Chapter 5
COMPLEX NUMBERS AND
QUADRATIC EQUATIONS
5.1 Overview
We know that the square of a real number is always non-negative e.g. (4)2 = 16 and
(– 4)2 = 16. Therefore, square root of 16 is ± 4. What about the square root of a
negative number? It is clear that a negative number can not have a real square root. So
we need to extend the system of real numbers to a system in which we can find out the
square roots of negative numbers. Euler (1707 - 1783) was the first mathematician to
introduce the symbol i (iota) for positive square root of – 1 i.e., i = −1 .
5.1.1 Imaginary numbers
Square root of a negative number is called an imaginary number., for example,
− 9 = −1 9 = i3, − 7 = −1 7 =i 7
5.1.2 Integral powers of i
i= −1 , i 2 = – 1, i 3 = i 2 i = – i , i 4 = (i 2)2 = (–1)2 = 1.
To compute in for n > 4, we divide n by 4 and write it in the form n = 4m + r, where m is
quotient and r is remainder (0 ≤ r ≤ 4)
Hence in = i4m+r = (i4)m . (i)r = (1)m (i)r = ir
For example, (i)39 = i 4 × 9 + 3 = (i4)9 . (i)3 = i3 = – i
and (i)–435 = i – (4 × 108 + 3) = (i)– (4 × 108) . (i)– 3
1 1 i
= 4 108 . 3 = 4 = i
(i ) (i) (i )
(i) If a and b are positive real numbers, then
− a × −b = −1 a × −1 b = i a × i b = − ab
(ii) a. b = ab if a and b are positive or at least one of them is negative or
zero. However, a b ≠ ab if a and b, both are negative.
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5.1.3 Complex numbers
(a) A number which can be written in the form a + ib, where a, b are real numbers
and i = −1 is called a complex number.
(b) If z = a + ib is the complex number, then a and b are called real and imaginary
parts, respectively, of the complex number and written as Re (z) = a, Im (z) = b.
(c) Order relations “greater than” and “less than” are not defined for complex
numbers.
(d) If the imaginary part of a complex number is zero, then the complex number is
known as purely real number and if real part is zero, then it is called
purely imaginary number, for example, 2 is a purely real number because its
imaginary part is zero and 3i is a purely imaginary number because its real part
is zero.
5.1.4 Algebra of complex numbers
(a) Two complex numbers z1 = a + ib and z2 = c + id are said to be equal if
a = c and b = d.
(b) Let z1 = a + ib and z2 = c + id be two complex numbers then
z1 + z2 = (a + c) + i (b + d).
5.1.5 Addition of complex numbers satisfies the following properties
1. As the sum of two complex numbers is again a complex number, the set of
complex numbers is closed with respect to addition.
2. Addition of complex numbers is commutative, i.e., z1 + z2 = z2 + z1
3. Addition of complex numbers is associative, i.e., (z1 + z2) + z3 = z1 + (z2 + z3)
4. For any complex number z = x + i y, there exist 0, i.e., (0 + 0i) complex number
such that z + 0 = 0 + z = z, known as identity element for addition.
5. For any complex number z = x + iy, there always exists a number – z = – a – ib
such that z + (– z) = (– z) + z = 0 and is known as the additive inverse of z.
5.1.6 Multiplication of complex numbers
Let z1 = a + ib and z2 = c + id, be two complex numbers. Then
z1 . z2 = (a + ib) (c + id) = (ac – bd) + i (ad + bc)
1. As the product of two complex numbers is a complex number, the set of complex
numbers is closed with respect to multiplication.
2. Multiplication of complex numbers is commutative, i.e., z1.z2 = z2.z1
3. Multiplication of complex numbers is associative, i.e., (z1.z2) . z3 = z1 . (z2.z3)
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4. For any complex number z = x + iy, there exists a complex number 1, i.e., (1 + 0i)
such that
z . 1 = 1 . z = z, known as identity element for multiplication.
1
5. For any non zero complex number z = x + i y, there exists a complex number
z
1 1 1 a − ib
such that z ⋅ = ⋅ z = 1 , i.e., multiplicative inverse of a + ib = = .
z z a + ib a 2 + b2
6. For any three complex numbers z1, z2 and z3 ,
z1 . (z2 + z3) = z1 . z2 + z1 . z3
and (z1 + z2) . z3 = z1 . z3 + z2 . z3
i.e., for complex numbers multiplication is distributive over addition.
5.1.7 Let z1 = a + ib and z2( ≠ 0) = c + id. Then
z1 a + ib (ac + bd ) (bc − ad )
z1 ÷ z2 == = 2 2
+i 2
z2 c + id c +d c +d2
5.1.8 Conjugate of a complex number
Let z = a + ib be a complex number. Then a complex number obtained by changing the
sign of imaginary part of the complex number is called the conjugate of z and it is denoted
by z , i.e., z = a – ib.
Note that additive inverse of z is – a – ib but conjugate of z is a – ib.
We have :
1. ( z ) = z
2. z + z = 2 Re (z) , z – z = 2 i Im(z)
3. z = z , if z is purely real.
4. z + z = 0 ⇔ z is purely imaginary
5. z . z = {Re (z)}2 + {Im (z)}2 .
6. ( z1 + z2 ) = z1 + z2 , ( z1 − z2 ) = z1 – z2
z1 (z )
= 1 ( z2 ≠ 0)
7. ( z1 . z2 ) = ( z1 ) ( z2 ),
z2 ( z2 )
5.1.9 Modulus of a complex number
Let z = a + ib be a complex number. Then the positive square root of the sum of square
of real part and square of imaginary part is called modulus (absolute value) of z and it
is denoted by z i.e., z = a 2 + b2
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COMPLEX NUMBERS AND
QUADRATIC EQUATIONS
5.1 Overview
We know that the square of a real number is always non-negative e.g. (4)2 = 16 and
(– 4)2 = 16. Therefore, square root of 16 is ± 4. What about the square root of a
negative number? It is clear that a negative number can not have a real square root. So
we need to extend the system of real numbers to a system in which we can find out the
square roots of negative numbers. Euler (1707 - 1783) was the first mathematician to
introduce the symbol i (iota) for positive square root of – 1 i.e., i = −1 .
5.1.1 Imaginary numbers
Square root of a negative number is called an imaginary number., for example,
− 9 = −1 9 = i3, − 7 = −1 7 =i 7
5.1.2 Integral powers of i
i= −1 , i 2 = – 1, i 3 = i 2 i = – i , i 4 = (i 2)2 = (–1)2 = 1.
To compute in for n > 4, we divide n by 4 and write it in the form n = 4m + r, where m is
quotient and r is remainder (0 ≤ r ≤ 4)
Hence in = i4m+r = (i4)m . (i)r = (1)m (i)r = ir
For example, (i)39 = i 4 × 9 + 3 = (i4)9 . (i)3 = i3 = – i
and (i)–435 = i – (4 × 108 + 3) = (i)– (4 × 108) . (i)– 3
1 1 i
= 4 108 . 3 = 4 = i
(i ) (i) (i )
(i) If a and b are positive real numbers, then
− a × −b = −1 a × −1 b = i a × i b = − ab
(ii) a. b = ab if a and b are positive or at least one of them is negative or
zero. However, a b ≠ ab if a and b, both are negative.
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5.1.3 Complex numbers
(a) A number which can be written in the form a + ib, where a, b are real numbers
and i = −1 is called a complex number.
(b) If z = a + ib is the complex number, then a and b are called real and imaginary
parts, respectively, of the complex number and written as Re (z) = a, Im (z) = b.
(c) Order relations “greater than” and “less than” are not defined for complex
numbers.
(d) If the imaginary part of a complex number is zero, then the complex number is
known as purely real number and if real part is zero, then it is called
purely imaginary number, for example, 2 is a purely real number because its
imaginary part is zero and 3i is a purely imaginary number because its real part
is zero.
5.1.4 Algebra of complex numbers
(a) Two complex numbers z1 = a + ib and z2 = c + id are said to be equal if
a = c and b = d.
(b) Let z1 = a + ib and z2 = c + id be two complex numbers then
z1 + z2 = (a + c) + i (b + d).
5.1.5 Addition of complex numbers satisfies the following properties
1. As the sum of two complex numbers is again a complex number, the set of
complex numbers is closed with respect to addition.
2. Addition of complex numbers is commutative, i.e., z1 + z2 = z2 + z1
3. Addition of complex numbers is associative, i.e., (z1 + z2) + z3 = z1 + (z2 + z3)
4. For any complex number z = x + i y, there exist 0, i.e., (0 + 0i) complex number
such that z + 0 = 0 + z = z, known as identity element for addition.
5. For any complex number z = x + iy, there always exists a number – z = – a – ib
such that z + (– z) = (– z) + z = 0 and is known as the additive inverse of z.
5.1.6 Multiplication of complex numbers
Let z1 = a + ib and z2 = c + id, be two complex numbers. Then
z1 . z2 = (a + ib) (c + id) = (ac – bd) + i (ad + bc)
1. As the product of two complex numbers is a complex number, the set of complex
numbers is closed with respect to multiplication.
2. Multiplication of complex numbers is commutative, i.e., z1.z2 = z2.z1
3. Multiplication of complex numbers is associative, i.e., (z1.z2) . z3 = z1 . (z2.z3)
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4. For any complex number z = x + iy, there exists a complex number 1, i.e., (1 + 0i)
such that
z . 1 = 1 . z = z, known as identity element for multiplication.
1
5. For any non zero complex number z = x + i y, there exists a complex number
z
1 1 1 a − ib
such that z ⋅ = ⋅ z = 1 , i.e., multiplicative inverse of a + ib = = .
z z a + ib a 2 + b2
6. For any three complex numbers z1, z2 and z3 ,
z1 . (z2 + z3) = z1 . z2 + z1 . z3
and (z1 + z2) . z3 = z1 . z3 + z2 . z3
i.e., for complex numbers multiplication is distributive over addition.
5.1.7 Let z1 = a + ib and z2( ≠ 0) = c + id. Then
z1 a + ib (ac + bd ) (bc − ad )
z1 ÷ z2 == = 2 2
+i 2
z2 c + id c +d c +d2
5.1.8 Conjugate of a complex number
Let z = a + ib be a complex number. Then a complex number obtained by changing the
sign of imaginary part of the complex number is called the conjugate of z and it is denoted
by z , i.e., z = a – ib.
Note that additive inverse of z is – a – ib but conjugate of z is a – ib.
We have :
1. ( z ) = z
2. z + z = 2 Re (z) , z – z = 2 i Im(z)
3. z = z , if z is purely real.
4. z + z = 0 ⇔ z is purely imaginary
5. z . z = {Re (z)}2 + {Im (z)}2 .
6. ( z1 + z2 ) = z1 + z2 , ( z1 − z2 ) = z1 – z2
z1 (z )
= 1 ( z2 ≠ 0)
7. ( z1 . z2 ) = ( z1 ) ( z2 ),
z2 ( z2 )
5.1.9 Modulus of a complex number
Let z = a + ib be a complex number. Then the positive square root of the sum of square
of real part and square of imaginary part is called modulus (absolute value) of z and it
is denoted by z i.e., z = a 2 + b2
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