Revision Notes: Complex Numbers & Quadratic Equations

# Complex Numbers and Quadratic Equations Class 11 Notes Maths Chapter 4

``` Page 1

KEY POINTS
? The imaginary number 1 i ? ? , is called iota
? For any integer k, i
4k
=1, i
4k+1
= i, i
4k+2
= –1, i
4k+3
= –i
? a b ab ? ? if both a and b are negative real numbers
? A number of the form z = a + ib, where a, b ? R is called a
complexnumber.
a is called the real part of z, denoted by Re(z) and b is called the
imaginary part of z, denoted by Im(z)
? a + ib = c + id if a = c, and b = d
? z1 = a + ib, z2 = c + id.
In general, we cannot compare and say that z1> z2 or z1< z2
but if b, d = 0 and a > c then z1> z2
i.e. we can compare two complex numbers only if they are
purely real.
? ? 0 + i 0 is additive identity of a complex number.
? –z = –a –ib is called the Additive Inverse or negative ofz = a + ib
Page 2

KEY POINTS
? The imaginary number 1 i ? ? , is called iota
? For any integer k, i
4k
=1, i
4k+1
= i, i
4k+2
= –1, i
4k+3
= –i
? a b ab ? ? if both a and b are negative real numbers
? A number of the form z = a + ib, where a, b ? R is called a
complexnumber.
a is called the real part of z, denoted by Re(z) and b is called the
imaginary part of z, denoted by Im(z)
? a + ib = c + id if a = c, and b = d
? z1 = a + ib, z2 = c + id.
In general, we cannot compare and say that z1> z2 or z1< z2
but if b, d = 0 and a > c then z1> z2
i.e. we can compare two complex numbers only if they are
purely real.
? ? 0 + i 0 is additive identity of a complex number.
? –z = –a –ib is called the Additive Inverse or negative ofz = a + ib

? 1 + i 0 is multiplicative identity of complex number.
? z a ib ? ? is called the conjugate of z = a + ib
?
0
1 i ?
?
1
2 2 2
1 a ib z
z
z a b
z
?
?
? ? ?
?
is called the multiplicative Inverse of
z = a + ib (a ? 0, b ? 0)
? The coordinate plane that represents the complex numbers is
called the complex plane or the Argand plane
? Polar form of z = a + ib is,
z = r (cos ? + i sin ?) where
2 2
r a b z ? ? ? is called the modulus
of z, ? is called the argument or amplitude of z.
? The value of ? such that, – ?< ? ?< ? is called the principle
argument of z.
? Z = x + iy, x > 0 and y > 0 the argument of z is acute angle given
by
tan
y
x
? ?
figure (i)
Page 3

KEY POINTS
? The imaginary number 1 i ? ? , is called iota
? For any integer k, i
4k
=1, i
4k+1
= i, i
4k+2
= –1, i
4k+3
= –i
? a b ab ? ? if both a and b are negative real numbers
? A number of the form z = a + ib, where a, b ? R is called a
complexnumber.
a is called the real part of z, denoted by Re(z) and b is called the
imaginary part of z, denoted by Im(z)
? a + ib = c + id if a = c, and b = d
? z1 = a + ib, z2 = c + id.
In general, we cannot compare and say that z1> z2 or z1< z2
but if b, d = 0 and a > c then z1> z2
i.e. we can compare two complex numbers only if they are
purely real.
? ? 0 + i 0 is additive identity of a complex number.
? –z = –a –ib is called the Additive Inverse or negative ofz = a + ib

? 1 + i 0 is multiplicative identity of complex number.
? z a ib ? ? is called the conjugate of z = a + ib
?
0
1 i ?
?
1
2 2 2
1 a ib z
z
z a b
z
?
?
? ? ?
?
is called the multiplicative Inverse of
z = a + ib (a ? 0, b ? 0)
? The coordinate plane that represents the complex numbers is
called the complex plane or the Argand plane
? Polar form of z = a + ib is,
z = r (cos ? + i sin ?) where
2 2
r a b z ? ? ? is called the modulus
of z, ? is called the argument or amplitude of z.
? The value of ? such that, – ?< ? ?< ? is called the principle
argument of z.
? Z = x + iy, x > 0 and y > 0 the argument of z is acute angle given
by
tan
y
x
? ?
figure (i)

? Z = x + iy, x < 0 and y > 0 the argument of z is ? ? ? ,where ? is
acute angle given by
ta n
y
x
? ?
figure (ii)
? Z = x + iy, x < 0 and y < 0 the argument of z is ? ? ? ,where ? is
acute angle given by
ta n
y
x
? ?
figure (iii)
? Z = x + iy, x > 0 and y < 0 the argument of z is ? ? ,where ? is
acute angle given by
ta n
y
x
? ?
a
? ? ? ?
a

( ) ? ?
? ? ?
? ?
? ?

? ? ? ? ?
? ? ? ? ?
? ? ? ?
Page 4

KEY POINTS
? The imaginary number 1 i ? ? , is called iota
? For any integer k, i
4k
=1, i
4k+1
= i, i
4k+2
= –1, i
4k+3
= –i
? a b ab ? ? if both a and b are negative real numbers
? A number of the form z = a + ib, where a, b ? R is called a
complexnumber.
a is called the real part of z, denoted by Re(z) and b is called the
imaginary part of z, denoted by Im(z)
? a + ib = c + id if a = c, and b = d
? z1 = a + ib, z2 = c + id.
In general, we cannot compare and say that z1> z2 or z1< z2
but if b, d = 0 and a > c then z1> z2
i.e. we can compare two complex numbers only if they are
purely real.
? ? 0 + i 0 is additive identity of a complex number.
? –z = –a –ib is called the Additive Inverse or negative ofz = a + ib

? 1 + i 0 is multiplicative identity of complex number.
? z a ib ? ? is called the conjugate of z = a + ib
?
0
1 i ?
?
1
2 2 2
1 a ib z
z
z a b
z
?
?
? ? ?
?
is called the multiplicative Inverse of
z = a + ib (a ? 0, b ? 0)
? The coordinate plane that represents the complex numbers is
called the complex plane or the Argand plane
? Polar form of z = a + ib is,
z = r (cos ? + i sin ?) where
2 2
r a b z ? ? ? is called the modulus
of z, ? is called the argument or amplitude of z.
? The value of ? such that, – ?< ? ?< ? is called the principle
argument of z.
? Z = x + iy, x > 0 and y > 0 the argument of z is acute angle given
by
tan
y
x
? ?
figure (i)

? Z = x + iy, x < 0 and y > 0 the argument of z is ? ? ? ,where ? is
acute angle given by
ta n
y
x
? ?
figure (ii)
? Z = x + iy, x < 0 and y < 0 the argument of z is ? ? ? ,where ? is
acute angle given by
ta n
y
x
? ?
figure (iii)
? Z = x + iy, x > 0 and y < 0 the argument of z is ? ? ,where ? is
acute angle given by
ta n
y
x
? ?
a
? ? ? ?
a

( ) ? ?
? ? ?
? ?
? ?

? ? ? ? ?
? ? ? ? ?
? ? ? ?

figure (iv)
? |z1 + z2| ? |z1| + |z2|
? |z1z2| = |z1|. |z2|
?
2
1 1
2 2
; ; ;
n
n
z z
z z z z z z z z z
z z
? ? ? ? ? ? ? ?
?
1 2 1 2
z z z z ? ? ?
? If   z1 = r1 (cos ?1 + isin ?1)
z2 =  r 2 (cos ?2 + isin ?2)
then z1z2 = r1r2[cos ( ?1 + ?2) + isin ( ?1 + ?2)]
? ?
1 1
1 2 1 2
2 2
cos ( ) s in( )
z r
i
z r
? ? ? ? ? ? ? ?
? For the quadratic equation ax
2
+ bx + c =0,
a, b, c ? R, a ? 0,if b
2
– 4ac < 0
then it will have complex roots given by,
2
4
2
b i ac b
x
a
? ? ?
?
? a ib ? is called square root of z = a + ib, ? a ib x iy ? ? ?
squaring both sides we get a + ib = x
2
– y
2
+ 2i(xy)
```

## Mathematics (Maths) for JEE Main & Advanced

209 videos|443 docs|143 tests

## FAQs on Complex Numbers and Quadratic Equations Class 11 Notes Maths Chapter 4

 1. What are complex numbers and how are they related to quadratic equations?
Ans. Complex numbers are numbers of the form a + bi, where a and b are real numbers and i is the imaginary unit, defined as the square root of -1. Complex numbers are used to solve quadratic equations because they can represent the solutions to equations that do not have real solutions. Complex numbers also have properties that make them useful in solving and manipulating quadratic equations.
 2. How do we add and subtract complex numbers?
Ans. To add or subtract complex numbers, we simply add or subtract the real and imaginary parts separately. For example, to add (3 + 2i) and (1 - 4i), we add the real parts (3 + 1) and the imaginary parts (2i - 4i) to get the result: 4 - 2i.
 3. How do we multiply complex numbers?
Ans. To multiply complex numbers, we use the distributive property and the fact that i^2 = -1. For example, to multiply (2 + 3i) and (4 - 5i), we multiply each term in the first complex number by each term in the second complex number and combine like terms. The result is (8 + 6i - 10i - 15i^2), which simplifies to (8 - 4i - 15) and further simplifies to (-7 - 4i).
 4. Can complex numbers be divided?
Ans. Yes, complex numbers can be divided. To divide two complex numbers, we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of a complex number a + bi is a - bi. This process allows us to eliminate the imaginary terms in the denominator and simplify the division. However, it's important to note that division by zero is not defined in complex numbers.
 5. How are complex numbers used to solve quadratic equations?
Ans. Complex numbers are used to solve quadratic equations that do not have real solutions. When solving a quadratic equation, if the discriminant (b^2 - 4ac) is negative, the solutions are complex numbers. The quadratic formula is used to find the complex solutions, where x = (-b ± √(b^2 - 4ac))/(2a). The term inside the square root, the discriminant, determines whether the solutions are complex or real.

## Mathematics (Maths) for JEE Main & Advanced

209 videos|443 docs|143 tests

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