Pi is a ratio, yet irrational?

Here's an interesting dilemma:

We know Pi is an irrational number; mathematicians have proven it to be so.

But its definition says that it is a RATIO of the circumference and the diameter of any circle.

Now, when you divide a rational number by another rational number, you get a rational number.

Doesn't this seem like a contradiction?

In the words of a certain visitor to my site:

In the equation where the circumference is divided by the diameter, when the circumference and diameter are rational values, why is it that the quotient can be an irrational quantity?




The solution

First of all, like a commenter pointed out, Pi being a ratio of two numbers does not mean it is rational. Pi has been established as irrational, and we know
Pi = C/d, where C is the circumference and d is the diameter of some circle.

It follows that either C or d or both have to be irrational!

This is kind of amazing to think about, but it's true: for any circle, either the circumference, or the diameter, or both are irrational (in the abstract world of mathematics...).

BUT (and herein lies the crux of the apparent contradiction)... in the real world where we have to use measurements, of course your measurements will always be rational numbers. ANY measurement you make is just an approximation anyway, and is rational, such as 14.52 cm or 8 1/16 in.

And, if you try to find the value of Pi using measurements from real world, all you get is a rational approximation to the value of Pi.

Even your calculator just gives you a rational approximation for Pi, to however many decimal digits it can show it. But that's all that is necessary for real world applications.
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