Complex Numbers AdditionSubtraction

Given two complex numbers

z1 = a1 + ib1 and z2 = a2 + ib2

        Then z1+ z2 = (a1 + a2) + i (b1+b2)

        And z1 - z2 = (a1 - a2) + i (b1- b2)

Example 1

Add the complex numbers

6+ 4i, 3 + 2i

Let z1 = 6+ 4i, a1 = 6, b1 = 4

     z2 = 3 +2i, a2 = 3, b2 = 2

Comparing with z1 + z2 = (a1 + a2) + i(b1+ b2)

                   6+ 4i + 3+ 2i = (6 + 3) + i (4 + 2)

                                         = 9 + 6i

Example 2

Add the complex numbers

-3-4i, 2+ 6i

Let z1= -3- 4i, a1= -3, b1 = -4

     z2= 2+ 6i,a2 = 2,b2 = 6

Comparing with z1 + z2 = (a1+ a2) +i (b1+b2)

                   -3-4i + 2 + 6i = (-3 + 2) + i (-4+ 6)

                                          = -1+2i

Example 3

Add the complex numbers

-5+ 6i, -5-6i

Let z1= -5+ 6i, a1= -5, b1 = 6

      z2= -5+ 6i, a2 = -5,b2 = -6

Comparing with z1 + z2 = (a1+ a2) +i (b1+b2)

                 (-5+ 6i) (-5-6i) = (-5 -5) +i (6- 6)

                                           = -10

Example 4

Add 4/3+2/7 i, - 11/2 - 5/6 i

Let z1 = 4/3+2/7 i,   a1= 4/3,    b1= 2/7

     z2 = - 11/2- 5/6 i, a1= -, b2= - 5/6

Comparing with z1 + z2 = (a1+ a2) +i (b1+b2)

   4/3+2/7 i - 11/2 - 5/6 i = (4/3 - 11/2 ) + i (2/7 - 5/6)

Example 5

Subtract 6+3i from 5+7i

Let z1 = 5+7i, a1= 5,  b1=7s

     z2 = 6+3i, a2 = 6, b2= 3

Comparing with z1 - z2 = (a1- a2) +i (b1-b2)

              (5+7i)- (6+3i) = (5-6) +i (7-3)

                                 = -1+ 4i

Example 6

Subtract - 506 + 217i from - 510 + 560i

Let z1 = - 510 + 560 i, a1= - 510,  b1= 560

     z2 = - 506 + 217i, a2 = - 506, b2= 217

     Comparing with z1 - z2 = (a1- a2) +i (b1-b2)

 (- 510 + 560i) - (- 506 + 217i)

                                      = (-510 - (-506) ) + i(560 - 217)

                                      = ( - 510+ 506) + i (343)

                                      = - 4+ 343i

Example 7

Subtract 12/7+ 17/4i from 2/3- 5/6i

Let z1 = 2/3- 5/6i,        a1= 2/3,   b1= -5/6

     z2 = 12/7+ 17/4 i,  a2 =, b2= 17/4

  Comparing with z1 - z2 = (a1- a2) +i (b1-b2)

(2/3 - 5/6 i) - (12/7 + 17/4 i) = (2/3 - 12/7) + i (- 5/6- 17/4)

Try these questions

1.    Add the following complex numbers

a.    -16+ 3i ; 7+ 4i

Answer: Let z1 = -16+3i, a1 = -16, b1 =3

     z2 = 7+4i,     a2 = 7,    b2= 4

Comparing with z1 + z2 = (a1+ a2) +i (b1+b2)

        (-16+3i) +(7+4i) = (-16 +7) +i (3+4)

                               = -9+7i

b.    

Answer:

c.    100 -14i; -6 -12i

Answer: 100 - 14i ; -6 -12i ;

Let z1 = 100 -14i a1= -100 b1 = -14

     z2 = -6 -12i       a2 = -6     b2= -12

Comparing with z1 + z2 = (a1+ a2) +i (b1+b2)

(100- 14i) + (-6-12i) = (100- 6) +i (-14-12)

                             = 94 - 26i


2.    Subtract the second number from the first number

a.    8 + 7i ; 6 - 41i

Answer: 8 + 7i; 6- 41i

Let z1=8+ 7i       a1= 8            b1 =7

     z2 = 6- 41i    a2 = 6            b2 = - 41

Comparing with z1 - z2 = (a1- a2) + i (b1- b2)

(8 + 7i) - (6 - 41i) = (8 - 6) + i (7 - (-41))

                         = 2 + i (7+ 41)

                         = 2 + 48i


b.    √7+ 13i ; 20 - 4√6 i

Answer: √7+ 13i; 20 - 4√6i

Let z1 = √7+ 13i,      a1 =,     b1 = 13

      z2 = 20 - 4√6i,     a2 = 20,      b2 = -4√6

Comparing with z1 - z2 = (a1 - a2) + i (b1 - b2)

( √7+ 13i) - (20 - 4√6i) = (√7 - 20) + i(13 - (-4√6))

= (√7 - 20) + (13 + 4√6)i


c.    - 9+ 10i ; -15 - 12i

Answer: -9 + 10i; -15 -12i

Let z1=-9 +10i,     a1= -9     b1 =10

     z2 = -15 –12i,  a2 = -15   b2 = -12

Comparing with z1 - z2 = (a1- a2) + i (b1- b2)

(-9 +10i)-(-15 - 12i) = (-9 - (-15) + i (10 - (-12))

                             = (-9 + 15) + i (10 + 12)

                             = 6 + 22i