## Revision Notes for Class 11 Mathematics Complex Numbers Quadratic Equations

Class 11 Mathematics students should refer to the following concepts and notes for Complex Numbers Quadratic Equations in standard 11. These exam notes for Grade 11 Mathematics will be very useful for upcoming class tests and examinations and help you to score good marks

### Complex Numbers Quadratic Equations Notes Class 11 Mathematics

**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/a^{2}+b^{2}+i-b/a^{2}+b^{2 }denoted by 1/z or Z ^{-} called the multiplicative inverse of z such that (a + ib) (a^{2}/a^{2}+b^{2}+i-b/a^{2}+b^{2)}=1+i0=1.

5. Modulus of a complex number z = a+ib , denoted by |z|, is defined to be the non – negative real number √a^{2}+b^{2}

,i.e|Z|=√a^{2}+b^{2}

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 = a^{2} + b^{2} 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 z_{1} = a + ib and z_{2} = c +id be any two complex numbers then, the sum z_{1} + z_{2} = (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 z_{1} and z_{2}, z_{1} + z_{2} = z_{2} + z_{1}.

ii. Addition of complex numbers satisfies associative law for any three complex numbers z_{1}, z_{2}, z_{3}, (z_{1} + z_{2}) + z_{3} = z_{1} + (z_{2} + z_{3}).

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 z_{1} = a + ib and z_{2} = c +id the difference z_{1} – z_{2} is given by z_{1} – z_{2} = z_{1} + (-z_{2}) = (a - c) + i(b - d).

5 Multiplication of two complex numbers Let z_{1} = a + ib and z_{2} = c + id be any two complex numbers. Then, the product z_{1} z_{2} is defined as follows:

z_{1} z_{2} = (ac – bd) + i(ad + bc)

6. Properties of multiplication of complex numbers: Product of two complex numbers is a complex number, the product z_{1} z_{2} is a complex number for all complex numbers z_{1} and z_{2}.

i. Product of complex numbers is commutative i.e for any two complex numbers z_{1} and z_{2},

z_{1} z_{2} = z_{2} z_{1}

ii. Product of complex numbers is associative law For any three complex numbers z_{1}, z_{2}, z_{3},

(z_{1} z_{2}) z_{3} = z_{1} (z_{2} z_{3})

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/ a^{2}+b^{2} + -b/ a^{2}+ b^{2 }, called the multiplicative

inverse of z such that

z x 1/z = 1

v. The distributive law: For any three complex numbers z_{1}, z_{2}, z_{3},

a. z_{1} (z_{2} + z_{3}) = z_{1}.z_{2} + z_{1}.z_{3}

b. (z_{1} + z_{2}) z_{3} = z_{1}.z_{3} + z_{2}._{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 z_{1} / z_{2 }is defined by 8. Identities for the complex numbers

i. (z_{1} + z_{2})² = z_{1}² + z_{2}² = 2z_{1}.z_{2}, for all complex numbers z_{1} and z_{2}.

ii (z_{1} - z_{2})² = z_{1}² - 2z_{1}z_{2} + z_{2}²

iii.(z_{1} + z_{2})³ = z_{1}³ + 3z_{1}²z_{2} + 3z_{1}z_{2}² + z_{2}³

iv (z_{1} - z_{2})³ = z_{1}³ = 3z_{1}²z_{2} + 3z_{1}z_{2}³ - z_{2}³

v z1² - z2² = (z1 + z2) (z1 – z2)

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