People created mathematics to understand and interpret nature. Some people developed measurement units to transfer a certain length to another, while others created numbers to understand their assets. Mathematics was formed as a result of such a great sequence.
Today, mathematics has been divided into billions of cells in order to make sense of certain subjects, various cells have grown within themselves and formed the basis of classical physics, modern physics, chemistry, biology and engineering, which is a magical combination of these.
In today's blog post, we will talk about an element we often hear about in this magical world: Integral!
What is the integral that most of us have heard of at least once in our lives? When there is no motivation, should you invert the number in front of a variable, multiply it by itself and increase the variable by one? If so, wouldn't the question "What good will this unnecessary procedure do for us?" come to our minds?
In order to answer these questions, we will discuss what the integral actually is, which we often think of as an unnecessary operation.
What is Integral?
Everything in the world is based on the harmony of words. For this very reason, even sciences, which at first glance seem to be the disciplines most unrelated to literature and language, actually contain language and literature within them. Of course, mathematics has also benefited from the blessings of language. Starting from here, we We will approach the integral through its literal meaning.
The word integral has passed into our language from English, and into English from the French word "integral". Although it is not widely used, its original Turkish equivalent is tümlev. These words, which have semantic integrity, are used in the meanings of complement, integration and combination and summarize the integral quite nicely.
Integral, from the point of view of algebra, is actually an addition operation. In other words, it is a simple addition operation like “two plus two equals four.” On the other hand, this addition process is such that you can objectively add everything with this process. Let's try to explain this ability to collect everything more clearly with an example;
Let's imagine we have two apples in our hands. If we pick up two more apples, the total number of apples we have will be four. This is a very simple addition process. But what if we want to know the surface area, volume or mass of apples? At this point, integral comes into play and tells us that we can find these elements through the addition process.
Imagine finding the area of a cell on the surface of an apple, then adding the area of all the cells on its surface to find the surface area of the apple. This is the integral. If we want to summarize it in one sentence, integral means converting the area of objects whose shape is not regular, such as a square (the area of a square is the product of its two sides) and whose area cannot be easily calculated, into millions (or even theoretically infinite) tiny calculable particles (e.g. squares), calculating these areas, and then It is the process of collecting these fields.
A Fundamental Element of Different Branches of Science and Engineering
Let's concretize this process with a small example; When you look very closely at a digital screen, you will see that the image on the screen is made up of small squares called pixels. By finding the area of the pixels on this digital screen and adding the number of pixels contained in the image, we can obtain the current area of the image on the screen. This addition process, known as Riemann sums, is just one example of the elements that the integral adds.
In order to show that the integral can even collect invisible elements, let's give another small example using the terms FM and AM in radio terminology. Frequency Modulation, which stands for FM, means “frequency modulation”, and Amplitude Modulation, which stands for AM, means “amplitude modulation”. Modulation means combining an information signal (for example, sound) with a carrier signal (frequency settings such as 103.5 that we set on radios). At this point integral comes into play.
We need to sum the carrier signal with the information signal moment by moment, and we can only express this algebraically by integral. The integral we use in this process is called the convolution integral. As can be seen, although the function of the integral is basically to integrate, combine and add, this integration process can take many different forms. As a result, although the integral consists of a basic addition operation, it does not fit into simple memorized rules, on the contrary, it is a game changer in science; It is one of the most basic mathematical elements used in informatics, electronics, machines, structures, biology, economics, statistics, all kinds of science and engineering.
Ahmet Cemal Kurtulmuş
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