Imaginary unit

Imaginary unit

 Imaginary unit: brief history

    Does every quadratic equation have a real solution?

   This is the first question,  where a fresh concept or a new branch of mathematics has started. Yes, it's the story behind complex analysis or complex numbers or imaginary numbers.

  The equation behind this brand new concept is:

                   

  

The problem was to find the roots of this equation. As we all know that, one of the main features of a real number is it's square is always positive. Here, the problem started when mathematicians tried to find the root of a negative real number. i.e. x=√(-1) =???

   This is the scene when the concept of complex numbers and imaginary unit begins. 

    The square root of (-1) becomes an icon and mathematicians represented it uniquely by the symbol "i". i.e.  i=√-1.

          

 Again, since every number has two possible square roots: one positive and one negative; (-1) demands  a negative square root also. This idea introduces "-i". i.e. , -i=-√-1.

  This is the brief  history of imaginary unit.

Properties of imaginary unit 'i':

  (1) i=√-1.

  (2) i×i=(√-1)×(√-1)=-1.

  (3) i×i×i=(i×i)×i=(-1)×i=-i.

  (4) i×i×i×i= (i×i)×(i×i)=(-1)×(-1)=1.

Importance of imaginary unit:

   Imaginary unit i is the key to complex numbers or complex analysis. It is one of the main pilar for this branch of mathematics. To represent a complex number 'z' we need  i. If the real part and imaginary part of a complex number 'z' is 'x' and 'y' respectively ; then , z=x+iy. 

   Clearly, to represent a complex number we need i. 

    Again to perform operations with complex numbers (like, sum, multiplication, subtraction, division) we use the properties of imaginary unit.

 To sum up, the study of complex numbers is impossible without imaginary unit.

So, we may say, "imaginary unit is the hero of complex analysis".