I just have a quick question that has bugged me for some time now.
Why is Pi irrational?
If Pi is the ratio of a circle's circumference and its diameter - that is two rational numbers of finite length - why is the result an irrational value?
If you were to divide a rational number by another rational number doesn't the resulting number have to be rational?
Obviously I am over looking something or not understanding something. Some insight into this would be much appreciated.
Thank you.
What makes you think that either of the circumference or diameter must be rational? It is possible for one or the other to be rational but not both.
If you were to divide a rational number by another rational number doesn't the resulting number have to be rational?
Obviously I am over looking something or not understanding something. Some insight into this would be much appreciated.
Thank you.
What I don't understand is that both circumference and diameter are a measurable length, which means they must be able to be given a finite value.
So if you were to take two pieces of string one for the circumference and one for the diameter and measured each piece of string individually with a ruler (or whatever equipment you like) you must be able to provide two exact measurements. Dividing one exact finite measurement by another exact finite measurement gives you a finite measurement, does it not?
That's true; but circumference and diameter are definitely measurable. Everything OP wrote in post #4 is correct, but the fact that the measurements of both circumference and diameter are exact and finite does not mean that both of them can be rational at the same time. At least one must be irrational.
well, that is a difficult question to answer. let's examine a similar situation, where our lines are straight: a square of side length one.
well, the diagonal is surely some distinct length, and so we ought to be able to measure it. and we can, we find it has (using the pythagorean theorem, for example)
a length of √2. but √2 is not rational, for if it were: say √2 = a/b, where a and b are integers with no common factor:
then .
but since has double the number of factors of 2 that a does, and likewise for b, we see that has an even number of factors of 2, and has an ODD number of factors of 2. but we are supposing they are the same integer, and this can't happen, factorization into primes is unique.
huh.
you see, what "rational" means, is that we can find "a common denominator", a unit small enough so that both 1 and √2 wind up being integers in that unit of measurement. and this never happens. the conclusion is: not all measureable finite numbers are rational. that is to say: the set of all "comparable" (that is commensurate) numbers isn't "big enough" to describe our notion of "measurement", it's not "fluid enough".
these are deep ideas....and rather hard to wrap your mind around: there are "gaps" in the rational number system, even though we can always find two rational numbers as close together as we can imagine.
the problem with your "ruler" is: how close together can you make the markings....how do you guarantee these are perfectly "accurate"? they would have to be "infinitely close together", and instead of having just individual marks, you have a blur. to properly express the idea of "measurement" we need a new kind of number, one that can vary perfectly fluidly, or continually, no matter how small our "scale". and in point of fact, we are faced with the distressing reality that we can't actually make a "perfect ruler", only "approximately perfect" ones (we have to choose some, perhaps very tiny, length as our "unit". i believe the current standard is based on how far light can travel in a certain amount of time, which makes the accuracy of length measurement dependent on the accuracy of time measurement. our current standard of time is based on the hope that the amount of time it takes between 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium-133 atom, is sufficiently regular).
some people believe that all these "theoretically unmeasurable" numbers are more than we need, and that numbers that we can construct geometrically, or at least calculate as accurately as we need with a computer, ought to suffice. such a view, however, would make calculus, and many branches of higher mathematics used frequently in science, without any sound logical footing, and on the basis of the utility of these methods alone, most people are reluctant to give up the irrational numbers.
Thank you very much Deveno. It is clearer to me now.
So essentially the irrationality of Pi boils down to the flaws of humankind and the decimal numbering system?
Would you say that, theoretically, if you were to use a different numbering system - say one that potentially had an extra value per decimal position
E.g. Instead of the possible values of each position being 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 you had 0, 1, 2, 3, 4, 5, X, 6, 7, 8. and 9.
Could you theoretically provide a rational value for Pi?
No! A rational number is defined to be an equivalence class of ordered pair of natural numbers (with caveats to exclude zero in one position), or in a more hand waving sense the ratio of two integers. Pi is irrational because it cannot be so represented, in fact since we are talking about rational/irrational we could conduct this discussion about sqrt(2). Restricting attention to sqrt(2) rather than pi means that an elementary proof that it is not rational is available.
There are number systems where pi is represented by a finite symbol (in fact a single digit) but that does not make it rational. In fact in the system I am thinking of the integers do not have a terminating representation.
CB
no, the number "10" is not to blame. we could have binary numbers, or base 37 numbers, and pi would still be irrational.
this is because no matter how we "represent" integers, they still have the same algebraic properties. the only way we could declare pi to be rational, would be to measure everything in units of pi, but that would have the disadvantage of making numbers like 1,2 and 3 irrational, which seems like the wrong way to go about things.
a more clever approach would be to use rational functions in pi, like:
this would actually get us a field, but it's maybe more complicated than we want (and what about "other" irrational numbers like √2, or √6? it's not at all clear if they would be part of this new number system, and yet they clearly correspond to "lengths").
the irrationality of pi is "intrinsic", it doesn't have to do with our "number base". a proof of this, is rather involved, and is usually reserved for an advanced topic of analysis (stuff beyond calculus). if you're really interested, you can try this link:
Proof that pi is irrational
but i warn you, none of the proofs are "easy", and most of them require a knowledge of calculus.
to give you an idea of how hard it is to prove pi is irrational, pi had been in use for several centuries before the first proof that it was irrational was given by Johann Lambert in 1761.