Quote:
Originally Posted by jyl
Interesting, irrational numbers.
Irrational number - Wikipedia, the free encyclopedia
On the OQ, I believe most circles have area that is an irrational number. But not all.
You may read that "all" circles have area that is an irrational number, e.g. here is a blogger at
Plato's Heaven: Can't square a circle
who states "all circles have a circumference and an area which is irrational."
But that seems evidently false to me. Suppose radius r = sqrt ( X / pi ) where X is any rational number. Then area is X = pi * r^2 and is, almost definitionally, a rational number.
Seems like an example of not believing everything you read on the Internet.
Does anyone see something wrong in my logic? |
you are venturing into an area of number theory that is fascinating
One usually thinks of infinity as being a static thing, but in actuality in the heirarchy of counting the infinity represented by the 3# of irrationals is greater than the infinity that represents the # of rationals
the heirarchy of infinities is represented by the Hebrew letter aleph
counting the # of things is called cardinality if you count all the rational #s the result is
. How do you count the # of rationals? easy, just expand on this and add the negatives
interestingly this # is identical to the # of whole #s {..., -3, -2, -1, 0, 1, 2, 3, ...0}
a similar technique can be used to find the cardinality of the irrational #s, First, to simplify things just concentrate on all the irrationals between 0 and 1. Now lets assume that the irrationals are denumberable; this would then mean that there would be some way to list all of them such that we would not leave any number out.
Part of one such example list appears on the diagram above. Since this is clearly an infinte set and since the decimal representation of every irrational must contain an infinte number of no repatitous digits, the diagram really extends indefinitely in both directions. Now take each digit along some diagonal of this list; in this example 0.13497.... This then gives us one of the irrational numbers. But if we now alter every single digit, say by adding 1 (wrapping 9's back to 0's), then we get another irrational number 0.24508.... But this new number can not possibly appear anywhere in our list since it must always have at least one digit with the wrong value no matter which row we may compare it against. We have just produced an irrational number which is not in our list, and therefore our original assumption that the irrational numbers are denumerable must be wrong.
At this point we have discovered some new level of infinity, and infinity which is somehow greater than the infinity of all the integers. For now lets call this new transfinite cardinal number by the symbol
. We also have the relation
<
.
As Cantor proved there is a series of alephs each larger than then previous. This leads one to question whether this series is itself complete or if there may be other levels of infinity which lie between the alephs. The conjecture that there are no intermediate infinities is most famously stated by Cantor's Continuum Hypothesis, expressed by the equation:
