Notes On Algebra of Complex Numbers - CBSE Class 11 Maths

Addition of two complex numbers

Consider two complex numbers z1 = a + ib, z2 = c + id .

z1 + z2 = (a + c) + i(b + d)

The sum of the real parts of the two given complex numbers gives the real part of the resultant complex number.

Similarly, the sum of their imaginary parts gives the imaginary part of the resultant complex number.

The resultant number is also a complex number.

Ex: (5 + 7i) + (6 + i)

Sol: (5 + 7i) + (6 + i) = (5 + 6) + (7 + 1)i

= 11 + 8i

Properties of addition of complex numbers

Closure Property: The sum of two complex numbers is a complex number.

z1 + z2 is a complex number for all complex numbers z1 and z2.

Commutative Property: Two complex numbers can be added in any order.

For any two complex numbers z1 and z2, z1 + z2 = z2 + z1.

Associative Property: In addition of three complex numbers, the numbers can be grouped in any order.

For any three complex numbers z1 , z2 and z3, (z1 + z2)+z3 = z1 +(z2 + z3).

Additive identity of a complex number: There exists a complex number 0 + 0i (denoted as 0), called the additive identity, such that for every complex number z, z + 0 = z.

Additive inverse of a complex number: For every complex number z = a + ib, there exists a complex number -a+i(-b) (denoted as -z), called the additive inverse or negative of z, such that z + (-z) = 0.

Subtraction of two complex numbers

Consider two complex numbers z1 = a + ib, z2 = c + id

z1 - z2

z1 + (-z2)

=(a+ib)+(-c-id)

=(a-c)+i(b-d)

Ex: (3 + 2i) - (2 - 7i)

(3 + 2i) - (2 - 7i) = (3 + 2i) + [-(2 - 7i)]

=(3 + 2i) + (-2 + 7i)

= 1 + 9i

Multiplication of two complex numbers

Consider two complex numbers z1 = a + ib,z2 = c + id .

z1 x z2 = (a + ib) x (c + id)

=(ac - bd) + i(ad + bc)


Ex: (1 + 4i) x (3 + 2i)

(1 + 4i) x (3 + 2i) = (1 x 3 - 4 x 2) + (1 x 2 + 4 x 3)i

=(3 - 8) + (2 + 12)i

= -5 + 14i

Hence, the product of two complex numbers is also a complex number.

Properties of multiplication of complex numbers:

Closure Property: The product of two complex numbers is a complex number.

z1z2 is a complex number for all complex numbers z1 and z2.

Commutative Property: Two complex numbers can be multiplied in any order.

For any complex numbers z1 and z2, z1z2 = z2z1

Associative Property: In multiplication of three complex numbers, the numbers can be grouped in any order.

For any three complex numbers z1,z2 and z3, (z1z2)z3 = z1(z2z3)

Multiplicative identity of a complex number: There exists a complex number 1 + 0i (denoted as 1), called the multiplicative identity, such that for every complex number z, z x 1= z.

Multiplicative inverse of a complex number: For every complex number z = a + ib, where a ≠ 0 and b ≠ 0, there exists a complex number 1/(a+ib) = a/(a2 + b2) +i -b/(a2 + b2) (denoted as 1/z or z-1), called the multiplicative inverse of z, such that z. 1/z = 1.

Distributive Property: For any three complex numbers z1, z2 and z3:

z1(z2 + z3) = z1z2 + z1z3

(z1 + z2)z3 = z1z3 + z2z3


Division of two complex numbers

Consider complex numbers z1 = a + ib , z2 = c + id ( z2 ≠ 0).

z1 ÷ z2 =  z1 x 1/z1

= (a + ib)/(c + id)

= ((a + ib)(c - id))/((c + id)(c - id))

= (ac - iad + ibc -i2bd)/(c2 - i2d2)

= (ac - iad + ibc + bd)/(c2 + d2)  [i2 = -1]

= ((ac + bd) + i(bc - ad)) /(c2 + d2)

∴ z1/z2 = (ac + bd)/(c2 + d2) + i(bc - ad)/(c2 + d2)

Ex: (2 + 5i) ÷ (1 + 2i)

= (2 + 5i)/(1 + 2i)

= ((2 + 5i)(1 - 2i)) / ((1 + 2i)(1 - 2i))

= (2 - 4i + 5i - 10i2)/(1 - 4i2)

= (2 + i + 10)/(1 + 4) (Since i2 = -1)

= (12 + i)/5

= (1/5)(12 + i)

Powers of 'i'

i2 = -1

i3 = i2.i = -i

i4 = (i2)2 = (-1)2 = 1

i5 = (i2)2 i= (-1)2 i= i

i6 = (i2)3 = (-1)3 = -1

i-1 = 1/i

    =(1x i)/(i x i) = i/i2

    = i/-1 = -i

i-2 = 1/i2 = 1/-1 = -1

i-3 = 1/i3 = 1/-i = 1/-i = -1xi / ixi = -i/i2 = -i/-1 = i

i-4 = 1/i4 = 1/1 = 1

WE notice from the above results that, for any integer k:

•  i4k = 1

• i4k+1 = i

• i4k+2 = -1

• i4k+3 = -i

Summary

Addition of two complex numbers

Consider two complex numbers z1 = a + ib, z2 = c + id .

z1 + z2 = (a + c) + i(b + d)

The sum of the real parts of the two given complex numbers gives the real part of the resultant complex number.

Similarly, the sum of their imaginary parts gives the imaginary part of the resultant complex number.

The resultant number is also a complex number.

Ex: (5 + 7i) + (6 + i)

Sol: (5 + 7i) + (6 + i) = (5 + 6) + (7 + 1)i

= 11 + 8i

Properties of addition of complex numbers

Closure Property: The sum of two complex numbers is a complex number.

z1 + z2 is a complex number for all complex numbers z1 and z2.

Commutative Property: Two complex numbers can be added in any order.

For any two complex numbers z1 and z2, z1 + z2 = z2 + z1.

Associative Property: In addition of three complex numbers, the numbers can be grouped in any order.

For any three complex numbers z1 , z2 and z3, (z1 + z2)+z3 = z1 +(z2 + z3).

Additive identity of a complex number: There exists a complex number 0 + 0i (denoted as 0), called the additive identity, such that for every complex number z, z + 0 = z.

Additive inverse of a complex number: For every complex number z = a + ib, there exists a complex number -a+i(-b) (denoted as -z), called the additive inverse or negative of z, such that z + (-z) = 0.

Subtraction of two complex numbers

Consider two complex numbers z1 = a + ib, z2 = c + id

z1 - z2

z1 + (-z2)

=(a+ib)+(-c-id)

=(a-c)+i(b-d)

Ex: (3 + 2i) - (2 - 7i)

(3 + 2i) - (2 - 7i) = (3 + 2i) + [-(2 - 7i)]

=(3 + 2i) + (-2 + 7i)

= 1 + 9i

Multiplication of two complex numbers

Consider two complex numbers z1 = a + ib,z2 = c + id .

z1 x z2 = (a + ib) x (c + id)

=(ac - bd) + i(ad + bc)


Ex: (1 + 4i) x (3 + 2i)

(1 + 4i) x (3 + 2i) = (1 x 3 - 4 x 2) + (1 x 2 + 4 x 3)i

=(3 - 8) + (2 + 12)i

= -5 + 14i

Hence, the product of two complex numbers is also a complex number.

Properties of multiplication of complex numbers:

Closure Property: The product of two complex numbers is a complex number.

z1z2 is a complex number for all complex numbers z1 and z2.

Commutative Property: Two complex numbers can be multiplied in any order.

For any complex numbers z1 and z2, z1z2 = z2z1

Associative Property: In multiplication of three complex numbers, the numbers can be grouped in any order.

For any three complex numbers z1,z2 and z3, (z1z2)z3 = z1(z2z3)

Multiplicative identity of a complex number: There exists a complex number 1 + 0i (denoted as 1), called the multiplicative identity, such that for every complex number z, z x 1= z.

Multiplicative inverse of a complex number: For every complex number z = a + ib, where a ≠ 0 and b ≠ 0, there exists a complex number 1/(a+ib) = a/(a2 + b2) +i -b/(a2 + b2) (denoted as 1/z or z-1), called the multiplicative inverse of z, such that z. 1/z = 1.

Distributive Property: For any three complex numbers z1, z2 and z3:

z1(z2 + z3) = z1z2 + z1z3

(z1 + z2)z3 = z1z3 + z2z3


Division of two complex numbers

Consider complex numbers z1 = a + ib , z2 = c + id ( z2 ≠ 0).

z1 ÷ z2 =  z1 x 1/z1

= (a + ib)/(c + id)

= ((a + ib)(c - id))/((c + id)(c - id))

= (ac - iad + ibc -i2bd)/(c2 - i2d2)

= (ac - iad + ibc + bd)/(c2 + d2)  [i2 = -1]

= ((ac + bd) + i(bc - ad)) /(c2 + d2)

∴ z1/z2 = (ac + bd)/(c2 + d2) + i(bc - ad)/(c2 + d2)

Ex: (2 + 5i) ÷ (1 + 2i)

= (2 + 5i)/(1 + 2i)

= ((2 + 5i)(1 - 2i)) / ((1 + 2i)(1 - 2i))

= (2 - 4i + 5i - 10i2)/(1 - 4i2)

= (2 + i + 10)/(1 + 4) (Since i2 = -1)

= (12 + i)/5

= (1/5)(12 + i)

Powers of 'i'

i2 = -1

i3 = i2.i = -i

i4 = (i2)2 = (-1)2 = 1

i5 = (i2)2 i= (-1)2 i= i

i6 = (i2)3 = (-1)3 = -1

i-1 = 1/i

    =(1x i)/(i x i) = i/i2

    = i/-1 = -i

i-2 = 1/i2 = 1/-1 = -1

i-3 = 1/i3 = 1/-i = 1/-i = -1xi / ixi = -i/i2 = -i/-1 = i

i-4 = 1/i4 = 1/1 = 1

WE notice from the above results that, for any integer k:

•  i4k = 1

• i4k+1 = i

• i4k+2 = -1

• i4k+3 = -i

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