PPT: Complex Numbers & Quadratic Equations | Mathematics (Maths) Class 11 - Commerce PDF Download

 Page 1


COMPLEX 
NUMBERS AND 
QUADRATIC 
EQUATIONS
Page 2


COMPLEX 
NUMBERS AND 
QUADRATIC 
EQUATIONS
Introduction to Complex Numbers
1
Beyond Real Numbers
Equations like x² + 1 = 0 have 
no real solutions since x² = -1 is 
impossible when x is real.
2
The Need for Expansion
We need to solve quadratic 
equations where the 
discriminant D = b² - 4ac is 
negative, which is impossible 
using only real numbers.
3
Mathematical Innovation
By including solutions to x² = -1, 
we create a complete 
mathematical framework that 
preserves algebraic properties 
while expanding our problem-
solving abilities.
Page 3


COMPLEX 
NUMBERS AND 
QUADRATIC 
EQUATIONS
Introduction to Complex Numbers
1
Beyond Real Numbers
Equations like x² + 1 = 0 have 
no real solutions since x² = -1 is 
impossible when x is real.
2
The Need for Expansion
We need to solve quadratic 
equations where the 
discriminant D = b² - 4ac is 
negative, which is impossible 
using only real numbers.
3
Mathematical Innovation
By including solutions to x² = -1, 
we create a complete 
mathematical framework that 
preserves algebraic properties 
while expanding our problem-
solving abilities.
Defining Complex Numbers
The Imaginary Unit
The symbol i represents :(-1), where i² = -1. This creates 
a solution to x² + 1 = 0, which has no real solutions.
A complex number has the form a + ib, where a and b are 
real numbers (e.g., 2 + i3, -1 + i:3).
Components of Complex Numbers
For a complex number z = a + ib:
Real part (Re z) = a
Imaginary part (Im z) = b
For z = 2 + i5, Re z = 2 and Im z = 5.
Two complex numbers z¡ = a + ib and z¢ = c + id are 
equal if and only if a = c and b = d.
Page 4


COMPLEX 
NUMBERS AND 
QUADRATIC 
EQUATIONS
Introduction to Complex Numbers
1
Beyond Real Numbers
Equations like x² + 1 = 0 have 
no real solutions since x² = -1 is 
impossible when x is real.
2
The Need for Expansion
We need to solve quadratic 
equations where the 
discriminant D = b² - 4ac is 
negative, which is impossible 
using only real numbers.
3
Mathematical Innovation
By including solutions to x² = -1, 
we create a complete 
mathematical framework that 
preserves algebraic properties 
while expanding our problem-
solving abilities.
Defining Complex Numbers
The Imaginary Unit
The symbol i represents :(-1), where i² = -1. This creates 
a solution to x² + 1 = 0, which has no real solutions.
A complex number has the form a + ib, where a and b are 
real numbers (e.g., 2 + i3, -1 + i:3).
Components of Complex Numbers
For a complex number z = a + ib:
Real part (Re z) = a
Imaginary part (Im z) = b
For z = 2 + i5, Re z = 2 and Im z = 5.
Two complex numbers z¡ = a + ib and z¢ = c + id are 
equal if and only if a = c and b = d.
Addition of Complex Numbers
1
Definition
For z¡ = a + ib and z¢ = c + id:
z¡ + z¢ = (a + c) + i(b + d)
2
Properties
Closure: Sum of complex numbers remains 
complex
Commutative: z¡ + z¢ = z¢ + z¡
Associative: (z¡ + z¢) + z£ = z¡ + (z¢ + z£)
3
Identity and Inverse
Additive identity: 0 + i0 (or simply 0), where z + 0 = z
Additive inverse: For z = a + ib, its inverse is -z = -a - ib, where z + (-z) = 0
Page 5


COMPLEX 
NUMBERS AND 
QUADRATIC 
EQUATIONS
Introduction to Complex Numbers
1
Beyond Real Numbers
Equations like x² + 1 = 0 have 
no real solutions since x² = -1 is 
impossible when x is real.
2
The Need for Expansion
We need to solve quadratic 
equations where the 
discriminant D = b² - 4ac is 
negative, which is impossible 
using only real numbers.
3
Mathematical Innovation
By including solutions to x² = -1, 
we create a complete 
mathematical framework that 
preserves algebraic properties 
while expanding our problem-
solving abilities.
Defining Complex Numbers
The Imaginary Unit
The symbol i represents :(-1), where i² = -1. This creates 
a solution to x² + 1 = 0, which has no real solutions.
A complex number has the form a + ib, where a and b are 
real numbers (e.g., 2 + i3, -1 + i:3).
Components of Complex Numbers
For a complex number z = a + ib:
Real part (Re z) = a
Imaginary part (Im z) = b
For z = 2 + i5, Re z = 2 and Im z = 5.
Two complex numbers z¡ = a + ib and z¢ = c + id are 
equal if and only if a = c and b = d.
Addition of Complex Numbers
1
Definition
For z¡ = a + ib and z¢ = c + id:
z¡ + z¢ = (a + c) + i(b + d)
2
Properties
Closure: Sum of complex numbers remains 
complex
Commutative: z¡ + z¢ = z¢ + z¡
Associative: (z¡ + z¢) + z£ = z¡ + (z¢ + z£)
3
Identity and Inverse
Additive identity: 0 + i0 (or simply 0), where z + 0 = z
Additive inverse: For z = a + ib, its inverse is -z = -a - ib, where z + (-z) = 0
Subtraction of Complex 
Numbers
1
Definition
For any two complex numbers z¡ and z¢, their difference is 
defined as:
z¡ - z¢ = z¡ + (-z¢)
This means we add the first complex number to the additive 
inverse of the second.
2
Example 
(6 + 3i) - (2 - i) = (6 + 3i) + (-2 + i)
= (6 - 2) + (3i + i)
= 4 + 4i