History of birth of Imaginary numbers
Historians do not dispute that the Egyptians were ahead of their contemporaries in knowledge. Various aspects of their knowledge emerged around their famous Kirti Pyramid. In mathematics, the shape of a pyramid or a cone has a special significance. What is obtained by cutting off the head of a cone or pyramid is called a frustum in mathematical terms.
The Egyptians had to determine the volume of this frustum at various stages of the construction of the pyramids. The ancient Egyptians, advanced in knowledge, themselves established a formula for determining the volume of the frustum. The formula was, Here, a and b denote the sides of the vertex and the length of the ground of the frustum respectively. However, determining the volume of a frustum by directly measuring its height, although easy nowadays, was quite a difficult task then.
So the Egyptians used the frustum's diameter for their convenience to determine its height. In that case the formula for determining their height was: Here C is half height.
The Egyptians once got into quite a quandary in determining the height of the pyramids by this formula. The base of the pyramid was 28 meters, the side of the apex was 4 meters and the height was 15 meters. Putting all the values into the equation gives height = √-63. No one knew then what the value of the square root of this negative number would be. The famous mathematician Heron of Alexandria came forward to solve this problem. He solved this problem by saying, √63 and √-63 are actually the same thing! As a result, there was no obstacle in determining the volume of the frustum. But that was actually wrong. And with this simple solution, Heron missed a golden opportunity to discover complex numbers. Even medieval mathematicians considered the square root of negative numbers to be impossible, imaginary or false. However, around the third century AD, Diophantus of Alexandria and Mahavir Acharya of India around 850 AD proposed the theoretical possibility of the square root of negative numbers. Between 1450 and 1600 AD, two famous mathematicians, Del Ferro and Cardano, revived the concept of the irrational number. It was not until the eighteenth century that the irrational number gained its desired recognition. And since then, it has become common to write complex numbers by combining real and unreal numbers. But mathematicians are very confused about the expression of complex numbers. Cardano was the first to introduce complex numbers into algebra. He expressed it as (a+√-b). But he failed to give any answer about this method of expressing complex numbers.
Just then Leonardo Euler arrived like an angel. In 1777 he was the first to denote the negative square root of one by the English letter i. He also suggested expressing complex numbers by a point in the Cartesian coordinate system. But unfortunately he could not give the basic theory of complex numbers. Euler's fear of negative numbers is responsible for this. At present, we can quickly solve a simple equation, such as: x+5=2. But it was too difficult an equation for Euler, an eighteenth-century mathematician. Because he didn't know how to use negative numbers in practice. Even once he said, negative numbers can be greater than infinity. So the general public, and even the mathematicians of the day, tried to avoid the problem of negative numbers.
However, five hundred years ago, an unprecedented mathematical problem arose in Europe, due to which mathematicians could no longer ignore negative numbers. The original proponent of the problem was the Italian mathematician Del Ferro. He was trying to solve the quadratic equation at that time. Its general form is (ax3+bx2+cx+d=0). Del Ferro makes a slight variation on this simple equation. He dropped the quadratic part of x in the equation and took the value of d i.e. the constant part to be negative. Note that if a quadratic term is absent in a quadratic equation, it is called a depressed quadratic equation. But as mentioned earlier, people still frowned upon negative numbers. So he cleverly rearranges the equation to write ax3+cx= d, where c and d are both positive integers. He then proceeded to find the value of x.
After many calculations, he found a simple solution to the equation he created, like a quadratic equation. But Del Ferro kept that solution hidden from the public eye. Finally, on his deathbed, he revealed the secret solution to his disciple Antonio Foir. After Foir discovered the solution, he became determined to defeat his old enemy, the famous Italian mathematician Fontana Tartaglia. He challenged Tartaglia to solve Del Ferro's mathematical problem. Tartaglia did not hesitate, he accepted the challenge. On the brink of the deadline, he not only found a solution to the problem, but on the contrary threw a counter-challenge. Because he discovers a significant flaw in Foir's and Del Ferro's solution. He shows mathematically, that for some specific values of a, c and d (eg: a=1, c=15, d=4) Fourier's solution is not rational. Because in that case it is necessary to find the square root of the negative number. The famous mathematician Cardano tried to solve the problem after learning about it. Although he discovered a rule for the practical application of negative numbers, he did not go very far in solving the problem at hand. After Cardano's death, his student Bombelli resumed research on the problem.
Bombelli was different in thought from his predecessors. He thought that if there was no solution with positive or negative numbers, there was another number that would solve the equation. So even though everyone ignores the square root of a negative number, he recognizes it as a number. But he gave √-1 the status of number instead of any other number or symbol. Then a new problem arose. Is √-1 actually a real number? So it was a very difficult task for Bombelli to place the newly discovered numbers on the number line. He knew that the number line had been changed several times before. Time has come for another revolutionary change. But before that, he tried to present Cardano's equation in a simpler way. Mathematically, he showed that adding the two terms in the existing solution to Tartaglia's problem no longer contained the term containing √-1. Then the entire solution becomes a real number. He then defined the two terms in the following way. Then he freed the cube from the equation and did some calculations to finally arrive at the following two equations. After solving the above two equations he got a=2 and b=1. Next, substituting the values of a and b into Tartaglia's equation gives the solution 4. And with this, the main problem of Cardano's equation is eliminated. But in that case the condition is the same. √-1 should be given number status. But in normal thinking he could not be placed anywhere on the number line. Because no one could imagine √-1 as a number in the real world. That is why it is called Imaginary number. But actually √-1 is a number. It exists on the number line of complex numbers. It even has its own pattern.
This idea came about 100 years after Bombelli's solution. In 1777, the Swiss mathematician Leonardo Euler expressed the square root of a negative number by changing the square root of the number in real or positive form and multiplying it by the square root of -1 (eg: √-63= √63*√-1). Later he denoted it more briefly by i (√63 i). As a result, a novel number system is formed by combining real and unreal i.e. real coefficients of √-1 (ie: 6+3i). Which is called complex number. Even Euler's equation, known as the most beautiful equation in the world, is arranged in terms of i. But the traditional problem of placing i on the number line has not been solved. The problem, to put it another way, is to find a number which, when multiplied by two, gives a negative number. But no such number was available in the real number line. For example, take the number 3. Multiplying 3 by 3 gives 9. That means whether you start from the right or left side of the number line, the result will change 180 degrees and always lie on the positive side of the number line.
To solve this problem, mathematicians introduced the idea of a number such that when the number is multiplied twice, the result rotates 90 degrees instead of 180 degrees. This is exactly how irrational numbers work. Increasing the exponent of a real number increases its value. But increasing the exponent or exponent of i to an irrational number does not increase its value, but creates an interesting pattern of its value. After every 4 multiplications the same number keeps repeating itself. And thus the complex number line begins. Where imaginary or imaginary number lines join at right angles to real number lines. This results in a two-dimensional number line. And with that, there is a radical change in people's basic thinking about numbers. But if an electrical engineer says j=√-1, then there is no reason to be upset. Because in solving various electrical problems they denote current flow with i. (Note, until 1894, current flow was expressed as C in a journal of electrical engineering. Then they first decided to express the name of the capacitor or container with C. And chose to use Cu (the first two letters of the word Current) as the signal for current flow. But then chemists objected to this because copper is represented by Cu in the periodic table. To clear up any doubts, the unit of current is named in honor of the French scientist Andre Marie Ampere. The French translation of current density is 'intensité du courant'. The initial i is designated as the current symbol). So to avoid calculation confusion, they denote the unreal numbers by j.
