COMPLEX NUMBERS: REALITY IN IMAGINARY CONCEPT

 

If we study carefully, the discussion here is fundamentally about ‘numbers’. However, the word ‘complex’ is attached to it, indicating a new category of numbers. As we know, the real application of mathematics lies in calculation and formulation, and these tasks are performed with the help of numbers. But mathematics does not consist solely of natural numbers; rather, it includes various types of numbers such as positive numbers, negative numbers, rational numbers, irrational numbers, etc., but each with its own application.

An important observation here is that all these types of numbers exist on a single number line, and most mathematical calculations are also tied to this number line. But a question arises here: Is a single number line sufficient for all real-world calculations? The answer to this question is: "No! A single number line is not enough." Just as we perceive three dimensions on land but only one dimension underwater, similarly, in mathematics, there are certain calculations where this number line becomes useless. In such situations, we need a new number line, or rather, an entirely new kind of number system. This number system is designed for highly complex calculations, so the term ‘complex numbers’ is used for it.

Applications of Complex Numbers in the Real World:-

Most applications of complex numbers are aimed at simplifying calculations involving multiple dimensions, such as in wireless communication systems. For instance, in electrical circuits, whenever capacitors and inductors are used, we employ complex numbers to simplify calculations related to circuit parameters. The primary reason for this is the multidimensional flow of electric current through these components. Similarly, electromagnetic waves exhibit a multidimensional flow in wireless communication, and calculating their parameters using just a single number line (which represents only one dimension) is insufficient. Hence, an extra dimension or an extra number line is introduced.

However, if this extra number line were identical to the real number line, then calculations would not work correctly. Unlike in physics, where concepts like vectors and scalars exist, mathematics requires a different approach. Therefore, an imaginary number line was created. On this imaginary number line, we encounter a new term: ‘i’ (iota). Its defining characteristic is that its square is a negative number, which is otherwise impossible. This is precisely why these numbers are called imaginary numbers.

Now the question arises: Why was the square taken as negative? The answer lies in the study of the motion of objects in free space. When mathematicians began analyzing such phenomena, they encountered highly complicated calculations. To solve them, they relied on quadratic equations, but a new problem emerged: the presence of negative numbers under the square root. To simplify these calculations, mathematicians invented iota (i). This invention proved immensely helpful.

In scenarios where movement is highly unpredictable, or where the three dimensions of physics fail to apply, this imaginary number iota comes into play. In wireless communication, information is transmitted through energy packets, which travel via waves that are not confined to any single fixed dimension. Thus, without the invention of the iota, wireless communication would have been impossible.

So, learn the applications of complex numbers in solving real-life problems. But it requires a lot of practice, and practice comes from continuously solving questions. That is why, on this website, you will find numerous practice questions with their detailed solutions related to probability and other topics of mathematics. These practice questions will be very useful for various competitive exams like IITJEE, GATE, CAT, GMAT, etc. So, find questions according to your topic and practice as much as you want.

CHECK HERE THE LIST OF ALL PRACTICE QUESTIONS