CBSE Class 11 Mathematics Complex Numbers Quadratic Equations Notes

Class XI

Chapter 5

Complex Numbers & Quadratic Equations

Chapter Notes

Top Definitions

1. A number of the form a + ib, where a and b are real numbers, is said to be a complex number.

2. In complex number z = a + ib, a is the real part, denoted by Re z and b is the imaginary part denoted by Im z of the complex number z.

3 √-1 =i is called the iota the complex number.

4. For any non – zero complex number z = a + ib (a ≠ 0, b ≠ 0), there exists

a complex number a/a²+b²+i-b/a²+b²  denoted by 1/z or Z ^- called the multiplicative inverse of z such that (a + ib) (a²/a²+b²+i-b/a²+b²⁾=1+i0=1.

5. Modulus of a complex number z = a+ib , denoted by |z|, is defined to be the non – negative real number √a²+b²

,i.e|Z|=√a²+b²

6. Conjugate of a complex number z =a+ib, denoted as z , is the complex number a – ib.

7. z=r(cos θ +isin θ) is the polar form of the complex number z=a+ib. here √r = a² + b² is called the modulus of z and θ = tan^-1(a/b) is called the argument or amplitude of z, denoted by arg z.

8. The value of θ such that –π < θ ≤ π, called principal argument of z. 

9 The plane having a complex number assigned to each of its points is called the complex plane or the Argand plane.

10.Fundamental Theorem of Algebra states that “A polynomial equation of degree n has n roots.”

Top Concepts

1. Addition of two complex numbers:If z₁ = a + ib and z₂ = c +id be any two complex numbers then, the sum z₁ + z₂ = (a + c) + i(b + d).

2. Sum of two complex numbers is also a complex number. this is known as the closure property.

3. The addition of complex numbers satisfy the following properties:

i. Addition of complex numbers satisfies the commutative law. For any two complex numbers z₁ and z₂, z₁ + z₂ = z₂ + z₁.

ii. Addition of complex numbers satisfies associative law for any three complex numbers z₁, z₂, z₃, (z₁ + z₂) + z₃ = z₁ + (z₂ + z₃).

iii. There exists a complex number 0 + i0 or 0, called the additive identity or the zero complex number, such that, for every complex number z, z + 0 = 0+z = z.

iv. To every complex number z = a + ib, there exists another complex number –z =–a + i(-b) called the additive inverse of z. z+(-z)=(-z)+z=0

4 Difference of two complex numbers: Given any two complex numbers If z₁ = a + ib and z₂ = c +id the difference z₁ – z₂ is given by z₁ – z₂ = z₁ + (-z₂) = (a - c) + i(b - d).

5 Multiplication of two complex numbers Let z₁ = a + ib and z₂ = c + id be any two complex numbers. Then, the product z₁ z₂ is defined as follows:
z₁ z₂ = (ac – bd) + i(ad + bc)

6. Properties of multiplication of complex numbers: Product of two complex numbers is a complex number, the product z₁ z₂ is a complex number for all complex numbers z₁ and z₂.

i. Product of complex numbers is commutative i.e for any two complex numbers z₁ and z₂,

z₁ z₂ = z₂ z₁

ii. Product of complex numbers is associative law For any three complex numbers z₁, z₂, z₃,

(z₁ z₂) z₃ = z₁ (z₂ z₃)

iii. There exists the complex number 1 + i0 (denoted as 1), called the

multiplicative identity such that z.1 = z for every complex number z.

iv. For every non- zero complex number z = a + ib or a + bi (a ≠ 0, b ≠ 0),

there is a complex number

a/ a²+b² + -b/ a²+ b² , called the multiplicative

inverse of z such that

z x 1/z = 1

v. The distributive law: For any three complex numbers z₁, z₂, z₃,

a. z₁ (z₂ + z₃) = z₁.z₂ + z₁.z₃

b. (z₁ + z₂) z₃ = z₁.z₃ + z₂._z3

7.Division of two complex numbers Given any two complex numbers z1 =

a + ib and z2 = c + id z1 and z2, where z2 ≠ 0, the quotient z₁ / z₂ is defined by 8. Identities for the complex numbers

i. (z₁ + z₂)² = z₁² + z₂² = 2z₁.z₂, for all complex numbers z₁ and z₂.

ii (z₁ - z₂)² = z₁² - 2z₁z₂ + z₂²

iii.(z₁ + z₂)³ = z₁³ + 3z₁²z₂ + 3z₁z₂² + z₂³

iv (z₁ - z₂)³ = z₁³ = 3z₁²z₂ + 3z₁z₂³ - z₂³

v z1² - z2² = (z1 + z2) (z1 – z2)

Please click the link below to download pdf file for CBSE Class 11 Mathematics - Complex Numbers _ Quadratic Equations Concepts.