0
$\begingroup$

This is a request for help, not an attempt to challenge anything.

Since $\pi$ is irrational, this tells me that it's impossible to express the distance around a circle in terms of the distance accross.

That boggles my mind, but maybe it should not.

I think crazy thoughts like: "this means that a the path of a circle around a unit lenth line segment has a non existant length".

Is there a way to accept that the number is irrational and not break from reason?

  • 7
    Knowing what [rational](http://en.wikipedia.org/wiki/Rational_number) and [irrational](http://en.wikipedia.org/wiki/Irrational_number) actually mean would be a good start. The irrationality of $\pi$ certainly does not mean that it's impossible to express the circumference of a circle in terms of the diameter.2012-02-22
  • 0
    apologies.. maybe i have to delete this2012-02-22
  • 0
    Why it tells you that it is impossible to express it? You can express it using $\pi$. You can't say that is is a nonexistant length it is just a length that is not a fraction. Like the length of the diagonal of the unit square. Nothing special.2012-02-22
  • 0
    What do you mean by non existant length... We have something existing number for circumference of circle... But we can't express it in decimal representation... and that is why we call it Irrational number..2012-02-22
  • 0
    Irrationality doesn't imply non-existence. In fact, Although mathematically non-rigorous, if the stars in the night sky were rational numbers, the dark background would be the irrationals. An irrational number simply implies inability to express it as a ratio of integers.2012-02-22
  • 0
    OK, i was thinking that C = PI * d so a circle that is 1 accross is PI around... and PI is irrational... what am I goofing up here?2012-02-22
  • 0
    The above seems right.2012-02-22
  • 0
    Nothing. A circle with radius $1$ has circumference $\pi$.2012-02-22
  • 1
    @PradipMishra We do not call numbers without decimal representation irrational. ($1/3=0.33333\ldots$ is rational but has no representation) We call numbers irrational if they are not rational, so they are not a ratio $a/b$ of some integers $a,b$.2012-02-22
  • 0
    You're not alone in the bewilderment. The idea of [incommensurability](http://en.wikipedia.org/wiki/Commensurability_(mathematics)) (what we now refer to as irrationality) was a struggle for the Ancient Greeks. They knew, for example, that the diagonal of a square was incommensurable with its side (a fact that we express today as "the square root of two is irrational"), but struggled for a long time to show that the area of a circle was incommensurable with the square on its diameter ("squaring the circle") which led them to doubt whether the concept of curvilinear area was meaningful at all!2012-02-22
  • 0
    $saviko1, I agree..2012-02-22
  • 3
    It's of interest to note that the modern connotation of "irrational" arose from the consternation expressed by those who first realized $\sqrt 2$ is *not* a *ratio* (of integers), at least according to some sources...2012-02-22
  • 1
    My experience is that most people graduate high school with only a vague and often confused notion of irrational numbers and the elemental properties of the reals. I attribute this to a bad education system, as anyone of average intelligence can understand it if they are willing to listen.2016-01-25

4 Answers 4