Complex Numbers and Quadratic Equations Class 11 Notes - Chapter 5

A complex number is a number that can be expressed in the form p + iq, where p and q are real numbers, and i is a solution of the equation \(x^{2} = −1\). \(\sqrt{-1}\;=\;i\) or \(i^{2}=-1\). Examples of complex numbers: 8 – 2i, 2 +31i, \(2+\frac{4}{5}i\), etc. Complex numbers are denoted by ‘z’.

General form of Complex Number: z = p + iq

Where,

  • p is known as the real part, denoted by Re z
  • q is known as the imaginary part, denoted by Im z

If z = 12 + 35i, then Re z = 12 and Im z = 35. If z1 and z2 are two complex numbers such that z1 = p + iq and z2 = r + is. z1 and z2 are equal if p = r and q = s.

Algebra of Complex Numbers

Addition of complex numbers

Let z1 = m + ni and z2 = o + ip be two complex numbers. Then, z1 + z2 = z = (m + o) + (n + p)i, where z = resultant complex number. For example, (12 + 13i) + (-16 +15i) = (12 – 16) + (13 + 15)i = -4 + 28i.

  • The sum of complex numbers is always a complex number (closure law)
  • For complex numbers z1 and z2: z2 + z1= z1 + z2 (commutative law) For complex numbers z1, z2, z3: (z1 + z2) + z3 = z1 + (z2 + z3) [associative law].
  • For every complex number z, z + 0 = z [additive identity]
  • To every complex number z = p + qi, we have the complex number -z = -p + i(-q), called the negative or additive inverse of z. [z + (–z) = 0]

Difference of complex numbers

Let z1 = m + ni and z2 = o + ip be two complex numbers, then z1 – z2 = z1 + (-z2). For example, (16 + 13i) – (12 – 1i) = (16 + 13i) + (-12 + 1i ) = 4 + 14i and (12 – 1i) – (16 + 13i) = (12 – 1i) + ( -16 – 13i) = -4 – 14i

Multiplication of complex numbers

Let z1 = m + ni and z2 = o + ip be two complex numbers then, z1 × z2 = (mo – np) + i(no + pm). For example, (2 + 4i) (1 + 5i) = (2 × 1 – 4 × 5) + i(2 × 5 + 4 × 1) = -22 + 14i The product of two complex numbers is a complex number (closure law)

  • For complex numbers z1 and z2, z1 × z2 = z2 × z1 (commutative law).
  • For complex numbers z1, z2, z3, (z1 × z2) × z3 = z1 × (z2 × z3) [associative law].

Let z1 = m + in and z2 = o + ip. Then,

  • z1 + z2 = (m + o) + i (n + p)
  • z1 z2 = (mo – np) + i(mp + on)
  • The conjugate of the complex number z = m + in, denoted by \(\overline{z}\), is given by z = m – in.

The Modulus and Conjugate of Complex Numbers

Let z = m + in be a complex number. Then, the modulus of z, denoted by |z| = \(\sqrt{m^{2}-n^{2}}\) and the conjugate of z, denoted by \(\overline{z}\) is the complex number m – ni.In the Argand plane, the modulus of the complex number m + in = \(\sqrt{m^{2}-n^{2}}\) is the distance between the point (m, n) and the origin (0, 0). The x-axis is termed as the real axis and the y-axis is termed as the imaginary axis.

 

Complex Numbers and Quadratic Equations Practice Questions

  1. Find the modulus and argument of the complex number \(\frac{1+i}{1-i}\)
  2. Convert the complex number in the polar form \(\frac{i-1}{cos\;\frac{\pi }{3}\;-\;sin\;\frac{\pi }{3}i}\)
  3. Solve the following equation: \(x^{3}-3x^{2}+2x-1\)
  4. Represent the given complex number in the polar form \(z = 1 + \sqrt{3}i\)
  5. Solve \(\sqrt{5}x^{2}+x+\sqrt{5}\)

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