What is Infinitesimals?

Considering what the opposite of infinity is, many people would characterize it as "zero". However, what we are looking for is a value that is not zero, but slightly larger than zero, but smaller than any other number imaginable. This concept is characterized as infinitesimal. The infinitesimals are used to express small values ​​that are nearly immeasurably large. Although they are very small, they still have some features such as slope and angle, so these values ​​are used in various fields.

Just like the concept of infinity, the concept of infinitesimals is an abstract concept. We can exemplify this situation as follows: Since there is always another number between every number, there are as many real numbers as we can think of (perhaps billions, trillions, or even more) between the integers 0 and 1. We can consider one of these numbers as 0.00000000000001. Although this may seem like an infinitesimal, it is actually quite large compared to many other numbers in the same range (since we can push the limits of our imaginations when it comes to real number generation). For example, the number 0.000000000000000000000001 is 10^9 times the previous value. That is, one billionth of the previous value. In addition, it is sufficient to divide these values ​​by a real number greater than 1 in order to obtain an infinitesimal number smaller than these numbers. Accordingly, like the infinite, the infinitesimal also exists only through abstraction.

In order to exemplify the infinitesimal calculation with the figure below, let's try to get the areas of the circles by adding the areas of the triangles inside them. When the circle on the right is divided into 4 triangles, a large amount of space remains inside the circle. According to this figure, it cannot be said that the sum of the areas of the triangles is equal to the area of ​​the circle. To reduce this error, more triangles with smaller sizes should be placed inside the circle. As a result, the error decreases, but again, the sum of the areas of the triangles is not exactly equal to the area of ​​the circle, so the result is again incorrect.

To avoid the error completely, the circle must be divided into an infinite number of triangles. Thus, an infinite number of bases of an infinite number of triangles will almost flawlessly cover the circumference of the circle. Derived from an infinite number of triangles, these bases are incredibly small, so the curvature between the triangle bases is almost gone. In this way, the concept of infinitesimals is well illustrated. If we continue on the figure on the left to explain the situation better: When the infinite number of triangles covering the circle are removed from the circle and joined at one point, the base of the new triangle formed is almost equal to the circumference of the previous circle. Accordingly, it is possible to almost find the area of ​​the circle in this way. When put into action, this judgment is correct, but in practice it is erroneous.

In order to reach a correct result by means of this path, the base must be infinitesimal. But no matter how thin a triangle is drawn, it is known that a smaller triangle exists. Although the resulting word "almost" is not very pleasant to mathematicians, many ignore it. Because according to the mathematical process, the result is not wrong. For this reason, the case of the infinitesimal for a long time caused controversy among mathematicians.