Complex Numbers and Quadratic Equations Class 11 Notes Mathematics Chapter 5

Lesson at a Glance

1. A number of the form a + ib, where a and b are real numbers and i = – root (-1 ), is called a complex number and a + ib is called standard form of complex number.

2. If z = a + ib then Re(z) = a and Im(z) = b, the coefficient of i

6. Two complex numbers z1 = a + ib and z2 = c + id are equal if and only if a = c and b = d i.e., Re(z1) = Re(z2) and Im(z1) = Im(z2).

We call it as “Equating real and imaginary parts” on both sides.

7. The conjugate of the complex number z = a + ib is denoted by z bar and is given by z bar = a – ib.

8. To express z1/z2, z2 ≠ 0, in the standard form a + ib, multiply and divide by z2 bar , the conjugate of denominator.

9. The modulus of the complex number z = a + ib is denoted

11. Multiplicative inverse (or Reciprocal) of non-zero complex number z i.e.

12. Every non-zero complex number z can be written as z = x + iy = r(cos θ + i sin θ)

(:. x = r cos θ and y = r sin θ.) This is called polar form of z where r = |z|, the modulus of z and θ = arg z, where – π < θ ≤ π.

This angle θ is always expressed in radians.

13. If θ is the argument of a complex number z = x + iy (as defined in 12), then θ is given by tan θ = y/x

θ mentioned in all the four cases (or as in the adjoining figure) listed below is s.t. 0 ≤ θ < π/2.

  • (i) If (x, y) lies in the first quadrant, then arg z = θ
  • (ii) If (x, y) lies in the second quadrant, then arg z = π – θ
  • (iii) If (x, y) lies in third quadrant, then arg z = – (π – θ) = θ – π.
  • (iv) If (x, y) lies in fourth quadrant, then arg z = – θ

14. Every real number is also a complex number. (… x = x + 0i)

15. A complex number of the form z = iy (y ≠ 0, y ∈ R) is called a purely imaginary number. (⇒ Real part of z = 0)

16. Every quadratic equation has two complex roots.

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