The purpose of this research paper is to introduce the topic of “Complex and Imaginary Numbers” and its applications. I chose the topic “Complex and Imaginary Numbers” because I am interested in mathematics that is hard to be pictured in your mind, unlike geometry or equations. An imaginary number is the square root of a negative number. That is why they are called imaginary, what René Descartes called them, because he thought such a number could not exist. In this paper, I will discuss how complex numbers and imaginary numbers were discovered, the interesting math of complex numbers, and how they are used in other areas of mathematics and science. Complex numbers are applied in engineering, control theory and improper integrals to take the place of certain imaginary values, as well as to simplify some explanations. 2. THe Concept

2.1 History of imaginary numbers

Long ago in ancient Greece, there was a society of mathematicians called Pythagoreans who believed the only numbers were natural numbers and positive rational numbers (Rusczyk 357). Later, Hippasus discovered irrational numbers such as√2, then 0 and negative numbers were introduced. This completes the real numbers set, but mathematicians at that time have not even thought of the set of imaginary numbers yet. The first appearance of complex numbers was made by Heron of Alexandria in the 1st AD (Complex Numbers-Wikipedia, the free encyclopedia). He merely replaced the inside of the square root as its positive, thinking he made an error. It was not until 1545 they were really used in mathematics when Cardano published “Ars Magna”, where he mentioned imaginary numbers. Later Bombelli set rules for arithmetic of imaginary and complex numbers in his book “Algebra”. Then, Euler assigned the letter “i” to denote the square root of -1. At the time of its discovery, many mathematicians like Descartes thought imaginary numbers to be useless and fictitious. 2.2 The mathematics of complex and imaginary numbers

A. How imaginary numbers were discoverd

As mathematics reawakened in Western Europe in the 13th century, many people became mathematicians. These mathematicians did not accept negative numbers and only thought positive numbers were entities. In the 16th century, two mathematicians tried to solve a cubic equation, Cardano and Tartaglia. In his book “Ars Magna” Cardano finds square root of negative solutions to some equations, and he calls these “fictitious” numbers (Quadratic and Cubic Equations-Dave’s Short Course on Complex Numbers). He also says that the sum of the solutions of a cubic equation is the negation of b, the coefficient of the x2 term. In one problem, he factors 10, so that their product would be 40 in order to factor the equation. He soon runs into a problem. He gets (5+ √-15) and (5- √-15). He could not finish the problem and did not go further into this problem. Later, Bombelli worked more into this problem and found the solution 4 to one of these problems. Later mathematicians understood that any algebraic equation higher than the 2nd degree could have an imaginary solution because of the Fundamental Theorem of Algebra. The Fundamental Theorem of Algebra states that an nth degree equation will have n solutions. When Gauss proved the Fundamental Theorem of Algebra, he said that each root has the form of a+bi. This is how complex numbers are written today, if the solution is real, then the i coefficient would be 0 and vice versa. b. Arithmetic with complex numbers

The letter “i” is equal to the square root of -1, so i2 is equal to -1. In a complex number, there is a real part and an imaginary part in the form of a+bi, where “b” is the imaginary part and “a” is the real part. In order to add or subtract complex numbers, add or subtract the real parts together, then add the imaginary parts. Example: (4+5i) + (7-2i) =11+3i To...