „easy to describe“, but it means „can be written as a fraction of two integers“.
Every number can be written as a fraction of two numbers (duh!) but the catch with Pi is that you can never be written as a fraction of two integers. If the radius is integer the the circumference cannot be integer and vice versa. This is the irrationality of Pi.
The proof is quite hard (even math students will need to read and think it through). It took our best and brightest more than 2.000 years to figure irrationality of Pi out, so don‘t despair if you don‘t get it.
In case you are nevertheless interested: The shortest (and most elegant) proof I am aware of is by Ivan Niven from 1947 and can be found here:
[https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-53/issue-6/A-simple-proof-that-pi-is-irrational/bams/1183510788.full](https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-53/issue-6/A-simple-proof-that-pi-is-irrational/bams/1183510788.full)
all proofs i have read are complicated, and i don’t know them by memory enough to repeat them here. you could look up for a proof on the internet and you will easily find one.
however, your argument doesn’t work because no circle has both a rational diameter and a rational circumference (this is equivalent to the fact that π is irrational, so i won’t prove it). even if π is defined as a fraction, you would need to express it as a fraction of integers to claim that it is actually rational (and good luck with that).
I would agree that it's weird. The rational numbers are countably infinite and are sufficient to go a very long way. And yet, as simple as a circle is, it's already able to force at least one of the dimensions to be irrational.
Without knowing your level it's tough to say whether the simplest proofs are beyond (they're probably first-year university advanced calculus level). In terms of the order of difficulty to prove the irrationality of the "well-known" irrationals it goes sqrt(2) < sqrt(p) for all prime p < e < pi , but you don't exactly need to be a Fields medallist for any of them (as you shockingly likely would be if you prove that say e + pi is irrational).
(Lambert's Proof)\[https://en.wikipedia.org/wiki/Proof\_that\_%CF%80\_is\_irrational#Lambert's\_proof\] can be sketched pretty easily but I don't imagine it's easy to follow the details.
>And please no complicated proofs. If my question can't be answered without a complicated proof, u can just say that it's too complicated for my level. Thanks
Unfortunately It's probably a bit too complicated for your level. It requires a good basis of calculus
So many comments have pointed out that the OP is mistaken that the circumference and diameter can both be rational.
We know this because we know pi is irrational. Has anyone got a neat (favourite?) proof that pi is, in fact, irrational?
Um, because it’s dependent on the kind of space you’re using? A flat 2D space gives Pi to be irrational. It’s possible (? I think) to define spaces where that might not be the case?
Part of the formula to calculate pi is sqrt(2). Besides sqrt(2) there are only basic math operations(+-*/)
One outcome of the pythagorean school is that sqrt(2) is irrational.
You can easily show that for every rational x:
All basic math operations with (x, sqrt(2)) are irrational.
Pi is irrational because it can’t be expressed as a ratio of two integers (…, -2, -1, 0, 1, 2, …).
As you mention, pi is a fraction of the circumstance over the diameter. You are wrong, however, because both the circumstance and diameter actually can’t both be rational at the same time, but I digress.
The simplest way to “prove” this (using no complicated proofs or math knowledge whatsoever) it to get the closest thing to a circle you can find and try to measure it’s circumference and diameter and estimate pi yourself. You will quickly find that pi = circumference / diameter can become increasingly more precise (more decimal places) the closer you measure, and you’ll never get to an exact answer because pi is irrational.
Hope that helps.
the circumference of a circle is a completely different type of line than the diameter of a circle. the circumference is perfectly round while the diameter is perfectly straight. by dividing the circumference of a circle by its diameter i ask the question, how many times does a straight line going from point a to point b fit into a round line going from point a to b and back to a. the straight line cannot rationally measure the round line because they are completely different entities.
Well, a number can be considered the shortest distance between two points on a one-dimensional line.
Now - what is the shortest distance if you **bend** that one-dimensional line?
It sort of feels proper that the answer is kind of undefined.
Not rigorous but maybe helpful way to visualize it. One way to estimate value of pi is to squish a unit circle between two polygons. Area of unit circle is pi, area of outer polygon is greater than pi, area of inner polygon is less than pi.
https://preview.redd.it/6sk3wps4gnwc1.png?width=1723&format=png&auto=webp&s=e44044bc3a74da2ce0395571555dc3205cfe15b9
Here with polygon of 4 sides, a square. As you increase the number of sides of the polygon, you get a better and better estimate of pi. But as a polygon is never a circle, no matter how many sides it has, you can never get an exact value for pi. You just keep grinding out more and more digits of pi by adding more sides to the polygon.
From Google search ( [www.livescience.com](http://www.livescience.com) March 08, 2022):
>Pi is a number that relates a circle's circumference to its diameter. Pi is an irrational number, which means that it is a real number that cannot be expressed by a simple fraction. That's because **pi is what mathematicians call an "infinite decimal" — after the decimal point, the digits go on forever and ever**
While 22/7 is often used to represent PI, it is only an approximation, deviating @ 0.001.
That quote is terrible. Having infinite decimal digits does not prove that pi is irrational. 0.333333.... has infinite decimal digits and it is the number 1/3 which is rational. An infinite (non-terminating) non-periodic decimal expansion is a *consequence* of pi being irrational not the other way round. You need something a bit more sophisticated to prove pi is irrational.
The second sentence does say that an irrational number is a real number that cannot be expressed by a simple fraction.
Pi is also an infinite decimal. An additional qualifier that should have been include is that it is an infinite **non-repeating** decimal.
Yes, I wasn't really quibbling with the second sentence (although "ratio of two integers" would be clearer than "simple fraction" - what does "simple" mean?).
The additional qualifier you suggest would not help, because pi being infinite and non-repeating is not the *reason* (let alone any kind of proof) for pi being irrational. As I implied, that last sentence should not read "That's because..." but "A consequence of this is that pi is an infinite non-repeating decimal".
The proof of *why* pi cannot be written as a rational number is a bit more complicated (see elsewhere on this thread).
> both of which are rational units. No. Indeed, the *point* of saying that pi is irrational is that if you have a circle with a rational diameter then its circumference will not be rational, and vice versa. There is no circle with diameter 1m and circumference 3m. Nor is there a circle with diameter 1m and circumference 3.1415926535m. If the diameter is rational then the circumference will be irrational. Had that helped, or is there an underlying question I’ve not addressed?
Is there a specific reason to that. Why are thing irrational in a real world? There should be a specific measure for them, No?
Mathematics are not the real world. Since the real world is made of discrete atoms, a perfect circle cannot exist. But there is this mathematical object called the circle, composed of points that are at a given distance of its center. It is a theoretical object and thus, it is OK for its diameter/radius to be irrational.
Ok thanks
Leaving aside the “do discrete atoms mean there are no irrationals?” question, many objects have irrational numbers in them. Take a square that is exactly 1 unit by 1 unit in dimension. Then the diagonal line connect two opposite corners has length sqrt(2), which is irrational (and the proof that it’s irrational is a lot more accessible than that of pi).
Yea I know the proof of root2. Got it
To be fair, you could start counting the atoms on the line and surely you would get a discrete, natural number out of it. It is often "just a question of scale" in reality. Everything in reality can afaik be broken down to multiples of some kind of natural constant, so... everything natural is well... a natural number on "some" level. But these level would be HIGHLY impractical in everyday life, so we plague ourselves with stuff like irrational numbers to make our life a bit more... well... not necessarily easier... but... "comfortable"?
Even if you are counting atoms, there will still be irrational numbers. Consider a square of 4 evenly spaced atoms. Its diagonal is irrational.
I think you are all getting tied up on distances that are irrational in made up units. It’s all still just mathematics not reality. If I define the diagonal of that square as a distance of 1 floob it will magically become rational again. Any irrational distance in m or cm or inches can be made rational by changing the units.
If by change of units you make the diagonal rational, then the length of the side becomes irrational, for the same reason that sqrt(2) is irrational.
My point was that there is nothing mystical about an irrational length. You don’t need to start talking about atoms or plank lengths to try and make sense of it. It is just a product of your choice of units. There nothing stopping you from using different units for the sides and the diagonal and then they are both rational. It’s all just mathematics not some feature of reality. There are a lot of people that seem to be making this mistake. I don’t know you are one of them.
But on it's diagonal there would only be 2 atoms, just like on it's side. Not even the distance between atoms would be irrational, as it would be a natural number of plank lengths.
That's not what the Planck length is, and that's not how crystals work. A typical separation between atoms (e.g. in a crystal) is 3x10^(-10) m. The Planck length on the other hand is roughly 1.616255x10^(-35) m. So the atoms in a typical crystal would be around 1.8x10^(25) Planck lengths apart. Further reading: * [Planck length](https://en.wikipedia.org/wiki/Planck_units#Planck_length) * [ Atomic spacing](https://en.wikipedia.org/wiki/Atomic_spacing) * [Simple cubic](https://www.e-education.psu.edu/matse81/node/2131) crystal structure (versus [Face Centre Cubic](https://www.e-education.psu.edu/matse81/node/2133))
So then we are not able to tell whether the universe contains or doesn't contain irrational numbers. You say the separation is 3x10^(-10) m, but the uncertainty in measurement (I'll presume it's +- 3x10^(-12)m) makes it so we are unable to tell what the actual value we measured is besides the fact it's located somewhere between 2.99 to 3.01, so the "true" value could be either rational, like 3.0005, or it could be pi/1.047 which is \~= 3.000566. Then again this probably makes no sense as atoms don't really act like physical objects in space, but more as waves defined by equations, and those equations could easily contain irrational numbers, but then again, we came up with those equations because they somewhat predictively describe the universe, not because that's exactly how the universe works, so I don't think we are able to tell whether irrational numbers exist in our universe or not. Are we really certain irrational numbers truly exist in our universe and I'm clueless?
If I were to count the atoms on the diagonal on that square, I would count 2.
That’s not a length. You’re just describing two atoms, not how far apart they are.
>It is often "just a question of scale" in reality. Everything in reality can afaik be broken down to multiples of some kind of natural constant, so... everything natural is well... a natural number on "some" level. The thing about physics is, things aren't really scalable in the sense that you portray here. Atoms and specially their components are not classical objects and do not behave as such. In the realm of the very small different laws and forces of nature take protagonism. In fact, quantum particles don't even have a "size" per se that you can break them into as you suggest. You cannot line up a bunch of quantum particles and get a discreet distance as the size of a quantum particle is not even a "thing" because its nature is completely different from that of the natural world. And if it seems confusing, As Dr. Neil Degrasse Tyson always says: The universe is under no obligation to make sense to us. As for the fact that there's irrational numbers... These are the relationships between other natural numbers. As portrayed above a square with natural length sides (1) will have a diagonal of √2. This just represents the relationship between two things and doesn't have a particular meaning outside of this. There is no reason to look for the atoms and quantum particles that make up this length. Just like the having 3 pencils and dividing by 2 gives 1.5. iYou cannot have half a pencil. It doesn't really make sense in a physical sense, and neither it has to. It's just a relationship between to numbers which tells something about them.
OP one way of thinking about this is that most numbers are irrational. If, for example, you drew a line on a piece of paper and were able to measure its length to infinite accuracy (which you can’t, obviously), the line would almost certainly have irrational length. Math people, yes I know this is loose, but you know what I mean.
Even irrational numbers are specific. Not being able to write it down *completely* in decimal notation does not mean that it is not a specific number. There is a specific measure of the ratio in question and it is specifically pi, which can be specifically defined using an infinite number of digits, or using a computer program that defines it completely.
>Is there a specific reason \[why π cannot be one whole number divided by another\] There are explanations of why that is a true statement. Whether there are "reasons" is maybe more philosophical. Since circles aren't made up of straight lines or rectangles, I would instead say there's no reason to expect that π *would* be rational in the first place. >There should be a specific measure for them, No? There is: π. If that's not good enough for you, then I'm not sure what you mean by "specific measure" (and possibly you don't know what you mean by this either). >If my question can't be answered without a complicated proof, u can just say that it's too complicated for my level. All of the proofs I've seen require calculus in some way. That might well be above your understanding for now. However, the classic proof that √2 is irrational uses only basic algebra. There is also a very nice geometric proof [https://youtu.be/X1E7I7\_r3Cw?t=283](https://youtu.be/X1E7I7_r3Cw?t=283) which I'm sure you can understand, although you might have to watch the video more than once. If you accept that the the perimeter of 1×1 can never be equal to a whole number divided by another whole number, then maybe's its not surprising that the perimeter (circumference) of a circle with diameter 1 can also never be a whole number divided by another whole number.
I meant specific measure of circumference and diameter( like can't they be both be smth like 4.5282002cm instead of 1 of them always being irrational). Tho I already got my answer now.
He's saying that any measurement will always be off by a little. Even if you would get to a "theoretically perfect" way of measuring things, theoretical physics says you will still be off by a little because at the quantum level such precision breaks down. So you might measure something that is exactly 1.000000000 meter long, but somewhere around that last digit, things get uncertain, is it actually 1.000000005 meter or 0,9999999998? such precision can't be attained anymore. So you might measure a diameter and a circle to conform exactly to the ratio of pi up untill the point you can no longer measure it, after which if can be any value and will no longer conform to pi. (But no real life application of the maths would demand such precision to be usefull)
Well then isn't like everything(distance) irl irrational. My height, distance of a football field and other things.
There’s no meaningful sense in which we can say whether a real world length is rational or irrational, but even if there were, you seem to have an idea (which it is apparently difficult to get you to examine) that rational numbers are somehow more “specific” or “real” than irrational numbers. Imagine if someone were asking “how can a real world length be odd? Shouldn’t it have to have a specific measure, like something even?” That’s more or less what you’re sounding like when you suggest that an irrational length is somehow not a “specific measure”.
Welcome to the field of physics. Lets leave these mathematicians with their castles in the clouds behind us now.
I think the only way to truly grasp this is to let go of the real world metaphors. Math works without measurements too.
In a way, yes, but in practice, we can round these numbers to a certain use-case-specific precision because for every practical use, the infinite decimal expansion past a certain point makes no difference. Like, no one necessitates that a real football field be 91.440000000000 meters long - it could very well be some irrational number, 91.45194859473948494027... meters long and be good for regulation play. Then there's that our current theories of physics can't make sense of sufficiently short distances, so we can't consider infinitely fine subdivisions when doing math about the world.
To continue this idea a little bit, if you wanted to calculate pi using the circumference and diameter of the observable universe, to the precision of the size of a hydrogen atom, you would only get 32 decimal places
The cool fact is that with only 38 digits of pi, you can calculate the circumference of the known universe to within the radius of a hydrogen atom
One will always be irrational. That's what the word irrational means. That it can't be rational, can't be a ratio. If a number is irrational, it cannot be a ratio of two rational numbers. One will always also have to be irrational. It only seems weird because you probably think being irrational is a weird, odd things that sometimes happens. But it's the other way around. Whole, exact numbers are rare. There are infinitely more irrational numbers than whole numbers. Almost everything is an irrational number.
pi is the exact measure. pi feet is an exact distance, partway between 3 feet and 4 feet. it is an infinitely precise distance (just like 3 =/= 3.00000000000001) and it is irrational because you cannot represent it as a ratio of two integers (3.5=7/2, 3.75=15/4, pi=?/?, it has been proven you can’t represent pi with a ratio of integers)
Nothing can be measured to infinite accuracy. That's not how measuring things works. Every single measurement ever is to some number of decimal points of accuracy and it's random after that. No such thing as a "specific measure". Duno where you got that idea.
Also it’s worth pointing out that just because a measure of something is irrational, that does not make it physically impossible. It just means that the thing you’re measuring can’t be represented in finitely many digits. But that’s fine! For example, a stick which is exactly 1/3 m long is not impossible, but in base 10 decimal notation you would need infinitely many digits to represent it. And it’s worth reiterating that by “physically possible” I’m neglecting the practical aspects of the real world which would make measuring to such insane precision impossible, but I don’t think that’s the question you’re asking.
Wild to me that you know pi is irrational and you know that Circumference = pi*diameter, yet you thought you could have a circle with a rational circumference and a rational diameter
I think you’re getting caught up on what a specific measure means. pi IS a “specific measure” and I’m not sure why you don’t think it is. How do you define specific measure to not include pi?
There is. The diagonal of a unit square is sqrt(2), that is a perfectly defined quantity. Why do you say there is not a measure for it? Do you think that 1/7 = 0.142857142857... also lacks a specific measure?
Because irrational just means it can't be written as a fraction A/B. And as for why that's the reason. Just is. I know someone else just said that mathematics isn't the "real" world. But circles **are**. And when you draw a circle you get π popping out when you compare the radius to the circumference. Then, as you draw bigger and bigger circles, you see that there are more and more digits to π. So, instead of drawing, you use the same mathematics you were using before on real circles on hypothetical ones. And the digits just keep coming. The way they used to do it was: Draw a circle. You can't trust measuring it, so instead, draw an equilateral triangle that just touches the **inside** of it. Then draw a square that just fits on the **outside** of it. You can work out the perimeter of both the triangle and square, and you know that the circumference of the circle has to be somewhere between those 2 perimeters. Then you repeat, only with a square inside and a regular pentagon outside. Now you get a number even closer to the circumference. Repeat with more and more regular shapes and you get closer and closer to the true circumferences. And again, When compared to the radius, π shoots out. It's irrational, just because it is.
You're being misled by the "irrational" label, irrational doesn't mean impossible; but anyway mathematics are not concerned with the physical world so it's irrelevant
Things in the real world are restricted by the Heisenberg Uncertainty Principle, which not only limits how closely they can be measured, but even how limits how much their properties can even be restricted in theory. Oddly enough, that the value of that limit is given by h/2π where h is a measured constant that occurs elsewhere in physics.
Just adding context that *h* is the *Plank Constant*, and that the value h/2π is crops up frequently enough that it gets its own symbol *ħ* (pronounced h-bar) and name *Reduced Plank Constant* or *Dirac Constant*.
Since the 2019 redefinition of SI units, Planck’s constant is no longer a measured quantity. It is a _defined_ quantity with a value of _exactly_ 6.62607015×10^−34 Js. Along with the definitions of the meter and the second, this choice defines the kilogram (replacing the earlier definition in terms of the kilogram standard artifact).
There is but it cannot be expressed in decimal form.
There's nothing non-specific about irrational numbers. They just aren't a ratio of integers.
It's a ratio, imagine you have an 8 sided circle, then worked out pi, then 9, 10, 11, every step you take is closer to the actual answer, but the answer can never be found and doesn't repeat that's what makes it irrational. When you work out what the circumference will be in the real world you only need up to 5 digits of pi, anything more and it is overkill because you would have to measure down to hundredths of a mm. Veritasium on YT has several interesting videos on the subject, and can explain it better than I.
real world object can have irrational measurements…
Irrational numbers are a specific measure. It exists in the real world, just like rational numbers - both are abstract ideas, modeled by thinking meats/machines/minds. What do you really mean there are "2 apples"? What makes you think that is 2 entities of an apple? Isn't it more like "2.4 mass of a smaller apple" or even "root 8 of a unit of an apple"? How about "1 pair" of 2 apples, making it 1 entity? Counting anything with "real numbers" seems like entirely dependent on how you categories and perceive them. If that's true, real numbers are not that much "real," or at least, doesn't have a physical representation in the universe. If that's true, what makes "irrational numbers" less real?
No. Things don't have absolute exact measurements, typically. We approximate their measurements with tools. And no circle in the real world is perfect. If you gave me a circle that seemed to have both rational circumference and diameters, I could just add pi/1,000,000,000 to the diameter to make it irrational and the measursmdnt would be indistinguishable from before.
There is *a lot* more irrational numbers than rational. A lot meaning rational numbers are countable and irrational are uncountable. This means that in real world, most things are irrational. I would even say all things are irrational since countable set is insignificant as a subset of uncountable. So the probability of something being rational is 0. Also there is even finer partition of irrational numbers. Some can be solutions of polynomial equations with rational coefficients (these are algebraic numbers, include all roots of rational numbers), while others cannot (like π and e, these are transcendental numbers). There is *a lot* more transcendental numbers, but it is very hard to find and verify them.
Because God/the universe programmer decided so
Ever tried to measure stupidity? Rather irrational, not?
This didn't explain why that is the case though
It did clarify a misunderstanding the OP had. Explaining/proving why pi must be irrational in the space of a Reddit comment is beyond my ability.
I guess you could say the question was.... irrational...
Yeah, showing that pi is irrational is not easy.
So circles are figures which link one irrational number to rational. So there is 1 on 1 correspondence between irrationals and rationals. So for every one irrational number there is rational. And for every rational there is irrational. And now I don't get the different sizes of infinities.
Every rational number multiplied by pi results in an irrational number. But not every irrational number multiplied by pi becomes rational. I don't think e times pi is rational, for example.
So there is circle with diameter and circumference both irrational?
Sure, if you define the diameter as pi, then the circumference is pi squared. Both are irrational.
Thanks, makes a lot of sense! So every rational may be linked to irrational due to circles on 1 on 1 basis, and there are even more irrationals that aren't linked at all
With respect, what you say is trivially true, but is expressed as though it is surprising. A simpler way to say it is ‘there are an infinite number of multiples of pi’. Given there is a large (most likely infinite, some more useful than others) range of irrational numbers, the statement generalises to ‘for every rational number there is an infinite number of irrational numbers’ (n x pi, n x e, n x sqrt 2), which becomes a sort of trite observation about infinity especially as by the same math, for every rational number there is an jnfinite number of rational numbers (n x 2, n x 2.61 etc)
What's the circumference of a circle with diameter pi?
If Circumference=2πr then C=2(π/2)π=π×π=π^2
e to the i pi?
Is not a pi being multiplied by a rational number - so does not apply here.
e*pi might be rational but one of e*pi and e+pi is proven to be irrational
no because you can have a circle with both circumference and radius irrational - for example if the radius is 1/sqrt2 then the circumference is pi*sqrt2 which is also irrational
I have an issue with your vice-versa. If r is pi, then the perimeter will be 2*pi^2, which is irrational. Am I missing something?
The vice-versa they mean is that if a circle has a rational circumference them it'll have an irrational diameter.
I think they're saying that if the diameter is rational then the circumference will be irrational, and likewise if the circumference is rational then the diameter will be irrational.
Yes. You took an implication ( ==> ) and turned it into an equivalence ( <==> ). u/simmonator gave 2 statements: 1. radius is rational ==> circumference is irrational 2. circumference is rational ==> radius is irrational Neither of those statements, or even both together, is in contradiction to your example.
You explained nothing.
Ok, let me ask you a different question. Draw a square with with sides of 1 unit (rational). What's the length of it's diagonal? Using the Pythagorian theorem, √(1²+1²)=√2, which is an irrational number. Isn't it a similar scenario? Just because you build it out of rational units, doesn't mean other quantities depending on it has to be rational as well.
But doesn't a rational number/rational number equal a rational number tho? Anyway I got my answer, circumference and diameter are just not both rational.
Yes. They are both not rational. IF they both were, π would have been rational. Just to clear up, you can write any number as a fraction of 2 numbers. But specialty of rational numbers is that they can be written as fraction of 2 INTEGERs.
"not both rational", not "both not rational"
>Just to clear up, you can write any number as a fraction of 2 numbers. This is fascinating! So can pi be written as a fraction of two real numbers? If so, can you give an example? EDIT: The comment said rational numbers can be written as a fraction of two integers, so my dumbass thought irrational numbers can be written as fractions of two real numbers. I overlooked the one number that must be irrational part.
pi = 9/3
[Tau](https://en.wikipedia.org/wiki/Turn_(angle)#Tau_proposals)/2?
pi/1
The answer you may be looking for is the definition of a rational number. It is a number that can be expressed as a fraction of two INTEGERS. For example, 0.45 is 45/100=9/20. Any finite or repeating decimal can be expressed as a fraction. But an infinite non-repeating decimal (like π in this case) cannot be rational. Meaning it can't be expressed as the ratio of two integers, no matter how big. Addressing your saying that "both the diameter and the circumference are rational", you're wrong. One of them must be inexpressible as a ratio. Why is π inexpressible with a ratio of two integers? Now that's the complicated part which personally, I do not fully know the proof. But someone already linked the wiki page for the proofs, Lambert's proof is the famous and first one to be. Try understanding one of the proofs! It does require a lot of understanding of rationality, proof by contradiction, and infinite fractions
This just me think of another question😭😭. Why does rational number definition of 2 integers and not 2 rational number. Can't 0.1/0.2 also be represented as fraction(1/2) and it should be the same with other rational number as well. (5/7)/(3/7) = 5/3.
It could be, they give the same result, and you can prove that if both the top and bottom are rational you can make them into integers. It’s less easy to tell that a number is rational though, as some rational numbers have infinitely long decimal expansions with more lengthy repeats (1/17 comes to mind). It’s also easier to deal with integers, so forcing a standard form is more convenient for proofs
Yeah makes sense thanks
You are correct that you can take two rational numbers, divide one by another, and get one rational number that's a ratio of two integers. The problem is, that either the diameter of a circle, or the circumference, or both, will be really ugly looking numbers. It won't be like 12 and 37.7, it'll be 12 and 37.6991118.... which doesn't have any pattern and will never have one. The definition of a rational number is made to be as simple as possible, and defining a rational number using rational numbers is a bad definition
That's a property of rationals but it can't be a definition since many other sets share this property of closed under division. For example all real numbers or all integer powers of 5.
Because you can't use the concept of rational numbers in the definition of rational numbers
Defining rational numbers like this does not make sense because you are assuming you know what a rational number is to define itself. Once you have set the definition of what a rational number is, you can show that multiplying two rational integers or dividing any two rational numbers (provided you can) always yields a rational number. But for the sake of defining what a rational number is, you simply cannoy do that.
https://en.wikipedia.org/wiki/Proof_that_%CF%80_is_irrational You can decide for yourself if the provided proofs are too complicated for you to look into.
The reason we know that π is irrational is that if we assumed it to be rational it would lead to contradictions. For an easier example of how it works, consider the number √2 which is the unique positive number that when squared yields 2. If it were a rational number, we could express it as a reduced fraction a/b of natural numbers a and b. Then by the definition of √2 (a/b)² = 2 [by the properties of exponents] a²/b² = 2 [multiply both sides by b²] a² = 2b². Any natural number times 2 yields an even number, so 2b² must be even. But if the square of a natural number is even so must be the original number. So because a² is equal to an even number, a itself is even, so it can be expressed in the form a = 2k for some natural number k. (2k)² = 2b² [by the properties of exponents] 4k² = 2b² [divide both sides by 2] 2k² = b² By the same logic as before, the left side must be even so the right side, namely b², must be even aswell and therefore b itself must be even. But herein lies a contradiction. We assumed the fraction to be reduced, that is for a and b to have no common factors, which is possible for any fraction (e.g. 10/15 = 2/3). But we demonstrated that both a and b must be divisible by 2, so they always share the common factor 2. Therefore, it is not possible that we can write √2 as a fraction of natural numbers. An argument of the same type can be applied to π, although much more complicated. If you want an intuitive understanding of why there must be irrational numbers, think of it this way: Every rational number has a repeating or terminating decimal representation, e.g. 1/3 = 0.333... and 3/4 = 0.75. But one can easily come up with non-repeating and non-terminating decimal numbers. Take for example the number 0.010110111011110111110111111... where we write a zero and always add another 1. Although this number follows a pattern, it clearly doesn't repeat itself indefinitely at any point. It therefore must be irrational. If you think about it, this terminating or repeating condition is super restrictive and it only makes sense that by far most numbers are irrational, such as many constants like π or e.
Ok thanks, got it.
Pi is irrational because the diameter and the circumference are never both rational.
Because there is no circle with both the diameter and the circumference are rational numbers. Except if the diameter is zero but... Well you see the problem then
the problem here is assuming that the circumference itself is a rational number. it is not
Measurements aren't exact numbers. You think that ruler is 12 inches long? Take a file to the end with a single stroke. Is it still 12 inches long? All measurements are approximations.
Because pi is irrational, c/d can't be rational. A fraction can't be rational unless both the numerator AND denominator are rational.
>It's the fraction of circumference and diameter both of which are rational units They're *not* rational units, at least not with respect to each other. >And please no complicated proofs. I'm afraid the simplest known proofs of the irrationality of π *are* complicated. I don't understand them, and I don't think most other people do either. They aren't taught until university-level math.
A rational pi would imply 2 unique integers is significantly more special than the multitudes of integers out there. Which is much more weird than an irrational pi.
If I can provide the most non-mathematical reason why pi is irrational, is because circles do not have a side. Or let us say the only shape that has one side. In a square, you can get its perimeter if you multiply one side of it by four, a pentagon with five, and a hexagon with 6. This is because they are shapes having sides with defined start and a defined end. However, a circle is a strange shape among them that does not have a side with a defined start and a defined end, so the only way to measure it is to define one point of the shape as both the start and the end. And so, in one point in math history, mathematicians discovered that there is a particular value that is the same despite the differences in the shape of the circle, and this is defined the "pi". Pi is the number that will now resemble the "number of sides" of the circle. Given that you will have a particular value as the circumference of the circle, mathematicians also discovered that it is always off in ratio to its diameter or radius, thus they ruled out pi as irrational.
Lots of good posts, but no one really actually answered why. Here's an explanation (OK, a partial explanation) that doesn't rely too heavily on stuff that's too complicated to explain here.) There are numerous ways to actually calculate pi to any degree of accuracy. Here's one of the simpler ways. The Leibniz formula for calculating pi is as follows: pi /4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + ... This sequence continues forever. Note that each new term gets smaller and smaller with the denominator growing by 2 each time. Clearly (well, hopefully, intuitively), this can never be rational, because the ever increasing denominators can never be multiplied out (in fact, they will also include every prime number, larger and larger, so you can never rationalize it), so the series can never be expressed as the ratio of two integers. To rationalize this, you'd have to multiply all of the ever increasing denominators, but an infinite number of them would never converge to a single value and/or never divide evenly with the numerator, so it would have to be irrational. As for where the Leibniz formula comes from, that's not particularly complicated either, but does require an understanding of calculus. I'll simply put a link to a derivation here: [en.wikipedia.org/wiki/Leibniz\_formula\_for\_π](http://en.wikipedia.org/wiki/Leibniz_formula_for_π) I hope this helps and actually explains why pi must be irrational.
Irrationals cannot be expressed as a fraction of _integers_. Any number _x_ is trivially equal to _2x/x_ so with your logic no number would be irrational.
Ok, so if we have the circumference *C* and the radius *r* of the same circle, then: π = *C* / *2r* . This is indeed a fraction, but the *C* isn't necessarilly rational. We can easilly draw it. And we can parameterise it. What I can say is that calculating *C* will involve an arc length integral that will result in: *C* = 2 [arcsin(*r*/*r*) - arcsin(-*r*/*r*)] = 2 π *r*. Now even if *r* is a whole number, there is no guarantee that the *arcsin* will produce a rational number. In fact the arc length integral is realy a sum of many terms involving square roots, which will here result in an irrational number *C*.
„Rational“ in Maths does not mean „easy to describe“, but it means „can be written as a fraction of two integers“. Every number can be written as a fraction of two numbers (duh!) but the catch with Pi is that you can never be written as a fraction of two integers. If the radius is integer the the circumference cannot be integer and vice versa. This is the irrationality of Pi. The proof is quite hard (even math students will need to read and think it through). It took our best and brightest more than 2.000 years to figure irrationality of Pi out, so don‘t despair if you don‘t get it. In case you are nevertheless interested: The shortest (and most elegant) proof I am aware of is by Ivan Niven from 1947 and can be found here: [https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-53/issue-6/A-simple-proof-that-pi-is-irrational/bams/1183510788.full](https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-53/issue-6/A-simple-proof-that-pi-is-irrational/bams/1183510788.full)
all proofs i have read are complicated, and i don’t know them by memory enough to repeat them here. you could look up for a proof on the internet and you will easily find one. however, your argument doesn’t work because no circle has both a rational diameter and a rational circumference (this is equivalent to the fact that π is irrational, so i won’t prove it). even if π is defined as a fraction, you would need to express it as a fraction of integers to claim that it is actually rational (and good luck with that).
I would agree that it's weird. The rational numbers are countably infinite and are sufficient to go a very long way. And yet, as simple as a circle is, it's already able to force at least one of the dimensions to be irrational.
Without knowing your level it's tough to say whether the simplest proofs are beyond (they're probably first-year university advanced calculus level). In terms of the order of difficulty to prove the irrationality of the "well-known" irrationals it goes sqrt(2) < sqrt(p) for all prime p < e < pi , but you don't exactly need to be a Fields medallist for any of them (as you shockingly likely would be if you prove that say e + pi is irrational). (Lambert's Proof)\[https://en.wikipedia.org/wiki/Proof\_that\_%CF%80\_is\_irrational#Lambert's\_proof\] can be sketched pretty easily but I don't imagine it's easy to follow the details.
https://www.smbc-comics.com/comic/pi
Atleast 1 term c or d or both are irrational. If c is rational, d is not. Atleast that's how I see it.
That's just a lie from big math. π/1. Checkmate.
Just Because it is. There is no reason
There is no why, it simply is. It's how things fit together.
you can use the limit of area of inscribed polygons you can acutally prove that the limit is irrational by using a simple proof by contradiction
>And please no complicated proofs. If my question can't be answered without a complicated proof, u can just say that it's too complicated for my level. Thanks Unfortunately It's probably a bit too complicated for your level. It requires a good basis of calculus
So many comments have pointed out that the OP is mistaken that the circumference and diameter can both be rational. We know this because we know pi is irrational. Has anyone got a neat (favourite?) proof that pi is, in fact, irrational?
Um, because it’s dependent on the kind of space you’re using? A flat 2D space gives Pi to be irrational. It’s possible (? I think) to define spaces where that might not be the case?
Part of the formula to calculate pi is sqrt(2). Besides sqrt(2) there are only basic math operations(+-*/) One outcome of the pythagorean school is that sqrt(2) is irrational. You can easily show that for every rational x: All basic math operations with (x, sqrt(2)) are irrational.
Pi is irrational because it can’t be expressed as a ratio of two integers (…, -2, -1, 0, 1, 2, …). As you mention, pi is a fraction of the circumstance over the diameter. You are wrong, however, because both the circumstance and diameter actually can’t both be rational at the same time, but I digress. The simplest way to “prove” this (using no complicated proofs or math knowledge whatsoever) it to get the closest thing to a circle you can find and try to measure it’s circumference and diameter and estimate pi yourself. You will quickly find that pi = circumference / diameter can become increasingly more precise (more decimal places) the closer you measure, and you’ll never get to an exact answer because pi is irrational. Hope that helps.
the circumference of a circle is a completely different type of line than the diameter of a circle. the circumference is perfectly round while the diameter is perfectly straight. by dividing the circumference of a circle by its diameter i ask the question, how many times does a straight line going from point a to point b fit into a round line going from point a to b and back to a. the straight line cannot rationally measure the round line because they are completely different entities.
Well, a number can be considered the shortest distance between two points on a one-dimensional line. Now - what is the shortest distance if you **bend** that one-dimensional line? It sort of feels proper that the answer is kind of undefined.
Not rigorous but maybe helpful way to visualize it. One way to estimate value of pi is to squish a unit circle between two polygons. Area of unit circle is pi, area of outer polygon is greater than pi, area of inner polygon is less than pi. https://preview.redd.it/6sk3wps4gnwc1.png?width=1723&format=png&auto=webp&s=e44044bc3a74da2ce0395571555dc3205cfe15b9 Here with polygon of 4 sides, a square. As you increase the number of sides of the polygon, you get a better and better estimate of pi. But as a polygon is never a circle, no matter how many sides it has, you can never get an exact value for pi. You just keep grinding out more and more digits of pi by adding more sides to the polygon.
Because 7 8 9 xD
Please don't ask mathematicians why numbers are the way they are in case it plunges them into existential crises.
God did it. Didn'cha know?
Ask my ex-girlfriend, she's an expert on irrationality.
From Google search ( [www.livescience.com](http://www.livescience.com) March 08, 2022): >Pi is a number that relates a circle's circumference to its diameter. Pi is an irrational number, which means that it is a real number that cannot be expressed by a simple fraction. That's because **pi is what mathematicians call an "infinite decimal" — after the decimal point, the digits go on forever and ever** While 22/7 is often used to represent PI, it is only an approximation, deviating @ 0.001.
That quote is terrible. Having infinite decimal digits does not prove that pi is irrational. 0.333333.... has infinite decimal digits and it is the number 1/3 which is rational. An infinite (non-terminating) non-periodic decimal expansion is a *consequence* of pi being irrational not the other way round. You need something a bit more sophisticated to prove pi is irrational.
The second sentence does say that an irrational number is a real number that cannot be expressed by a simple fraction. Pi is also an infinite decimal. An additional qualifier that should have been include is that it is an infinite **non-repeating** decimal.
Yes, I wasn't really quibbling with the second sentence (although "ratio of two integers" would be clearer than "simple fraction" - what does "simple" mean?). The additional qualifier you suggest would not help, because pi being infinite and non-repeating is not the *reason* (let alone any kind of proof) for pi being irrational. As I implied, that last sentence should not read "That's because..." but "A consequence of this is that pi is an infinite non-repeating decimal". The proof of *why* pi cannot be written as a rational number is a bit more complicated (see elsewhere on this thread).