What is the level of precision in the universe?

[aquaregia]

When I was first learning programming I was surprised that computers only hold numbers to 32 or 64 bits of precision (sometimes more). What I am wondering is to what precision does stuff in the universe happen?

For example: if you had a super slow motion microscope and could zoom into a single atom of hydrogen bounce off of something, to what level of precision would the angle be accurate compared to what the relevant formulas say it should be?

Would a quantum computer calculate something to this degree of precision? Because I remember reading something that said that using a quantum computer it would be possible to model how something would happen in reality EXACTLY rather than how most computers do an approximation.

 

[gmax137]

"to what level of precision would the angle be accurate compared to what the relevant formulas say it should be?"

The angle is exactly accurate to itself - the real world wins. If you have a formula to predict the angle, it might be right to a few decimal places. Formulas are nearly always less accurate than the precision of your calculator (or at least, other effects are neglected that are more important than the round off of your calculator). For example, if you are predicting the height a cannon ball reaches, you don't (generally) worry about where the moon is at the time of the shot. But if you did, it would affect your result to a very small degree. I suspect that effects like that are generally more important than roundoff in the 15th decimal place.

You do not need a quantum computer to calculate to greater precision than the bits in your calculator. You can write a program to calculate a result to as many bits as you like, you just need to do everything "bit by bit" so to speak. Mathematicians working in Number Theory, for example, may manipulate numbers such as 2^n-1 where n is a big integer, and 2^n-1 may contain hundreds of digits. They may want to add or multiply two such numbers. You can't do that on your hand calculator for n greater than about 40. But you can write a program to get the actual answer (you could even do it with excel on a PC if you wanted to). The double-precision in laguages such as Fortran offer a convenient way to do this to some extent - invoking DP makes the compiler use a different algorithm for arithmetic, in which more digits are carried accurately.

 

[aquaregia]

The angle is exactly accurate to itself - the real world wins. If you have a formula to predict the angle, it might be right to a few decimal places. Formulas are nearly always less accurate than the precision of your calculator (or at least, other effects are neglected that are more important than the round off of your calculator).
For example, if you are predicting the height a cannon ball reaches, you don't (generally) worry about where the moon is at the time of the shot. But if you did, it would affect your result to a very small degree. I suspect that effects like that are generally more important than roundoff in the 15th decimal place.

Well, you would take into account ALL things that were relevant to that calculation, that means all the moons gravity, that means the gravity caused by a raindrop falling on Jupiter, everything.

How can a formula be less accurate than a calculator? A formula tells you what it is by definition, a formula gives you an exact answer to infinite decimal places, if you put the sqrt2 or pi into a formula you get an exact answer a calculator always rounds as far as I know. From what I understand for a calculator to store a irrational number in digits would require an infinite state machine.

If you have a perfectly flat mirror and you beam a laser at it at a certain angle it should reflect off of the mirror at an angle so that it is symmetrical. You can do a simulation do a of a particle bouncing off a flat wall on a computer but the computer will never give you the exact answer since the numbers only have so much precision. My question is viewing the universe as doing the "simulation" how much precision does it have?

Also: is a quantum computer an infinite state machine?

Edit: I think the answer to my question has to do with whether or not space and time are continuous and Plank lengths, because if it is discrete then the precision of anything that happens in it would be limited just like in a regular computer, but if their is no minimum length of time and space from which all other times and space are made up of then that means that particles would bounce off the wall with infinite precision.

So, based on the Plank length how many bits of precision would you need to calculate the exact movement of atom bouncing off?

 

[Ben Niehoff]

You can write a program that uses arrays of integers to store a single number. Then you can explicitly write out various algorithms for doing math with these numbers. In this way, you can calculate to any precision you want.

In fact, many such packages already exist and are usually called "bignum" packages.

 

[gmax137]

One of the points I as trying to make is this - the "formulas" are not exact - they are approximations. Take gravity as an example: Newtons formula for the force between two masses is F = G mM/r^2, right? You can calculate that to as many places as you wish, but it IS NOT the exact force. Einstein's general relativity calculates a slightly different answer, and his is closer to the "truth" as shown by the precession of Mercury, which Newton's formula does not predict. But is GR "correct" ?? - no, someday someone else will come up with a "formula" more accurate still. The point is, these formulas do not describe reality, they model reality (to a greater or lesser accuracy).

If you do not believe this, take a look at the derivation of "formulas" in the physics books. You will find frequently that the derivations include simplifications - for example, series are often truncated at two terms. This allows for a compact "formula" while acknowledging that the inputs (things like the values of the two masses, the distance, and the value of "G") are all imperfectly known.

Now if you are contemplating other formulas (like, the circumference of a circle is pi * D) then you are in an ideal "world" where the circle is in a plane, etc, and numbers are continuous, and so on. In this world there is no quantum granularity, it is math not physics.

 

[Jimmy Snyder]

I think there is some question as to whether numbers greater than ~10^{100} have any physical significance. This number is very roughly the ratio between the largest and the smallest, the oldest and the newest, the most and least massive, and the number of particles in the universe.