Boundary between discrete&continous

[cheungyl]

what is the real boundary(or difference) between discrete&continous(as the title)? my major is physics and i find that scientists are dealing with the different treatment to these two kinds of phenomenons, but what is the real boundary? by this i mean what they actually are and how they ARE DIFFERENT from each other? we know that there is something like a transition,but will transitions really be found? I consider this a math problem so i write it here. is any mathematician working with this? (forgive my poor english)

 

[farleyknight]

I am not an expert on this subject, but probably the best way to describe how they are different (in terms of elementary math) is to consider a number like pi or Euler's e or the square root of 2. These numbers cannot be expressed in terms of a finite set of other numbers. You need an infinite sum of numbers to express it in exact terms. Without an infinite set of terms, you can only produce an approximation.

There are definitely others, but finite vs infinite is perhaps the easiest analogy.

Surely some one can chime in about Cantor's uncountability proof.

 

[cheungyl]

i know what u mean. Discrete in physics is born from infinity inf maths. thank u !

 

[awkward]

Personally, I would say that continuous mathematics has to do with limiting processes. Continuity, limits of sequences, differentiation, integrals-- these all have to do with limiting processes, one way or another.

 

[CellCoree]

I can provide some insight into the boundary between discrete and continuous phenomena. In physics, discrete refers to things that can be counted or measured in distinct, separate units, while continuous refers to things that can be measured on a spectrum or without distinct units. For example, the number of atoms in a substance is discrete, as they can be counted individually, while the temperature of a substance is continuous, as it can be measured on a spectrum without distinct units.

The real difference between discrete and continuous phenomena lies in their fundamental nature. Discrete phenomena are characterized by a finite, countable number of distinct units, while continuous phenomena are characterized by an infinite, uncountable range of values. This fundamental difference leads to different approaches in how we study and understand these phenomena in science.

In terms of transitions between discrete and continuous phenomena, there is ongoing research and discussion in both physics and mathematics. In physics, there are theories such as quantum mechanics which bridge the gap between discrete and continuous phenomena, and in mathematics, there are concepts such as fractals that blur the boundary between discrete and continuous. However, it is important to note that these transitions may not always be clearly defined and may vary depending on the specific phenomenon being studied.

In summary, the boundary between discrete and continuous phenomena is a fundamental aspect of how we understand the world around us. While there may not be a clear, definitive answer to where this boundary lies, scientists and mathematicians continue to study and explore this concept in order to further our understanding of the universe.