Nonstandard calculus

In mathematics, nonstandard calculus is a special way of doing calculus. It uses very tiny numbers called infinitesimals. These tiny numbers help make calculus ideas easier to understand.

For a long time, many mathematicians used infinitesimals. But they didn’t have a clear way to explain how they worked.

Later, some mathematicians thought infinitesimals were too unclear. But in 1960, a mathematician named Abraham Robinson showed that infinitesimals could be used in a clear and exact way. He used ideas from other mathematicians like Edwin Hewitt and Jerzy Łoś. This made nonstandard calculus very useful in modern mathematics. According to Howard Keisler, Robinson’s work solved a problem that had lasted for over 300 years. It was a big advance in math during the 20th century.

History

The history of nonstandard calculus began 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 helped with 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 strong mathematical foundation again through a method called nonstandard analysis. This new approach made calculus rules easier to understand.

Motivation

To find the slope of a line on 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 very small 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, ( \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 calculus using very small numbers called infinitesimals. It talks about continuity on page 125, the derivative on page 45, and the integral on page 183, all using these small numbers. The epsilon, delta method is introduced later, on page 282.

Main article: Elementary Calculus: An Infinitesimal Approach

Definition of derivative

The hyperreals help us understand very small numbers, called infinitesimals. These numbers are smaller than any normal number but still bigger than zero. In nonstandard calculus, we can find the derivative of a function more easily. We do not need long limiting processes. Instead, we use a simple formula to find the exact rate of change at any point.

The formula uses the standard part function standard part function. This function finds the closest real number to a hyperreal number. This makes calculations with very small numbers much easier.

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.

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 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 closely at how this function behaves around a number like π (pi). By using special “infinite” numbers, we see that the function changes values very near to π. This shows that the function is not continuous there and helps us understand why it jumps between values.

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

Limit

In nonstandard calculus, we can understand limits using something called the standard part function st. This helps us describe how a function's value gets very close to a certain number when the input gets very close to a point.

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

One good way to show the intermediate value theorem is by using tiny numbers, thanks to Robinson's idea. This theorem says that if you have a smooth line that goes from positive to negative (or vice versa) between two points, there must be a spot in the middle where the line crosses zero.

We can show this by splitting the space between the two points into lots of tiny pieces using something called a hyperinteger. By checking where the line is positive and using the transfer principle, we can spot exactly where the line hits zero. This way makes the proof easier by cutting out complicated thinking steps.

Basic theorems

In nonstandard calculus, we study functions using special numbers called infinitesimals. These numbers are very small. If we have a function f between two points a and b, we can use a tool called the transfer operator. This creates a new function *f. It helps us understand how f behaves.

One important idea shows that if a function f is differentiable at a point a, we can use infinitesimals to find its derivative. This gives us a new way to see how functions change. It makes some calculus ideas easier to use.