# What simple rigorous ways are there to incorporate infinitesimals into the number system and are they then useful for basic Calculus?

Here's about the simplest way, but the explanation gets a little long...

To achieve this, we must be able to "cancel out" common factors that exist between the denominator and the numerator.

Consequently, we think about:

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Here's a little discussion of application...

Is basic calculus useful in this field?

Has anyone benefited from this?

In a way, it takes the "standard part" and uses infinitesimal arithmetic to replace the limit process.

I apologize for the somewhat informal nature of this discussion, but I do not particularly like this "standard part" mechanism, and it would be fairly simple, albeit time-consuming, to make it more rigorous.

If:

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One simple rigorous way to incorporate infinitesimals into the number system is through non-standard analysis, which was developed by Abraham Robinson. In this approach, infinitesimals are treated as actual numbers, called hyperreal numbers, which are larger than zero but smaller than any positive real number. These hyperreal numbers can be used to extend the real number system and provide a framework for calculus.

Non-standard analysis allows for a more intuitive treatment of calculus concepts, such as limits, derivatives, and integrals. It provides a formal framework to reason with infinitesimals and allows for more concise and elegant proofs in calculus. However, it is important to note that non-standard analysis is an alternative approach to traditional calculus based on limits, and both approaches are mathematically equivalent.

In basic calculus, the use of infinitesimals can simplify calculations and provide a more intuitive understanding of concepts. For example, infinitesimals can be used to define derivatives as ratios of infinitesimal changes, leading to a more direct interpretation of the derivative as the instantaneous rate of change. Infinitesimals can also be used to define integrals as sums of infinitesimal contributions, making it easier to understand the concept of area under a curve.

Overall, incorporating infinitesimals through non-standard analysis can be a useful tool in basic calculus, providing a different perspective and facilitating calculations and conceptual understanding.

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One simple and rigorous way to incorporate infinitesimals into the number system is through non-standard analysis, specifically using the hyperreal numbers. In this approach, infinitesimals are treated as numbers that are smaller in magnitude than any standard real number but are not zero.

Hyperreal numbers extend the real numbers to include infinitesimal and infinite quantities while maintaining the usual rules of arithmetic. This allows for a more intuitive and rigorous treatment of concepts like limits, derivatives, and integrals in calculus.

Hyperreal numbers can be useful in basic calculus for providing a more intuitive framework for understanding concepts such as instantaneous rates of change (derivatives) and areas under curves (integrals). They can help bridge the gap between algebraic manipulations and geometric interpretations, making calculus concepts more accessible and intuitive for students.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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