Basics of Complex Numbers, Real and Imaginary Parts, Iota

• Complex Numbers is the largest and the complete set of numbers, consisting of both real and unreal numbers.
• A complex number is of the form
  i2=-1.
• A complex number is usually denoted by the letter ‘z’.
• ‘a’ is called the real part, and ‘b’ is called the imaginary part of the complex number.
• The notion of complex numbers increased the solutions to a lot of problems. For instance, had complex numbers been not there, the equation x2+x+1=0 had had no solutions.

  Because of ‘i', we can now extend the square root to be defined for negative numbers as well. If y is a negative number, then y can be written as -|y| where |y| is the absolute value of y. Then, the square root of y is defined to be equal to be

  y=i√|y|

Algebra of Complex Numbers, Conjugate, Modulus and Argument, Euler's Form

• Complex Numbers have different algebraic laws as well. Let z1=a+ib,z2=c+id.
• z1+z2=(a+c)+i(b+d)
• z1-z2=(a-c)+i(b-d)
• z1.z2=(ac-bd)+i(ad+bc)
• z1/z2=((ac+bd)+i(ad-bc))/(c2-d2)
• 1/z1=(a-ib)/(a2-b2)
• Conjugate : Conjugate of a complex number z=a+ib is defined as

z=a-ib.

• Modulus of z is written as |z| and is defined to be equal to √(a2+b2)
• Thus, z.Conjugate(z)=|z|2.
• Commutative, Associative, and Distributive Laws of algebra apply to Complex Algebra as well.
• Complex Numbers can be represented pictorially on a plane known as Argand Plane.

• The plane consists of a Real Axis, and an Imaginary Axis.
• The angle that z makes with the real axis (counterclockwise) is called the argument of z, and the length of line segment joining z and the origin is equal to |z|, also sometimes written as ‘r’.
• Different representations of z,
  
  
  iθ [Euler’s Form]
• Observe that, eiθ=r(cosθ + i sinθ)
• Multiplication and Division using using Euler’s Form:
  1=r1eiθ1, z2=r2eiθ2
  1z2=r1r2ei(θ1+θ2)

z1/z2=(r1/r2)ei(θ1-θ2)

• Conversion between various forms.
  
  -1(|b/a|); then θ=α,π-α,α-π,-θ depending upon z being in first, second, third or fourth quadrant respectively. This is called the principal argument.
• Approaches to solving questions:
• Expand every complex number as a real and imaginary part.
• Use Euler’s form for every Complex Number
• Use Algebraic Identities, and solve in general, keeping in mind that complex numbers have a real and an imaginary part.

• Another approach to dealing with Complex Numbers is to treat them like free vectors. Free Vectors are vectors that can be moved around without changing direction, in other words, that do not really matter upon the origin. The approach that was used so far was treating complex number as a point.
• Any comple number such as 2+3i can be any one of the infinite number of vectors

• eiθ is multiplied with the vector which is to be rotated, and rotates the vector by “θ” anticlockwise. Clockwise rotation is obtained by multiplying with e-iθ

• m, n are unit vectors; and we have neiθ=m, m e-iθ=n

Here's a quick look at some of the questions we will solve

Complete Tutorial Document with Examples and Solved Problems (There is an MCQ Quiz after this)