Introduction to Complex Numbers
* A number whose square is negative is called an imaginary number.
* A number of the form a + ib, where a and b are real numbers, is called a complex number and is represented by Z.
In the complex number z = a + ib,
* a: Real part of the complex number, denoted by Re z
* b: imaginary part of the complex number, denoted by Im z
* If the imaginary part of a complex number is zero, then the number is called a purely real number.
* If the real part of a complex number is zero, then the number is called a purely imaginary number.
Algebra of Complex Numbers
* Complex numbers are closed under addition, subtraction, multiplication and division.
* Complex numbers satisfy the commutative and the associative properties of addition and multiplication.
* For any three complex numbers, multiplication is distributive over addition.
* For any integer k:
* i4k = 1
* i4k + 1 = i
* i4k + 2 = – 1
* i4k + 3 = – i
Modulus and Conjugate of a Complex Number
* The modulus of the complex number z = a + ib is defined as |z| = |a + ib| = \sqrt{a² + b²}.
* The conjugate of the complex number z = a + ib is given as z ̅ = \overline{a + i b} = a – ib.
Polar Representation of Complex Numbers
* A plane having complex numbers corresponding to each of its point is called complex plane or the Argand plane.
x- coordinate of a point corresponds to the real part, and
y- coordinate corresponds to the imaginary part of complex number.
* The modulus of z = x + iy can be geometrically interpreted as the distance of point P (x, y) form origin
(0, 0) in the Argand plane.
* The polar form of complex number z = x + iy is
r (cos Θ + I sin Θ), where:
* r: Modulus of z = |z| = \sqrt{x² + y²}
* θ: Argument or amplitude of z, denoted by arg z
arg ( z_{1} \times z_{2} ) = arg ( z_{1} ) + arg ( z_{2} )
arg ( \frac{ z_{1} }{ z_{2} } ) = arg ( z_{1} ) – arg ( z_{2} )
Quadratic Equations
General form of quadratic equation ax² + bx + c = 0:
* a, b and c are real numbers
* a ≠ 0
Discriminant (D): b² – 4ac
* Roots of quadratic equation ax2 + bx + c =0, where
b2 – 4ac > 0, are \frac{- b + \sqrt{ b^{2} - 4ac } }{ 2a} and \frac{- b - \sqrt{ b^{2} - 4ac } }{ 2a}.
* The roots of the quadratic ax2 + bx + c = 0,
Where b2 – 4ac = 0, are \frac{- b}{ 2a} and \frac{- b}{ 2a}.
* The roots of quadratic equation ax2 + bx + c =0, where b2 – 4ac < 0 are,
\frac{- b + i \sqrt{4ac - b^{2} } }{ 2a} and \frac{- b + i \sqrt{4ac - b^{2} } }{ 2a}.
* The fundamental theorem of algebra states that a polynomial equation has at least one root.