In previous post, on WHY STUDY COMPLEX NUMBER (http://math2ever.blogspot.com/2013/05/why-study-complex-number.html), writer has explained the reasons learning complex numbers which also answering some hard-to-answer students' questions.To understand the basic of complex number, refer to the following video. It is one of my favorite video when come to complex number.
Imaginary numbers are multiples of i. Like 2i, 4i, 2.881i etc... They arise because not all equations can be solved by using only real numbers. Real numbers are all numbers ranging from - infinity to + infinity.
For example, we cannot solve x^2 + 1 = 0 using real numbers. The square of a positive number is positive. The square of a -ve number is also positive. Thus, the square of x is always positive and on adding a positive number to 1, we can never hope to get 0.
This is where imaginary numbers step in. If we "define" the square root of -1 to be i, then we can solve the above equation by saying x = i. Any equation of the form x^2 + a^2 can now be solved. The answer will be x = i*square_root(a).
Some equations can not be solved with just real numbers or just imaginary numbers. But they can be solved with a sum of real and imaginary number. Such a "composite" number is called a complex number. It's of the form a + ib where a and b are real numbers.
The application of complex numbers apply to many branches of science. Engineering is full of them, especially electrical engineering.
A real number can be represented graphically on the x-axis of a graph. How do we represent imaginary numbers or complex numbers? The trick is to realise that multiplying any real number a with i two times leads to i*i*a = -a. Thus, it represents a rotation of 180 degrees. So i alone must represent a rotation of 90 degrees. Thus, imaginary numbers can be represented by the y-axis, which is perpendicular to the x axis. Complex numbers, being the sum of a real part and an imaginary part, can be represented by a point on the plane defined by the x and y axis. Thus, a + ib is synonymous with the point (a,b) on the xy plane.
P.S. I disagree about complex numbers having no physical representation. I just gave a physical representation. Do real numbers have a physical representation? Have u ever "seen" or "felt" 1 or 2 or 3.14159? You might have seen 1 apple or 2 apples but they aren't numbers. Numbers would remain in existence even if apples became extinct.
