Is there a word that relates to infinitesimal in the way that zero

In summary, the conversation touches on the concept of infinitesimal and how it relates to zero and infinity. It is mentioned that in the hyperreal number system, there are two types of non-real numbers and their reciprocal relationship with each other. In the ordinary Real number system, infinitesimal and infinite numbers do not exist, leading to the use of limits in Calculus and Analysis. The concept of "hyperreals" is also mentioned, which requires a deeper understanding to define.
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Pjpic
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Is there a word that relates to infinitesimal in the way that zero relates to infinity?
 
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Pjpic said:
Is there a word that relates to infinitesimal in the way that zero relates to infinity?

I don't think that zero does relate to infinity.

In the hyperreal number system, there are two kinds of non-real numbers: infinitesimal and infinite. The reciprocal of any infinitesimal number is infinite. The reciprocal of any infinite number is infinitesimal. The inverse of zero is still undefined.
 
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In the ordinary Real number system, there are neither infinite nor infinitesmal numbers. That's why most Calculus and Analysis deal with derivatives in terms of limits rather than infinitesmals. It takes some deep theory to be able to define the "hyperreals" that tac-tics refers to.
 

FAQ: Is there a word that relates to infinitesimal in the way that zero

Is there a mathematical term that is related to infinitesimal and zero?

Yes, there is a term called "infinitesimal zero" or "infinitesimal element" that is used in mathematical analysis to describe a number that is smaller than any positive real number but still greater than zero.

How is the concept of infinitesimal related to the concept of zero?

The concept of infinitesimal is closely related to zero in that it is a number that is infinitely close to zero, but not equal to it. It is used to describe quantities or values that are approaching zero, but never actually reach it.

Can you give an example of a real-world application of the concept of infinitesimal and zero?

One example is in calculus, where infinitesimal values are used to calculate derivatives and integrals. Another example is in physics, where infinitesimal changes in variables are used to model and analyze physical systems.

Are there any other terms that are often used interchangeably with infinitesimal and zero?

Yes, some other terms that are often used interchangeably with infinitesimal and zero include "infinitely small," "vanishingly small," and "infinitely close to zero."

Is there any controversy surrounding the concept of infinitesimal and zero?

Yes, there has been some controversy and debate among mathematicians and philosophers about the legitimacy and usefulness of infinitesimal values. Some argue that they are essential for understanding and solving certain mathematical problems, while others argue that they are logically inconsistent and should not be used.

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