Infinitesimal

Imagine a number that is not zero. But it is also not big. It is smaller than any number you can name. We call these tiny numbers infinitesimals.

Long ago, a man named Archimedes used them. He used them to find the area of shapes. Later, people like Newton and Leibniz used them too. They used them to help build calculus.

Some people thought these numbers were not real. But they are part of special number systems. One system is called the hyperreal numbers. These systems let us use tiny and huge numbers together. It is a way to see the very small.

An infinitesimal is a number that is not zero. But it is smaller than any regular number you can name.

These tiny numbers were used to build calculus. Calculus is a way to study how things change. Early math experts like Newton and Leibniz used them. They used tiny amounts to find slopes or areas.

Some people did not like these ideas. In the 1600s, some leaders even banned them. Later, mathematicians found better ways to do math without them. They used something called limits. A limit looks at what happens as a number gets very close to a point.

But infinitesimals came back in the 1900s. A man named Abraham Robinson showed they could be used in a formal way. He used the hyperreal numbers. This is a special number system. It includes both tiny infinitesimals and huge infinite numbers.

In this system, the tiny and huge numbers are linked. They are like opposites. If you have a huge number, its flip side is a tiny one. This helps us study the very small and the very large at once.

Imagine an object so tiny that it is smaller than any measurement you could ever take. It is not zero, but it is closer to zero than any regular number you can name. In mathematics, we call such a quantity an infinitesimal. The word comes from a 17th-century Latin term meaning the "infinitieth" item in a sequence. While these numbers do not exist in our standard real number system, they do exist in others. One such system is called the hyperreal number system. In this system, infinitesimals and infinite numbers are linked as reciprocals of one another.

How do these tiny numbers work in math? They are often used to build things like the derivative, which is a way to measure change. Early mathematicians thought of a derivative as a ratio of two infinitesimal quantities. You can also use them to find areas by adding up an infinite number of them. This is similar to how a shape might be made of many thin slices. Some people use them to look at the slope of a line. Even if a number is infinitely small, it can still have properties like an angle. This allows math to handle things that are nearly zero without actually being zero.

History shows that people have wondered about the very small for a long time. In the 3rd century BC, the Greek mathematician Archimedes used a method of indivisibles. He used this to find the areas of shapes and the volumes of solids. Later, in the 1655 book Treatise on the Conic Sections, John Wallis introduced the expression 1/∞. This symbol was a way to represent an infinitesimal. Around 1670, thinkers like Gottfried Wilhelm Leibniz and Nicolaus Mercator helped introduce modern concepts. Leibniz used these tiny amounts to help invent calculus.

Not everyone agreed that infinitesimals were the right way to do math. In 1632, clerics in Rome even issued a ban on them. In the 1700s, famous mathematicians like Leonhard Euler used them routinely. However, in the 1800s, others like Cauchy and Weierstrass moved toward using limits instead. They wanted to use standard real numbers to solve problems. This led to many years of debate about whether infinitesimals were real or just ideas. It was not until the 20th century that they regained popularity. In 1961, Abraham Robinson developed nonstandard analysis to show they could be used formally.

Today, we can see how these ideas connect to the world around us. In everyday speech, an infinitesimal object is something too small to be measured. In math, it is a way to bridge the gap between zero and the smallest real number. The hyperreal system uses these numbers to create a special kind of continuum. This includes both the tiny infinitesimals and the huge infinite numbers. This connection helps mathematicians study growth and change in a very precise way. It turns a mysterious idea into a tool for deep discovery.

In mathematics, an infinitesimal is a non-zero quantity that is closer to zero than any non-zero real number. While these quantities do not exist in the standard real number system, they are central to other systems. For example, the hyperreal number system includes both infinitesimal and infinite quantities. These two types of numbers are reciprocals of one another. This means an infinitesimal is essentially the inverse of an infinite number.

In common speech, an infinitesimal object is something so small that it cannot be distinguished from zero by any available means. However, in a mathematical context, it remains a distinct entity that is smaller than any standard real number.

To understand how these numbers function, one can look at how they build complex mathematical structures. In the development of calculus, the derivative was originally conceived as a ratio of two infinitesimal quantities. This allows mathematicians to measure change by comparing two incredibly small amounts. Another application is found in integration, where an infinite number of infinitesimals are summed to calculate an area. This is similar to the idea of taking infinitely thin building blocks to form a shape. Even when a quantity is infinitely small, it can still retain certain properties like a slope or an angle. This allows for a way to handle change and motion without the values simply becoming zero.

There are several ways to categorize number systems that include these tiny quantities. One type is an ordered field, which obeys the usual axioms of the real number system that can be stated in first-order logic. An example of this is the field of Laurent series with a finite number of negative-power terms. Another category is a real closed field. This is a stronger condition where the system includes all first-order properties of the real number system. This includes the existence of a root for every odd-degree polynomial, such as a cube root. A third, even stronger category involves systems that have all first-order properties of the real number system regardless of the relations used. This could include well-defined sine functions for infinite inputs.

The history of the infinitesimal stretches back to ancient Greek thought. The Eleatic School discussed infinitely small quantities, and Zeno of Elea proposed the dichotomy paradox. This paradox considered the relationship between a finite interval and an interval approaching an infinitesimal size. In the 3rd century BC, Archimedes provided a logically rigorous definition in his work, *The Method of Mechanical Theorems*. He used a method of indivisibles to find the areas of regions and the volumes of solids. Archimedes also defined the Archimedean property, which describes a system that contains no infinite or infinitesimal members.

During the 17th century, the concept underwent significant development and controversy. In 1655, John Wallis introduced the expression 1/∞ in his book *Treatise on the Conic Sections*. This symbol represented the reciprocal of infinity, acting as a symbolic representation of an infinitesimal. Around 1670, the modern concept was introduced by either Nicolaus Mercator or Gottfried Wilhelm Leibniz. Leibniz used infinitesimals as a basic ingredient in his version of calculus. He relied on the law of continuity, which suggests that what works for finite numbers also works for infinite numbers. He also used the transcendental law of homogeneity to replace unassignable quantities with assignable ones.

Despite their utility, infinitesimals faced intense scrutiny and even bans. In 1632, clerics in Rome issued a ban on infinitesimals due to religious and political controversies. Later, Bishop Berkeley attacked their use in his work *The Analyst*. By the 19th century, mathematicians like Augustin-Louis Cauchy, Karl Weierstrass, and Georg Cantor sought to replace infinitesimals with the concept of limits. They developed the (ε, δ)-definition of limit to provide a more rigorous foundation using standard real numbers. For a time, many philosophers and mathematicians viewed infinitesimals as mere pseudoconcepts that lacked mathematical reality.

The 20th century brought a major shift in how these quantities were viewed. In 1961, Abraham Robinson developed nonstandard analysis, which provided a formal treatment of infinitesimal calculus. This work built upon earlier research by Edwin Hewitt in 1948 and Jerzy Łoś in 1955. Robinson's hyperreal numbers implemented an infinitesimal-enriched continuum. This proved that the ideas used by Newton and Leibniz could be made mathematically rigorous. Following this, mathematicians developed surreal numbers. This system is the largest ordered field and includes both hyperreal cardinal and ordinal numbers.

Today, the study of infinitesimals connects many different branches of mathematics. They relate to the study of growth rates in functions, a field explored by Paul du Bois-Reymond. His work inspired mathematicians like Émile Borel and Thoralf Skolem, who developed non-standard models of arithmetic. The concept of the infinitesimal bridges the gap between the finite and the infinite. It allows for the study of the continuum in ways that standard real numbers cannot always capture. Whether through hyperreals or surreal numbers, these tiny quantities remain a powerful tool for understanding the structure of the mathematical universe.