How about $$\sqrt{2}$$ or the square root of any other prime number? All you seem to be doing is fixating on the fact that $$\pi$$ doesn't contain any infinitely repeating sequences of digits, when that's a property of irrationals in general (of which in turn there are infinitely many).
Pi - No Patterns, because Pi is the pattern
- Thread starter Quantum Quack
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How about $$\sqrt{2}$$ or the square root of any other prime number? All you seem to be doing is fixating on the fact that $$\pi$$ doesn't contain any infinitely repeating sequences of digits, when that's a property of irrationals in general (of which in turn there are infinitely many).
The question you asked was:
"Do you know of any other computational number that can derive such an array of digits that demonstrate no pattern forming?"
I suggested sqrt(2). Can you find some pattern in its digits? It's certainly computational. Newton gave an algorithm for finding square roots over 300 years ago. The square roots of other nonsquare numbers form an infinite class of candidates for you to consider. As do e, the base of the natural log; or phi, the golden ratio. What is your objection to those examples?
If I give you the binary sequence 10101010101010101... that alternates, that's a pattern. Surely you would not call such a sequence random. The meaning of a random sequence is that no pattern ever shows up. If there's a pattern it's not random. You are operating from a definition of "random" that's the opposite of what everyone else means by that word.
Find the pattern in sqrt(2) and then we can talk about this. If there's a pattern then the number's not random. Patterns are the conceptual opposite of randomness. You're just making up your own definition, like defining "up" to mean "down" and then claiming that gravity causes things to fall up.
No it's actually the opposite. "Almost all" real numbers show no pattern whatsoever, where almost all has its usual technical meaning. The set of numbers having a pattern has measure zero in the real numbers. Numbers with patterns are extremely unusual. It's just the case that they're generally the number's we're familiar with, like 1/3 = .33333... Rational numbers are very rare in the scheme of things, as are numbers showing patterns.
Pi is typical in the sense that it's irrational and its digits show no pattern.
The vast majority of real numbers show no pattern. This has been mathematically proven. Pi isn't very special at all. The only reason people think pi is special is because it's the first "mysterious" irrational number people are exposed to in high school. But in the scheme of things pi is just a typical irrational.
Only if you define "up" to be "down" and toss in a little crankology like "universal structure" and "holistically." Those are phrases with no mathematical meaning in this context.
I've already suggested sqrt(2); the square roots of all the other non-square natural numbers; the constant e; and the constant phi. But we know mathematically that the patternless numbers are by far the more common real numbers; and that the numbers that show any pattern at all are exceedingly rare.
Precisely accurate Areas and Volumes can only be calculated using straight lines.
Any measurement of the area of a curve, is only an approximation made by adding up shapes with straight lines until they nearly fill the curve.
Am I correct?
Any measurement of the area of a curve, is only an approximation made by adding up shapes with straight lines until they nearly fill the curve.
Am I correct?
No, not correct. Ever since Newton and Leibniz developed the calculus in the late 1600's we have mathematical techniques to determine the precise areas and volumes defined by curves. For example the length of the circumference of the unit circle is precisely 2pi. This is proven in freshman (or high school) calculus by integrating the formula for the unit circle. Calculus students are routinely tasked to determine areas and volumes bounded by various curves.
You are correct that the underlying idea is to approximate an arbitrary shape with a collection of rectangles (or cubes in 3D). The great advance in calculus is that we can define the limit of this process to produce exact answers.
Archimedes had the basic idea of approximation by straight-line figures a couple of thousand years ago.
Newton and Leibniz developed specific algorithmic procedures for determining areas and volumes. And the full logical rigor underlying calculus was only completely worked out in the late 1800's. Today we can start with the axioms of set theory and rigorously develop the theory of the integral.
It's taken humanity a long time to nail this down; but today it's a done deal. The answers we get for areas and volumes are exact and the procedures are logically rigorous. And we understand the theory and practice well enough to teach it to high school students.
Archimedes not only knew how to do these approximations, in many cases he knew how to find the precise limit as the number of approximating shapes approached infinity, using clever tricks rather than the formal tools provided by modern calculus (I wish they would teach high school students more about these sorts of things before rushing them through the elementary calculus cookie cutter).
Define precise? [In the context of the area or volume of a circle or sphere]
or
How precise is "precise"?
or
How precise is "precise"?
Smart guy this Archimedes...
It's as precise as precise gets, the limit of a sequence of approximations as the approximations come arbitrarily close to perfection.
Would infinitesimal precision sound about right?
As compared with absolute precision.
Example:
3.999999999999999~ can not ever equal 4
You mean with an infinite sequence of 9's after the decimal point? Yes it would equal exactly 4, otherwise there would be a number inbetween them. What would that number be?
Interesting so then pi actually has a finite resolution yes?
I have no idea...
other than it would have to be infinitesimally greater than zero
What is "finite resolution" supposed to mean?
Hard for me to explain except by examples...
example of finite resolution
number 3.67
the resolution value is finite as it has a finish and can be considered as resolved
example of infinite resolution
number : 3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 59230 78164 06286 20899 86280 34825 34211 70679 ....
The resolution is infinite as the resolution value gets smaller and smaller and smaller infinitely therefore the number is unresolved, unresolvable
I think there is a distinct Zeno of Elea, over tone about this discussion
Pi has no expression in terms of a finite number of digits, it's not a rational number. On the other hand, any number with an infinite sequence of repeating decimals is rational and can be expressed as a fraction of two natural numbers.
Example:
$$x=18.1437373737373737\ldots\\100x=1814.37373737373737\ldots\\99x=100x-x=1796.23\\9900x=179623$$
Therefore $$x=18.1437373737373737\ldots=\frac{179623}{9900}$$
Once you have the number pi, then of course you can use it instead of "cookie cutting" the circle.
But can you calculate the number pi without cookie cutting?
If Archimedes had had the decimal point instead of having to use fractions,
he could have calculated pi to many decimal places.
Finding the necessary square roots using fractions must have been a monumental task.
His degree of accuracy would be a matter of how much time he wanted to devote to it.
But can you calculate the number pi without cookie cutting?
If Archimedes had had the decimal point instead of having to use fractions,
he could have calculated pi to many decimal places.
Finding the necessary square roots using fractions must have been a monumental task.
His degree of accuracy would be a matter of how much time he wanted to devote to it.
now that's a good question...can we?
I'm not sure if this is what you're cryptically referring to, but Archimedes did have a means of calculating $$\pi$$ to as many decimal places as he wanted (since they didn't use decimals, what he did was squeeze $$\pi$$ between arbitrarily tight fractions). He did it by drawing inscribed and circumscribed hexagons around a circle, then splitting them up into polygons with 12 sides, then splitting those up into polygons with 24 sides, etc. etc., and using recursion relations to find the perimeters of the inscribed and circumscribed polygons at every step.
Also, regardless of the actual value of $$\pi$$, it was well known by the time of Archimedes (I'll bet it was already known even in Euclid's time) that the area of a perfect circle was exactly $$\pi r^2$$ after an infinite amount of "cookie cutting". He'd do a clever slice job to calculate various volumes and surface areas, then he'd go about with much more sophisticated reasoning to prove that there was no other possible answer using a "double reductio ad absurdum". For instance if he calculated the volume of a cone, he'd then prove that the volume couldn't be any less, and it couldn't be any more.
There are zillions of ways to go about calculating $$\pi$$, both geometrically and using more abstract techniques (i.e. Taylor series, Fourier series, Wallis product...)
I know this is going to sound absurd coming from a person who is essentially math illiterate but:
assume a double reduction absurdo situation:
we have a square with a total finite volume
Inside this square we have a circle and given it's closed state could also be stated as being a finite volume.
Now if we calculate the volume outside the circle [with in the box] and then calculate the volume of the circle [with in the box], I think you will find that the circle itself doesn't exist, however the volumes of the two do.
assume a double reduction absurdo situation:
we have a square with a total finite volume
Inside this square we have a circle and given it's closed state could also be stated as being a finite volume.
Now if we calculate the volume outside the circle [with in the box] and then calculate the volume of the circle [with in the box], I think you will find that the circle itself doesn't exist, however the volumes of the two do.
Why would the circle not exist?