Nonstandard calculus

In mathematics, nonstandard calculus is a way of doing calculus using tiny numbers called infinitesimals. These infinitesimals help make some calculus ideas easier to understand and work with. Before the 1870s, many mathematicians used infinitesimals in their work, even though they didn’t have a solid way to explain why they worked.

For many years after that, some mathematicians thought infinitesimals were too vague and didn’t make sense. But in 1960, a mathematician named Abraham Robinson showed that infinitesimals could be used in a clear and exact way. He built on ideas from other mathematicians like Edwin Hewitt and Jerzy Łoś. This discovery made nonstandard calculus a powerful tool in modern mathematics. According to Howard Keisler, Robinson’s work solved a problem that had lasted for over 300 years and was one of the big advances in math during the 20th century.

History

The history of nonstandard calculus started with the use of very small numbers, called infinitesimals, in calculus. Both Gottfried Leibniz and Isaac Newton used these tiny numbers in the 1660s to help build the foundations of calculus. Other mathematicians like John Wallis, Pierre de Fermat, Isaac Barrow, and René Descartes also contributed to these early ideas.

Later, some mathematicians preferred using limits instead of infinitesimals. Karl Weierstrass created a way to base calculus on limits, which became known as standard calculus. In the 1960s, Abraham Robinson gave infinitesimals a solid mathematical foundation again through a method called nonstandard analysis. This new approach used advanced logic to create numbers that include infinitesimals, making calculus rules easier to understand.

Motivation

To find the slope of a line for a curve, we look at how small changes in the input affect the output. Imagine you have a function like ( y = x^2 ). If you change ( x ) by a tiny amount, called ( \Delta x ), you can see how much ( y ) changes, called ( \Delta y ). By dividing these changes, you get an approximation of the slope.

In traditional calculus, we make ( \Delta x ) smaller and smaller until it becomes zero to find the exact slope. In nonstandard calculus, instead, ( \Delta x ) is considered an infinitesimal — a number so small it's closer to zero than any real number, but not actually zero. This way, we can keep the simple calculations without needing to take limits, making the ideas behind calculus easier to understand.

The link between these two methods shows how both can describe the same result, just in different ways.

Main article: infinitesimal calculus

Keisler's textbook

Keisler's book Elementary Calculus: An Infinitesimal Approach explains important ideas in calculus using infinitesimals. On page 125, it talks about continuity using these tiny values instead of the epsilon, delta method. The derivative is shown on page 45 with infinitesimals, and the integral is defined on page 183 the same way. The epsilon, delta method is introduced much later, on page 282.

Main article: Elementary Calculus: An Infinitesimal Approach

Definition of derivative

The hyperreals help us understand tiny numbers, called infinitesimals, which are smaller than any normal number but still bigger than zero. These infinitesimals surround every real number like a thin cloud. In nonstandard calculus, we can find the derivative of a function without using long limiting processes. Instead, we use a simple formula that gives us the exact rate of change at any point.

The formula uses the standard part function standard part function, which finds the closest real number to a hyperreal number. This makes calculations with infinitesimally small numbers much easier and more direct.

Continuity

A real function f is continuous at a standard real number x if for every hyperreal x' infinitely close to x, the value f(x' ) is also infinitely close to f(x). This idea comes from Cauchy's definition of continuity from his 1821 textbook.

The definition can also apply to nonstandard points. A function f is microcontinuous at x if whenever x' is infinitely close to x, the value of f at x' is also infinitely close to the value of f at x. This way of thinking about continuity uses fewer details than the common (ε, δ)-definition taught in regular calculus classes.

Example: Dirichlet function

The Dirichlet function is a special math function that helps us understand continuity. It gives the number 1 if a value is a rational number (like fractions) and 0 if it is an irrational number (like the square root of 2).

Using nonstandard analysis, we can look at how this function behaves very closely around a number like π (pi). By using special "infinite" numbers, we see that the function changes values very near to π, showing that it is not continuous there. This helps us understand why the function jumps between values.

Main article: Dirichlet function Main articles: standard definition of continuity, hypernatural, transfer principle

Limit

In nonstandard calculus, the idea of a limit can be understood using something called the standard part function st. This helps us say that the limit of a function as it gets very close to a certain point is another number.

If the difference between x and a is very tiny (infinitesimal), then the difference between the function's value at x and the limit L is also very tiny. This gives a clear way to work with limits without using complex rules.

Limit of sequence

A sequence is a list of numbers, like 2, 4, 6, 8, and so on. Sometimes we want to know what these numbers get closer to as we go further in the list. We call this the limit of the sequence.

In regular calculus, we use a special way to find limits called the (ε, δ)-style. But in nonstandard calculus, we have another way using something called "hypernatural" numbers. This method is simpler because it doesn't need as many steps. Both ways help us understand what happens to numbers in a sequence when we go really far out.

Intermediate value theorem

A powerful way to prove the intermediate value theorem is by using infinitesimals, thanks to Robinson's approach. This theorem says that if you have a continuous function that changes sign between two points, there must be a point in between where the function equals zero.

The proof starts by dividing the interval into very many equal parts using something called a hyperinteger. By looking at where the function is positive and using the transfer principle, we can find a specific point that gives us the zero we need. This method makes the proof simpler by reducing complex logical steps.

Basic theorems

In nonstandard calculus, we study functions using special numbers called infinitesimals, which are incredibly small. If we have a function f defined between two points a and b, we can use a tool called the transfer operator to create a new function *f that helps us understand the behavior of f in more detail.

One important result shows that if a function f is differentiable at a point a, then its derivative can be understood using these tiny numbers, called infinitesimals. This gives us a different way to look at how functions change, making some calculus ideas easier to work with.