Another Boring Math Question

D

Dundee

Guest
I have been listening in my car to some Math webinars, Specifically on the History of math and numbers and how such things as Pi and E were derived, Actually quite interesting, but something occurred to me.

Pi is defined as the ratio of a circles circumference to its diameter. And Pi is also said to be never ending....

Equivalently, its decimal representation never ends and never settles into a permanently repeating pattern. Its decimal (or other base) digits appear to be randomly distributed, and are conjectured to satisfy a specific kind of statistical randomness

OK, so lets grab a length of string 100mm long and form a perfect circle,
In a physical sense, its length and diameter are clearly defined in front of you, and are most definitly finite.

Take the reverse, if you draw a circle with a diameter of 100mm, multiply it by Pi you get the circumference. But if Pi is never ending then the calculated number for the circumference has to be at best an approximation, and if Pi is endless, then the circles circumference can never be defined accuratly.

Only, the circles circumference as shown in the first example is finite. In that case 100mm.
So, how can Pi be infinite, when the circumference and diameter are clearly not Infinite.

Doesnt this imply that Pi at some point stops and is finite?
 
H

HAL

Guest
..Doesnt this imply that Pi at some point stops and is finite?..

No. What is happening is that by adding more decimal places you are gradually becoming more accurate.

Remember Pi is derives from the fraction 22/7. It doesn't matter what the units are, can be miles or millimetres. The ratio stays the same.

There are lots of times when converting from inproper fractions to decimal fractions can catch you out.due to the 'rounding up' of the last place.
 
D

Dundee

Guest
But if you look at an example where you try to travel 1km. But each stepyou halve the difference left. Do step 1 you travel 50m 3 you travel half whats left. So 25m step 3 12.5 m and so on. With this you never get to 100m you only get half each remainder. This is effectively what is happening with Pi. It never finishes as each conservative decimal plave is only ever a fraction of the remainder. So
Pi is said to have an infinite number of decimal places never reaching an end point. So 1 x infinite is an approximation.
So pi x d is only ever an approximation.
So working back a 100mm length of string is not infinite. It is 100mm. The diameter is also finite. So how can the ratio of two finite numbers be an infinite string of decimal places.
Tje implication woring the other way is the corcle never joins. It only ever closes the space by a proportion of tbe remaining distance
 
H

HAL

Guest
That is Zeno's paradox. And it is a load of baloney.

It can be shown to be so by numerous methods.

So pi x d is only ever an approximation.

No. It is correct. But as it is a ratio you have to stick with the improper fractions. The moment you plug in values and use decimals you get into the realm of approximation.

Essentially, if you know the exact diameter then ( using Pi x D) you can get a very close approximation of the circumference. But not exact. Or if you know the Circumference you can get an approximation of the diameter. If you need greater accuracy, just use more decimal places. I use eight places if I need anything close.

It is rather like charging a capacitor up to supply voltage. You can't do it.

Remember that the Ancient Greeks didn't have much to do with their spare time. :smile:
 
D

Dundee

Guest
So what about this though. As you said if you know the diameter you can only get an approximation of the circumstances. But although you cant calculate it due to the nature of Pi. It is a finite number. You just dont know what it is. A circle most definitely has a finite d and c.
So how can Pi be an infinite number of decimals.
Its like a Tangent. It gets closer and closer to the axis but never touches.

If pi is infinite and never stops.
Dosnt that mean pi x d never has a finite answer. So the circles circumference never quite joins mathematically.
But in the real world it does. So pi cant have an infinite number of decimals
 
H

HAL

Guest
You can always do the calculation if you have a century or two to spare.
But to save you the trouble...



31.4 trillion

In honor of Pi Day, today March 14 (represented as 3/14 in many parts of the world), we're excited to announce that we successfully computed π to 31.4 trillion decimal places—31,415,926,535,897 to be exact, or π * 1013

But if you write Pi as 3 1/7 you will always be correct.But you won't be able to get rid of the last 7th.

But our calculators work in bits. which are much more easy to manipulate.


..So how can Pi be an infinite number of decimals...

It may not be. But no one has yet managed to reach a point where it begins to repeat.
 
  • Like
Reactions: Dundee