Complex numbers

Complex numbers

complex numbers

    History of complex numbers:

 What is the root of the equation: x²+9=0.

 This is not a tough question in the present time. But this is the main cause behind invention of complex numbers. 

       

complex numbers

   We all know that , one of the main feature of a real number is it's square is always positive. But the problem  started when the mathematicians tried to find the square root of a negative real number.

  Yes, It was Heron of Alexandria who thought about this topic for the first time, probably in 1st century, 50 AD.

He was trying to find out the value of √(81-114) .But he gave up. After this for a long time nobody showed interest about this.

  But in 1500's when solutions of 3rd and 4th degree polynomial equations were discovered, mathematicians realised the necessity of square root of a negative number.  

  Finally in 1545 , Girlamo cardano, a famous mathematician wrote a book (title: Ars Magna) on the imaginary numbers. He solved the equation: x(10-x)=40. His solution was: (5 +√-15) and (5-√-15).

But he personally did not like to work with the imaginary numbers. So he did not work more on the complex numbers.

  Later in 1637, Rane Descartes came up with the  standard form(a+ib) of complex numbers. 

complex number:  Definition:

The square root of a negative real number is called a complex number. 

  In other words, a complex/ imaginary number is a number of the form: p+iq, where p and q are real and 'i' is considered as the imaginary unit or  "iota". Here , i=√-1, be the root of : x²+1=0.

    examples: 3+5i, -3+5i, -3- 5i, 3-5i,...etc.

   In a complex number, z=x+iy, x is called the real part of z and y is called the imaginary part of  z. 

   The order pair (x,y) of z=x+iy,  represents the complex number z. If x=0, then the number (0,y) is purely imaginary and if y=0, then the number (x,0) is purely real.

  complex numbers: Geometrical representation:

       

Complex numbers

Geometrically, a complex number (x+iy)  represents a point (x,y) in the complex plane or Argand plane. Here, we take (0,0) as origin and x- axis as the real axis and Y axis as the imaginary axis.

complex numbers formulas:

Modulus of a complex number:

Let,  (x+iy) be a complex number; where x,y are real numbers and i=√-1.Then the  positive square root of (x² + y²) is called the modulus of (x+iy) and denoted by mod(z) or |z|.

As for example, modulus of (3+4i) is √(3² + 4²) = √(9+16) = 5.

   Geometrically, modulus of a complex number is the distance of the  complex number  from the origin in the complex plane.

Amplitude or argument of a complex number:

Let, z=x+iy is a complex number and |z| not equals to zero. Then the value of  θ for which both the equations , x=|z|cosθ and y= |z|sinθ are satisfied; is called the amplitude or argument of z and denoted by arg(z) or amp(z).

 So it is clear that more than one value of θ can satisfy the equations. So, more than one value of argument may exist. But, the value of θ which also satisfy -π< θ(< or=)π , is called the principal value of argument. The value of argument of a complex number z is obtained from, y/x =tan(θ).
   There is a rule to find out the amplitude of a given complex number correctly. If the complex numbers be such that,
        (1) z=x+iy  then, arg(z) = tan⁻'(y/x) ;
        (2) z=-x+iy  then, arg(z)= π- tan⁻'(y/x) ;
        (3) z=-x-iy  then, arg(z) = -π+ tan⁻'(y/x) ;
        (4) z =x-iy  then, arg(z) = -tan⁻'(y/x) .

complex numbers calculator

  

 If you find out any incorrect information or know anything more about this , please write it in the comment section!