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Addition of two complex numbers

Consider two complex numbers z_{1} = a + ib, z_{2} = c + id .

z_{1} + z_{2} = (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.

z_{1} + z_{2} is a complex number for all complex numbers z_{1} and z_{2}.

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

For any two complex numbers z_{1} and z_{2}, z_{1} + z_{2 = }z_{2} + z_{1}.

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

For any three complex numbers z_{1} , z_{2} and z_{3}, (z_{1} + z_{2})+z_{3} = z_{1} +(z_{2} + z_{3}).

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 z_{1} = a + ib, z_{2} = c + id

z_{1} - z_{2}

= z_{1} + (-z_{2})

=(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 z_{1} = a + ib,z_{2} = c + id .

z_{1} x z_{2 }= (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.

z_{1}z_{2} is a complex number for all complex numbers z_{1} and z_{2}.

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

For any complex numbers z_{1} and z_{2}, z_{1}z_{2 = }z_{2}z_{1}

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

For any three complex numbers z_{1},z_{2} and z_{3}, (z_{1}z_{2})z_{3} = z_{1}(z_{2}z_{3})

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 â‰ 0 and b â‰ 0, there exists a complex number 1/(a+ib) = a/(a^{2} + b^{2}) +i -b/(a^{2} + b^{2}) (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 z_{1}, z_{2} and z_{3}:

z_{1}(z_{2} + z_{3}) = z_{1}z_{2} + z_{1}z_{3}

(z_{1} + z_{2})z_{3} = z_{1}z_{3} + z_{2}z_{3}

**Division of two complex numbers**

Consider complex numbers z_{1} = a + ib , z_{2} = c + id ( z_{2 }â‰ 0).

z_{1 }Ã· z_{2 = }z_{1 }x 1/z_{1}

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

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

= (ac - iad + ibc -i^{2}bd)/(c^{2} - i^{2}d^{2})

= (ac - iad + ibc + bd)/(c^{2} + d^{2}) [i^{2} = -1]

= ((ac + bd) + i(bc - ad)) /(c^{2} + d^{2})

âˆ´ z_{1}/z_{2} = (ac + bd)/(c^{2} + d^{2}) + i(bc - ad)/(c^{2} + d^{2})

Ex: (2 + 5i) Ã· (1 + 2i)

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

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

= (2 - 4i + 5i - 10i^{2})/(1 - 4i^{2})

= (2 + i + 10)/(1 + 4) (Since i^{2} = -1)

= (12 + i)/5

= (1/5)(12 + i)

Powers of 'i'

i^{2} = -1

i^{3} = i^{2}.i = -i

i^{4} = (i^{2})^{2} = (-1)^{2} = 1

i^{5} = (i^{2})^{2} i= (-1)^{2} i= i

i^{6} = (i^{2})^{3} = (-1)^{3} = -1

i^{-1} = 1/i

=(1x i)/(i x i) = i/i^{2}

= i/-1 = -i

i^{-2} = 1/i^{2} = 1/-1 = -1

i^{-3} = 1/i^{3} = 1/-i = 1/-i = -1xi / ixi = -i/i^{2} = -i/-1 = i

i^{-4} = 1/i^{4} = 1/1 = 1

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

â€¢ i^{4k} = 1

â€¢ i^{4k+1} = i

â€¢ i^{4k+2} = -1

â€¢ i^{4k+3} = -i

Addition of two complex numbers

Consider two complex numbers z_{1} = a + ib, z_{2} = c + id .

z_{1} + z_{2} = (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.

z_{1} + z_{2} is a complex number for all complex numbers z_{1} and z_{2}.

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

For any two complex numbers z_{1} and z_{2}, z_{1} + z_{2 = }z_{2} + z_{1}.

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

For any three complex numbers z_{1} , z_{2} and z_{3}, (z_{1} + z_{2})+z_{3} = z_{1} +(z_{2} + z_{3}).

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 z_{1} = a + ib, z_{2} = c + id

z_{1} - z_{2}

= z_{1} + (-z_{2})

=(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 z_{1} = a + ib,z_{2} = c + id .

z_{1} x z_{2 }= (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.

z_{1}z_{2} is a complex number for all complex numbers z_{1} and z_{2}.

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

For any complex numbers z_{1} and z_{2}, z_{1}z_{2 = }z_{2}z_{1}

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

For any three complex numbers z_{1},z_{2} and z_{3}, (z_{1}z_{2})z_{3} = z_{1}(z_{2}z_{3})

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 â‰ 0 and b â‰ 0, there exists a complex number 1/(a+ib) = a/(a^{2} + b^{2}) +i -b/(a^{2} + b^{2}) (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 z_{1}, z_{2} and z_{3}:

z_{1}(z_{2} + z_{3}) = z_{1}z_{2} + z_{1}z_{3}

(z_{1} + z_{2})z_{3} = z_{1}z_{3} + z_{2}z_{3}

**Division of two complex numbers**

Consider complex numbers z_{1} = a + ib , z_{2} = c + id ( z_{2 }â‰ 0).

z_{1 }Ã· z_{2 = }z_{1 }x 1/z_{1}

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

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

= (ac - iad + ibc -i^{2}bd)/(c^{2} - i^{2}d^{2})

= (ac - iad + ibc + bd)/(c^{2} + d^{2}) [i^{2} = -1]

= ((ac + bd) + i(bc - ad)) /(c^{2} + d^{2})

âˆ´ z_{1}/z_{2} = (ac + bd)/(c^{2} + d^{2}) + i(bc - ad)/(c^{2} + d^{2})

Ex: (2 + 5i) Ã· (1 + 2i)

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

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

= (2 - 4i + 5i - 10i^{2})/(1 - 4i^{2})

= (2 + i + 10)/(1 + 4) (Since i^{2} = -1)

= (12 + i)/5

= (1/5)(12 + i)

Powers of 'i'

i^{2} = -1

i^{3} = i^{2}.i = -i

i^{4} = (i^{2})^{2} = (-1)^{2} = 1

i^{5} = (i^{2})^{2} i= (-1)^{2} i= i

i^{6} = (i^{2})^{3} = (-1)^{3} = -1

i^{-1} = 1/i

=(1x i)/(i x i) = i/i^{2}

= i/-1 = -i

i^{-2} = 1/i^{2} = 1/-1 = -1

i^{-3} = 1/i^{3} = 1/-i = 1/-i = -1xi / ixi = -i/i^{2} = -i/-1 = i

i^{-4} = 1/i^{4} = 1/1 = 1

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

â€¢ i^{4k} = 1

â€¢ i^{4k+1} = i

â€¢ i^{4k+2} = -1

â€¢ i^{4k+3} = -i