Complex Numbers Properties

As in the case of Real Numbers, the closure, associative, and commutative properties hold true for complex numbers.

Let C be the set of complex numbers. It is denoted as

C = {z | z = a + ib where a, b ɛ R}

  1. Closure Property of Addition
    If z1, z2 ɛC then z1 + z2ɛ C.

  2. Closure Property of Multiplication
    If z1, z2 ɛC then (z1)(z2) ɛC.

  3. Associative Property of Addition
    If z1, z2, z3 ɛC then
    z1 + (z2 + z3) = (z1 + z2) + z3.

  4. Associative Property of Multiplication
    If z1, z2, z3 ɛC then
    z1((z2)(z3)) = ((z1)(z2))z3.

  5. Commutative Property of Addition
    If z1, z2 ɛC then
    z1 + z2 = z2 + z1

  6. Commutative Property of Multiplication
    If z1, z2 ɛC then
    (z1)(z2) = (z2)(z1)

  7. Additive Identity
    There exists for every z ɛC a z0 ɛC such that
    z + z0 = z0 + z = z
         z0 = 0 + 0i
    Since z = a + ib
      z + z0= (a + ib) + (0 + 0i)
               = (a + 0) + i(b + 0)
               = a + ib = z.
    z0 = 0 + 0i is called the identity element of C.

  8. Additive Inverse
    For every z ɛC there exists a z1 ɛC such that
    z + z1 = z1 + z = z0
          Consider z = x + iy
                 Let z1 = -x - iy
         Then z + z1 = x + iy + (-x - iy)
                          = (x - x) + i(y - y)
                          = 0 + 0i = z0
    We denote z1 = -z = -x - iy

  9. Multiplicative Identity Property
    For every z ɛC, there exists a z1ɛ C such that
    z ∗ z1 = z1 ∗ z = z
    Let z = a + ib,         a1 = a, b1 = b
         z1= 1 + 0i,         a2 = 1, b2 = 0
    Then z∗ z1 = ((a)∗ (1) - b ∗ 0) + i((a) ∗ (0) + (1) ∗ (b))
                      = a + ib = z
    ∴ z1 = 1 + 0i = 1 is called the multiplicative identity element of C.

  10. Multiplicative Inverse Property
    For every z ɛC other than z0, there exists an element z1 ɛC such that z1.z = z.z1 = z1= 1
    Let z ɛC such that z = x + iy where x ≠ 0, y ≠ 0. Then x2 + y2 ≠ 0.
    Let z1 =
    z.z1 = (x + iy)∗


  11. Distributive Property
    If z1, z2, z3 ɛC then
    z1.(z2 + z3) = z1.z2 + z1.z3

Try these questions:

  1. Show that the sum of 5 + 8i and 6 - 4i is a complex number and name the property used.
    Answer: The sum of 5 + 8i and 6 - 4i is
    5 + 8i + 6 - 4i = (5 + 6) + i(8 - 4)
                         = 11 + 4i
    11 + 4i is a complex number.
    The closure property of addition is used.

  2. Show that the product of -16 + 2i and 11 - 2i is a complex number and name the property used.
    Answer: The product of -16 + 2i and 11 - 2i is
      (a1+ib1) ∗ (a2+ ib2) = (a1a2-b1b2) + i(a1b2+ a2b1)
    (-16 + 2i) ∗ (11 - 2i) = [(-16) ∗ (11) - (2) ∗ (-2)] + i[(-16) ∗ (-2) + (2) ∗ (11)]
                                 = [-176 + 4] + i[32 + 22]
                                 = -172 + 54i   a complex number
    This property is called closure property of multiplication.

  3. If z1 = -26 + 29i, z2 = -12i + 27, z3 = 22 + 3i. Show that
    z1 + (z2 + z3 ) = (z1 + z2 )+ z3
    Name the property.

    Answer: z1= -26 + 29i
    z2 = -12i + 27
    z3 = 22 + 3i
    z1 + (z2 + z3) = (z1 + z2) + z3
    ( -26 + 29i) + [(-12i + 27) + (22 + 3i)] = [(-26 + 29i) + (-12i + 27)] +(22 + 3i)
      (-26 + 29i) + [(27 + 22) + i(-12 + 3)] = [(-26 + 27) + i(29 – 12)] + (22 + 3i)
                              -26 + 29i +[49 - 9i]= [1 + 17i] + 22 + 3i
                            (-26 + 49) + i(29 - 9) = (1 + 22) + i(17 + 3)
                                              23 + 20i = 23 + 20i
    This is the associative property of addition.


  4. If z1 = -7 + 3i, z2 = 11 + 14i, show that z1.z2 = z2.z1 and name the property used.
    Answer: z1 = -7 + 3i,    z2 = 11 + 4i
    z1 ∗ z 2 = z2 ∗ z1
    (-7 + 3i) ∗ (11 + 4i) = (11 + 4i) ∗ (-7 + 3I)
    [(-7) ∗ (11) - (3) ∗ (4)] + i[(-7) ∗ (4) + (11) (3)]
                                                           = [(11) ∗ (-7) - (4)∗ (3)] + i[(11)∗ (3) + (4) ∗ (-7)]
                          [-77 - 12] + i[-28 + 33] = [-77 -12] + i[33 - 28]
                                                -89 + 5i = -89 + 5i
    This is the commutative property of multiplication.

  5. Find the additive inverse of -25 + 13i.
    Answer: The additive inverse of z is -z
    ∴   z = -25 + 13i
        -z = -(-25 + 13i) = 25 - 13i
    25 - 13i is the additive inverse of -25 + 13i
    verification: z + (-z) = (-z) + z = 0 + 0i
    (-25 + 13i) + (25 - 13i) = (-25+25) + i(13-13)
                                    = 0 + 0i
    (25 - 13i) + (-25 +13i) = (25-25) + i(-13+13)
                                    = 0 + 0i

  6. Find the multiplicative inverse of 4i - 3. Answer:

  7. If z1 = 14 - 2i, z2 = 1/2+ 3i, z3 = 6 - 1/9i show that z1(z2∗ z3) = (z1∗ z2)z3 and name the property used.

    Answer: If z1 = 14 -2i, z2 = 1/2+ 3i, z3 = 6 -1/9i
    show that z1∗ (z2∗ z3) = (z1∗ z2) ∗ z3
    Consider the right-hand side of the equation.
    (z1 ∗ z2) ∗ z3
    (z1∗ z2) = (14 - 2i)(1/2 + 3i)
               = [14 ∗ 1/2 - (-2) ∗ 3] + i[14 ∗ (3) + 1/2 ∗ (-2)]
               = (7 + 6) + i[42 - 1]
               = 13 + 41i
    (z1∗ z2) ∗ z3 = (13 + 41i) ∗ (6 – 1/9 i)
                      = [13 ∗ (6) – 41 ∗ (-1/9)] + i[13 ∗ (-1/9) + 6 ∗ (41)]
                      = [78 + 41/9 ]+ i[-13/9 + 246]


    8. Consider the left-hand side of the equation

    z1 ∗ (z2∗ z3)

    (z2 ∗ z3) = (1/2 + 3i) ∗ (6 – 1/9i)

                 = [1/2 ∗ (6) – 3 ∗ (-1/9)] + i[1/2 ∗ (-1/9) + 3∗ 6]

                 = (3 + 1/3) + i[-1/18 + 18]

                  = 10/3 + 323/18i

    z1 ∗ (z2 ∗ z3) = (14 - 2i)(10/3 + 323/18i)

                       = (14(10)/3 - (-2)323/18) + i[14(323/18 + 10/3 ∗ (-2)]

                       = [140/3 + 323/9] + i[2261/9 - 20/3]

    We see that

    z1∗ (z2∗ z3) = (z1∗ z2) ∗ z3

    This is called the associative property of multiplication.