Theos-Talk Archives (March 2005 Message tt00162)

Bart wrote:
<< Pi is defined as a ratio: the ratio between the circumference and the 
diameter of a circle. >>

Steve wrote:
<< Yes and no. Pi shows up in all sorts of equations, not just the 
definition of a circle. >>

I guess Princeton must be wrong, then, because they define pi:

"the ratio of the circumference to the diameter of a circle"
www.cogsci.princeton.edu/cgi-bin/webwn

You are confusing "ratio" and "fraction". Regular (simple) fractions 
REQUIRE that both numbers be integers. Ratios do not. Every fraction is 
also a ratio, but a ratio is not necessarily a regular fraction.

Pi is a constant, even though we cannot accurately measure it. The 
distance of empty space between two molecules is enough to throw off our 
measurements, as someone else pointed out earlier, and you commented on 
in this reply.

By the way, Princeton is not the only source that defines pi as being a 
ratio. Here are some other definitions, after which I'll comment on 
other things you said, and bring up another topic for argument.

"The designated name for the ratio of the circumference of a circle to 
its diameter."
www.shodor.org/interactivate/dictionary/p.html

"Pi is the number of times the diameter divides into the circumference 
of a circle. It is approximately 3.14159 times. ( 3.14)"
library.thinkquest.org/C004647/util/glossary.html

"ratio of the circumference to the diameter of a circle (3.1415926...)"
www.allmath.com/glossary.asp

"Pi is a mathematical constant equal to approximately 3.1415926535897932."
www.csidata.com/custserv/onlinehelp/VBSdocs/vbs0.htm

"The ratio of the circumference of a circle to its diameter; a number 
having a value to eight decimal places of 3.14159265."

investigate.conservation.org/expeditions/guyana/glossary.htm

"p - The symbol for Pi is actually a greek letter. Pi is used to 
represent the ratio of a circumference of a circle to its diameter."
math.about.com/library/blp.htm

"The ratio of the circumference of a circle to its diameter 
(approximately 3.14159)."

www.macromedia.com/support/flash/action_scripts/objects/math_object.html

<< What I was getting at is, if it were a ratio, you could divide one 
number by another and get pi, ... >>

Yes and no ... if it were a rational ratio, you could divide one number 
by another and get pi, but pi is an irrational ratio, but still a ratio. 
Just because accurate calculations do not use ratios of real numbers 
does not make this definition wrong. Pi is still a ratio, it is just not 
a ratio that we can write as a ratio between two actual numbers.

<< People with nothing better to do have calculated pi to hundreds of 
decimal places ... >>

MILLIONS AND MORE, you mean ... computer programs, which were programmed 
by people with nothing better to do have taken it to unknown (to me) limits.

Do you know of two numbers which, if one were divided into another, 
would yield an accurate value for pi to any number of decimal places?

I can give you two numbers that would yield an accurate value of pi to a 
specific number of places. Just tell me how many digits you want it to 
be accurate and I'll give you two numbers.

For example, if you wanted two numbers that would give you pi to two 
decimal places, I'd tell you to use 314 over 100, or 157 over 50. But 
that's not what you mean.

Mathematicians have proved that pi is irrational. In other words, it 
cannot be written as a regular fraction, with both numbers integers.

But all that proves is that pi cannot be written as a integer-ratio. 
That does not mean that it's not a ratio. It is still the ratio between 
the circumference and the diameter of a circle.

And pi is considered to be a constant, even though every math student 
uses approximations. That's why, when a teacher asks for an exact 
answer, you give the answer including pi as part of your answer. If he 
asks for an approximate answer, you decimalize the answer.

Okay, new topic for argument:

Is 1 a prime number?

Some math teachers will tell you that 1 is a prime number because it can 
only be evenly divided by one and by itself.

Most math teachers will tell you that 1 is not prime, because it cannot 
be divided by BOTH one and itself, it can only be divided by one, itself.

I'm part of the latter group. Part of my reasoning is that if 1 was 
prime, the "Sieve of Eratosthenes" would not work, because part of the 
"Sieve" method of finding prime numbers is to circle the first prime 
number, cross out every multiple of that prime, then circle the next 
prime number, cross out every multiple of that prime number, etc.

For more on my arguments, see www.Mazes.com/primes/one.html, including 
the fact that Euclid said that one is not a number, and used that as 
part of some of his proofs.

John, webmaster
www.GodLovesEveryone.org and www.MAZES.com

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