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’.

Also Refer: Number system 

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.

  1. The sum of complex numbers is always a complex number (closure law)
  2. 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].
  3. For every complex number z, z + 0 = z [additive identity]
  4. 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.

Also Refer: Algebraic Operations On Complex Numbers

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.

To know more about Modulus and Conjugate of Complex Numbers, visit here.

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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