First of all, finite representation is mainly used to deal with a limited number of elements or objects. For example, when we deal with finite sets, we can use finite representations to represent the elements in the sets. In this case, the advantages of finite representation are intuitive, simple, easy to understand and operate. However, the disadvantage of finite representation is that it can't handle an infinite number of elements or objects effectively.
On the contrary, infinite representation is mainly used to deal with an infinite number of elements or objects. For example, when we deal with an infinite sequence or series, we can use infinite representation to represent the items in the sequence or series. In this case, the advantage of infinite representation is that it can effectively deal with infinite elements or objects, and can further study the properties and laws of infinity. However, the disadvantage of infinite representation is that it is usually more complicated and more difficult to understand and operate than finite representation.
In addition, finite representation and infinite representation have different applications in different fields of mathematics. For example, in algebra, finite representation is often used to represent finite dimensional spaces such as matrices and vectors; In analysis, infinite representation is often used to represent infinite dimensional spaces such as functions and series.
Generally speaking, finite representation and infinite representation have their own advantages and disadvantages and are suitable for different application scenarios. Understanding the characteristics and applications of these two representations can help us better understand and solve mathematical problems.