At what point was it determined that pi was an irrational number?

Question

Most people know that pi is the ratio of the cricumference of a circle to its diameter. What is not intuitive, however, is why this number is necessarily a non-repeating number. We know now by the various uses of trigonometry and calculus how to determine pi. When, however, did mathematicians know the nature of pi, and how? I mean, I could measure the circumference of a circle (say, take a string of known length and make a circle with it), then measure its diameter, and I will maybe get 3.14, but of course, my measurement would not be exact enough to know that it is an irrational number. SO....how did they do it?

For those who question me stating that it is an irrational number, the definition of an irrational number is and number that can be EXACTLY expressed as the ratio of two numbers. Now, if someone can give me an a and a b that when divided gives you 3.141592. . . ., then I stand corrected.

Answer 1

It was rigorously proven that pi is irrational by Lambert in 1770.

Here is the proof:
http://www.mathpages.com/home/kmath313.htm

Answer 2

Mathemeticians have tried to find a repeating pattern for pi after the decimal point but have been unable to find such a pattern no matter how many decimal places they examined. Finally, in 1768, a mathemetician by the name of Johann Lambert proved that there could not be a repeating pattern for pi !! Thus the number is irrational and its digits go on endlessly.

Answer 3

The real reason why it's irrational is because it cannot be represented by any combination of real numbers. Even the best formula used today figures pi accurately for several billion places, but it is not perfect in the end. Because there is no real formula that is 100 percent "The number" it is irrational.

Edited: As I stated earlier, there is a formula used by most computers to generate the number. It's good for several billion places, but it does fail in the end. I couldn't begin to input it here, it's quite involved. the first iteration of the first fraction actually generates 3.14159265. That means it uses large numbers.

How do we know it doesn't repeat? because it hasn't yet. unfortunately that's the only answer people can give. There was even a book published where someone had a theory that after so many places of pi the numbers formed a picture of a perfect sphere. This was suppose to be God's proof of existence. Truth is, humans have looked at it for thousands of years and it's still largely a mistery.

Answer 4

you're maximum appropriate in asserting which you will ensure pi from the section and radius or diameter as A=pi*r^2. yet pi is likewise the ratio of the circumference of the circle to its diameter. pi = c/d so it incredibly is quite calculated with a string and ruler.

Answer 5

I'm not quite sure but I think it was Archimedes (or some other mathematician) who came up with pi.

Irrational? Are you sure? The value of pi is said to be the ratio of the of a circle's circumference to its diameter.
A rational number can be expressed as a/b.
Let the circumference be "a" and the diameter be "b". And the value of pie as circumference/diameter, or a/b. Thus, it is a rational number based on my knowledge and analysis.

Answer 6

So far pi has been calculated to at least 1 million decimal places and it is not repeating.

The definition of a rational number is one that can be expressed as a fraction a/b where a and b are integers. This means the decimal must either be finite, or repeating.

As far as we have calculated, pi is neither of these, and is therefore irrational.

Answer 7

22 / 7 twenty two divided by 7

Answer 8

at what point? The day they named it pi

Answer 9

As far as I know, it has not been determined yet. I believe there is still room for doubt as to whether it is a nonrepeating decimal.

Frankly, I don't know how you would prove that it isn't.