Uncovering the Mystery of Pi's Value

[Universe_Man]

Pi is the most interesting number. I was thinking about the value for Pi that is accepted by the Scientific and Mathematic community, which starts off as 3.141592653... and so on to infinity.

My question is, How did they get this value of Pi as the most approximate? Did they use extremely fine implements of measuring, or is there a clever mathematical way of finding what the exact decimal values are out to a certain number of places? I don't know. Also, since pi goes on into infinity, does that mean that the circumference of a circle has no exact value?

 

[Jameson]

Check out this Wikipedia page on the history of Pi. There are many, many methods of calculating it to billions of digits. http://en.wikipedia.org/wiki/Pi

 

[Hurkyl]

does that mean that the circumference of a circle has no exact value?

No. It merely means that the circumference of a circle of diameter 1 cannot be written exactly as a terminating decimal number. Its exact value is pi.

 

[rcgldr]

Still trying to understand the methods used for optimizing extended precision math myself.

Some links:

http://numbers.computation.free.fr/Constants/PiProgram/pifast.html

http://home.istar.ca/~lyster/pi.html

http://www.apfloat.org

 

[Gokul43201]

...does that mean that the circumference of a circle has no exact value?

If you believe that the circumference has no exact value because it is a diameter times pi, then you should also believe that the diameter (or twice the radius) has no exact value, since that is just the circumference over pi (and 1/pi also has a non-terminating decimal representation). But the radius is just the distance between the center and a point on the circle. From this you must conclude that the distance between two points has no definite value. Virtually all of geometry then becomes meaningless.

All these problems are overcome by replacing your incorrect notion about pi with the statement in Hurkyl's post.

 

[rcgldr]

Also, since pi goes on into infinity, does that mean that the circumference of a circle has no exact value?

If you believe that the circumference has no exact value because it is a diameter times pi ... All these problems are overcome by replacing your incorrect notion about pi with the statement in Hurkyl's post.

He was asking a question, not stating a belief or notion.

To answer the question, the conventional methods used to decribe numbers have a weakness for radicals (like square root of 2) or transcendental (like pi) numbers, as these values can't be represented as a fixed point number with a finite number of digits, or as a fraction with a finite number of digits. So in the math world, they are just written as symbols, like pi, or descibed with mathematical terms, like square root of 2, or 4 times the inverse tangent of 1.

In the case of standard geometry, it's not possible to create a straight line that is pi times longer than another line.

On a number line, every real value is a (infinitely small) point on the line. With this analogy, pi is an exact point on the number line, as well as the square root of 2, or a simple integer like 1. There's no issue with these values on the number line, the issues occur when we try to come up with a means to describe values using fixed point number or fractions.

 

[rhj23]

every real value is a (infinitely small) point on the line.

is there such a thing as a point on a line that is not infinitely small?

 

[Gokul43201]

He was asking a question, not stating a belief or notion.

Point noted. Sorry for the misrepresentation.

 

[rcgldr]

every real value is a (infinitely small) point on the line.

is there such a thing as a point on a line that is not infinitely small?

No, which is why I put it in paranthesis for those few readers that may not understand the point about points. (almost sorry for the bad pun).

 

[rcgldr]

Other than e and pi, are there any other common transcendentals?

 

[Curious3141]

Other than e and pi, are there any other common transcendentals?

Liouville's constant $$L = \sum_{n=0}^{\infty} {10^{-n!}} = 0.1100010...$$ has '1's in every decimal place that's a factorial, and zeros elsewhere. It's the first number to be proven to be transcendental.

Champernowne's number, formed by concatenating decimal representations of the naturals. $$0.123456789101112...$$, proven to be transcendental.

$$e^{\pi}$$ known to be transcendental, can be easily proven with Gelfond's theorem. $${\pi}^e$$ is suspected but not known to be transcendental.

$$2^{\sqrt{2}}$$, Hilbert's number, known to be transcendental, provable by Gelfond's theorem.

There are many other numbers that are suspected but not known to be transcendental, e.g. $$\zeta(3)$$, Feigenbaum's constant, Catalan's constant, etc.