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)