Representation of infinitesimals in different ways

The four types of infinitesimals discussed are dx, hyperreal numbers, surreal numbers, and nilpotent infinitesimals. Each type is represented in a different way, such as using the cardinality of the set of natural numbers for dx or a transfinite number for surreal numbers. However, they all share the common characteristic of being infinitely small compared to other numbers.
  • #1
Mike_bb
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Hello.

There are 4 types of infinitesimals:

1) dx=1/N, N is the number of elemets of the set of the natural numbers (letter N is used to indicate the cardinality of the set of natural numbers)

2) Hyperreal numbers: ε=1/ω, ω is number greater than any real number.

3) Surreal numbers: { 0, 1, 2, 3, ... | } = ω , { 0 | 1, 1/2, 1/4, 1/8, ...} = ε, where ω is a transfinite number greater than all integers and ε is an infinitesimal greater than 0 but less than any positive real number.

4) Nilpotent infinitesimals are numbers ε where ε2 = 0

Infinitesimals are represented in different ways. What is difference between them?

Thanks.
 
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  • #2
Mike_bb said:
Infinitesimals are represented in different ways. What is difference between them?
An infinitesimal difference, perhaps?
 
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  • #3
PeroK said:
An infinitesimal difference, perhaps?
Yes
 

Related to Representation of infinitesimals in different ways

1. What is an infinitesimal?

An infinitesimal is a mathematical concept that refers to a quantity that is infinitely small, but not equal to zero. It is often denoted by the symbol "dx" or "dy" and is used in calculus to represent the change in a variable.

2. How are infinitesimals represented in standard calculus?

In standard calculus, infinitesimals are represented using the limit concept. This means that the infinitesimal is defined as the limit of a function as the independent variable approaches zero. This representation is also known as the "epsilon-delta" definition.

3. What is the difference between standard calculus and non-standard calculus?

Standard calculus uses the limit concept to represent infinitesimals, while non-standard calculus uses a different approach called the "hyperreal" approach. In non-standard calculus, infinitesimals are treated as real numbers that are smaller than any other real number, but not equal to zero.

4. How are infinitesimals represented in non-standard calculus?

In non-standard calculus, infinitesimals are represented using a number system called the hyperreal numbers. This system includes infinitely small numbers, called infinitesimals, as well as infinitely large numbers, called hyperreals. Infinitesimals are defined as numbers that are infinitely close to zero but not equal to it.

5. Why are infinitesimals important in mathematics?

Infinitesimals are important in mathematics because they allow us to accurately describe and analyze continuously changing quantities, such as velocity and acceleration. They also play a crucial role in the development of calculus and its applications in various fields, including physics, engineering, and economics.

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