Is there any gap between numbers?
How numbers are accurate to represent the continuity?
Are they continuous as the nature is?
Well, numbers are continuous, whether or not nature is analogue or digital depends on who you ask. You're question is actually something I've been thinking about for a while, as have many other mathematicians and philosophers, going all the way back to Parmenides and Zeno. Zeno set fourth a set of paradoxes that "proved" change was impossible.
Think of an arrow flying to a target. Before it got all the way to the target, it would get 1/2 the way, before that it would get 1/4 the way, before that 1/8, 1/16, 1/32, and so on to infinity. Of course, this would not be the case if you got to the point where space and time became indivisible. (The "atom" -Not the same as the atom we are familiar with now.-) However, if the atom was the smallest amount of space and time, then zero motion would be able to take place over the span of an atom. Since all motion would be the sum of changes through a number of atoms of space and time, and each individual atom has 0 motion within itself, there should be zero motion. (0+0+0+0+0+0... =0)
On the other hand, it could be analogue, in which case the arrow would have to travel an infinite distance to get to the target. In fact, any motion would take an infinite distance in space and time. But since the arrow itself can be divided an infinite amount of times as well, this only proves that an infinite series can take up a finite amount of space. (Integral calculus wasn't invented for a long time after this paradox arose, hence it's long lifespan.)
And so, it seems that nature is analogue, not that I'm an expert or anything. Numbers indeed reflect this. Even more, a Cartesian coordinate plain captures this effect. (You can plug in any number you can imagine into a function that is defined at all points and find it on a graph.)
Still, this leads to some weird stuff. There are an infinite amount of numbers between 0 and 1, which is equal to the amount of numbers between 0 and 100. (Infinity.) So what does it mean to say that the value of something is "1" or "2"? Sure, these numbers and others make sense when you have digital things. (5 fingers, 5 rocks, etc.) But if I had some sort of material that I could see splitting apart an infinite number of times, what does it mean to say I have 3 units of such a material? That I have 3 times infinity of it?
There are gaps between rational numbers but the irrational numbers then fill up the spaces between rational numbers so that there is a real number assigned to every point on the "number line".Is there any gap between numbers?
But there is no reason to believe that the real numbers, or any mathematical system is going to perfectly mirror any physical system. Any assignmemt of numbers to a physical situation can be at best an approximation- measurements themselves are only approximations.How numbers are accurate to represent the continuity?
??? What reason do you have to believe that nature is continuous? Mathematics has a specific definition of "continuous" that can be proved in many situations. What definiton of "continuous" do you want to use for nature? And how are you going to prove that nature, or any part of it is continuous?Are they continuous as the nature is?