All numbers are irrational numbers according to the HUP because of how math is

TRVP_ANGELIf i had a hypothetical camera or microscope that could zoom in forever on the location where the needle tip struck the number line, there would be a 100% probability that the needle would point to an irrational number that had a decimal like 0.581497239406~~>>.

ShadowXVXNo, there would be a 99.9...% probability. Congratulations, you have proven that .9... is not equal to 1.

little1337.9999... is equal to one. Otherwise calculus would be fucked

ShadowXVXLimits state what a function approaches as it nears an input, not what a function is equal to at a given input. I would go so far as to say that calculus works specifically because .9... is not equal to 1; that is, because if a function is not defined at an input, we can still (often) determine what value that function approaches as it approaches that input (another way of wording it: we can often determine the [defined] value of f(x+e), given infintesimal e, even if f(x) is undefined).

little1337Not exactly, the convergence of a function does determine what that function is equal to at infinity, so in this case, after an infinite number of 9's.

If you look at this as a sum of 0.9+0.09+0.009+..., it converges to the value 1 the same way the sum 1/2^n = 2 or the sum of the recipricals of powers(1/n+ 1/n^2 +1/n^3+...) equals n/n-1

Here is the proof from Wikipedia for 0.999...=1 :


For the accuracy of the proof, the number 0.999...9, with n nines after the decimal point, is denoted 0.(9)n. Thus 0.(9)1 = 0.9, 0.(9)2 = 0.99, 0.(9)3 = 0.999, and so on. As 1/10n = 0.0...01, with n digits after the decimal point, the addition rule for decimal numbers implies

0.(9)n + 1/10^n = 1

and

0.(9)n < 1,


for every positive integer n.

One has to show that 1 is the smallest number that is no less than all 0.(9)n. For this, it suffices to prove that, if a number x is not larger than 1 and no less than all 0.(9)n, then x = 1. So let x such that

0.(9)n less than or equal to x less than or equal to 1 (NS Wouldn't let me use less than or equal to sign)



Therefore,
0 less than or equal to 1-x less than or equal to 0.(9)n=1/10n

This implies that the difference between 1 and x is less than the inverse of any positive integer. Thus this difference must be zero, and, thus x = 1; that is

0.999... = 1




**This post was edited on Jul 20th 2020 at 1:56:13pm

**This post was edited on Jul 20th 2020 at 2:05:33pm

little1337If you look at this as a sum of 0.9+0.09+0.009+..., it converges to the value 1 the same way the sum 1/2^n = 2 or the sum of the recipricals of powers(1/n+ 1/n^2 +1/n^3+...) equals n/n-1

little13370.(9)n + 1/10^n = 1

and

0.(9)n < 1,


for every positive integer n.

little1337One has to show that 1 is the smallest number that is no less than all 0.(9)n. For this, it suffices to prove that, if a number x is not larger than 1 and no less than all 0.(9)n, then x = 1.

ShadowXVXConverges as in approaches. It's a limit, not an equivilence.




Yes






No. When working with infinitesimals (which we necessarily must be if we are even considering the concept of .9...), the fact that there is no real number between x and y does not mean that x=y. (in this case, x=1 and y=.9...) This is where the concept of hyperreals is applicable. The difference between the two is a nonzero infinitesimal hyperreal.

>inb4 b-but hyperreals don't count!

Then .9... does not exist. The entire concept of a repeating decimal such as .9... is a concept based upon hyperreals and infintesimals. Otherwise .9... does not even exist.

One way to define .9... would indeed be .(9)n=1-1/(10^n) as you said (though one could define it in other such ways as well, for example, if .(9)n is defined as 1-1/(1000^n) then .(9)∞ is still .9..., albeit a different .9... that, when working with hyperreals, is not the same as the other .9...) in which case .9...=1/(10^∞). And yeah sure ∞ isn't a number, but again, .9... doesn't really exist anyway, it's a concept involving infintesimals (and in the case that we define it this way, infinites). Also relevant is the fact that two hyperreals that look the same when written out are not necessarily equal. For example, not all 1/0 are equal, not all 0/0 are equal, not all ∞ are equal, not all .9... are equal.

**This post was edited on Jul 21st 2020 at 12:23:29am

TRVP_ANGELI have a small theory about rational/irrational numbers that id like to get feedback on if possible:


Ive searched online to see if the idea has been voiced or written or championed by anybody else but i was not successful in finding anything. If any of you know of any sources with this idea please link them.


Everybody knows the difference between rational and irrational numbers. I was thinking about a number line on a table that goes from 0 to 1. Say i had a hypothetical needle with a tip that was a true point, as in, it had no cross sectional area at the end of it, and i dropped it onto the number line such that it stuck somewhere between 0 and 1. If i had a hypothetical camera or microscope that could zoom in forever on the location where the needle tip struck the number line, there would be a 100% probability that the needle would point to an irrational number that had a decimal like 0.581497239406~~>>............... that went on forever, because the amount of irrational numbers that exist represents a larger infinity than the amount of rational numbers that exist; for every rational number that exists, an infinite number of irrational numbers exist. No matter how many times you drop this hypothetical needle, the probability that it will strike a rational number on the number line is zero.


The idea of an irrational number existing in reality has some contradiction with the Uncertainty Principle (HUP). A corollary of the HUP is that the amount of information present in a volume or particle or system is finite; if you try to pack infinite information into a finite space, you end up creating a black hole. But the black hole comes before you are ever able to pack infinite information into the space; you will never be able to cram an infinite amount of information into a space that is not also infinite.


Back on topic, the idea of an irrational number presents a problem because a number with an infinite amount of numbers to the right of a decimal place ,
for example 0.5837378219.....~~>> keeps giving you a little bit more information every decimal place you record. You would need a notebook with an infinite number of pages to write down the decimals that come out of an irrational number. As you zoom in on the needle point where it strikes the number line, you can keep harvesting more information every time you zoom in 10X.


In other words, a complete irrational number represents an infinite amount of information, therefore irrational numbers, like many other things in mathematics, might not exist in reality.


So what makes the pretty little rational point 0.5 different from the irrational monstrosity 0.72143953750874...~~>>>
from the perspective of the hypothetical needle point?
From the surface it seems like the irrational number represents much more information than the rational number 0.5. But using HUP as the ultimate arbiter we know this cannot be the case. It makes no sense for one point to hold more information than another point on a numberline.
The way i would reconcile this is to say that 0.5 is actually 0.5000000~~>>>. You have to keep using your camera to zoom in on this point location repeatedly, forever, and every zero you add represents the adding of more information.
So the point 0.5 which would conventionally be considered rational, I believe is actually the irrational number 0.500000~~>> because the zeros go on forever and for every zero (just like for every decimal place in a typical irrational number) you verify by zooming in on the point, you gain information on the precise location of the point.


According to the HUP, taking irrational numbers to extreme decimal places (Like the mile of pi) is pointless because very quickly the precision of the universe gets outpaced by the precision of your hypothetical model. But if you are just considering the issue mathematically, it may make more sense to consider ALL numbers to be irrational numbers.


Anyways, ive been thinking about this for a while and wanted to see if anybody could add perspective or maybe just murder my poor little theory

DolanReloadedWhat sort of choad dinosaur would post such obscene mental pornography.

freestyler540Numbers are relative to what you are counting.
Think of this; cant you have 1.5 trees. But you can have 0.25 apples if you have cut an apple 3/4. Now if you weigh the apple, you can have 35.67g.
Pi is simply the relationship between the radius of a circle and the arc of that circle. Its irrational because the relationship isnt perfect, not because its uncertain. The circle is complete, so its certain.
Uncertainty principle simply states that when you try to measure a subatomic particle, either you can calculate its position or its momentum, not both. When you try, you get a function innequality.
Think of it like this: you want to know the speed of a car with only a camera. If you take a still shot, the car in the image will be blurry. But if you try to pan the camera while the car passes, the car is sharp but everything else becomes blurry.

TheMailManThis mess of a comment has literally nothing to do with anything OP is discussing

freestyler540Elaborate

freestyler540Smartest thread I have seen in a long time.

freestyler540You, you Trump supporting, inbred troll.

TheMailManWhy don't you? Wtf are you talking about 1.5 trees and the weight of an apple?

You then go on quote the most basic explanation ever of the uncertainty principle ever for no reason, it doesn't have anything to do with the OP's theory

little1337I haven’t looked much into hyper real numbers so I’ll have to do that before getting into this, either way impressive maths friend

**This post was edited on Jul 21st 2020 at 8:04:33am