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Algebra Of Complex Numbers

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  • Sum of two complex numbers is a complex number i.e., the sum of two complex numbers can be expressed in the form A + iB where A and B are real.


  • Let z1 = a + ib and z2 = c + id be two complex numbers where a, b, c and d are real numbers. Then the sum of the complex numbers = z1 + z2 = a + ib + c + id

                                                                  = a + c + i(b + d)

                                                                  = A + iB

    Where A = a + c and B = b + d are real.

    Hence the sum of two complex numbers is a complex number.

    Example:

    If z1 = 3 + 2i and z2 = 4 - 3i are two complex numbers. Show that their sum is also a complex number.

    Solution:

    The sum of the complex numbers = z1 + z2 = 3 + 2i + 4 - 3i

    = 7 - i which is also a complex number. (Proved)

  • If z = a + ib (a, b are real) be a given complex number then the complex number (-a) + i(-b) is known as the negative of z and is denoted be (-z).


  • Difference of two complex numbers is a complex number i.e., the difference of two complex numbers can be expressed in the form A + iB where A and B are real.


  • Let z1 = a + ib and z2 = c + id be two complex numbers where a, b, c and d are real numbers. Then the difference of the complex numbers = z1 - z2 = a + ib - (c + id)

    = a - c + i(b - d)

    = A + iB

    Where A = a - c and B = b - d are real.

    Hence the difference of two complex numbers is a complex number.

    Example:

    If z1 = 5 + 3i and z2 = 3 + 2i are two complex numbers. Show that their difference is also a complex number.

    Solution:

    The difference of the complex numbers = z1 - z2 = (5 + 3i) - (3 + 2i)

                                                                  = 2 + i which is also a complex number.

                                                                          (Proved)

  • Product of two complex numbers is a complex number i.e., the product of two complex numbers can be expressed in the form A + iB where A and B are real.


  • Let z1 = a + ib and z2 = c + id be two complex numbers where a, b, c and d are real numbers. Then the product of the complex numbers = z1z2 = (a + ib)(c + id)


                             = ac - bd + i(bc + ad)

                             = A + iB

    Where A = ac - bd and B = bc + ad are real.

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

    Product of more than two complex numbers is also a complex number.

    Example:

    If z1 = 2 + i and z2 = 3 + 2i are two complex numbers. Show that their product is also a complex number.

    Solution:

    The product of the complex numbers = z1z2 = (2 + i)(3 + 2i)

    = 6 + 4i + 3i + 2i2

    = 6 + 7i - 2 (putting i2 = -1)

    = 4 + 7i which is also a complex number. (Proved)

  • Quotient of two complex numbers is a complex number i.e., the quotient of two complex numbers can be expressed in the form A + iB where A and B are real.


  • Let z1 = a + ib and z2 = c + id be two complex numbers where a, b, c, d are real numbers and c, d ≠ 0.

    Then the quotient

    Algebra of Complex Number

                                                                   = A + iB

    Algebra of 
                                   Complex Number

    Hence the quotient of two complex numbers is a complex number.

    Example:

    If z1 = z1 = 1 + i and z2 = 1 - i are two complex numbers. Show that their quotient is also a complex number.

    Solution:

    The quotient
    Algebra of Complex Number
    which is also a complex number. (Proved)

  • Any integral power of a complex number is a complex number i.e., any integral power of a complex number can be expressed in the form A + iB where A and B are real.Let z = x + iy be a complex number where x, y are real.


  • If n is a positive integer then, zn = z.z.z.....to n factors

                       = (x + iy)(x + iy).....to n factors

                        = A + iB

    (Since the product of more than two complex numbers is also a complex number), where A and B are real.

    Now if n is a negative integer, let n = -m (where m is a positive integer), then

    Algebra of Complex Number
    Hence any integral power of a complex number is a complex number.

    Example:

    If z = 1 + i and n = 2 then show that zn is also a complex number.

    Solution:

    zn = z2 = (1 + i)(1 + i) = (1 + i)2 = 1 + 2i + i2

                = 1 + 2i - 1 = 2i which is also a complex number. (Proved)

  • Any root of a complex numbers is a complex number i.e., any root of a complex number can be expressed in the form A + iB where A and B are real.


  • Let z = x + iy be a complex number (where x, y are real and x ≠ 0, y ≠ 0) and n is a positive integer.

    If the nth root of z be m then,

    n√z = m or n√x+ iy = m or x + iy = mn (1)

    Hence it is clear that mn is real when m is purely real quantity and mn is either purely real or purely imaginary quantity when m is purely imaginary quantity.

    Again x ≠ 0, y ≠ 0, hence equation (1) is satisfied if and only if m is an imaginary number of the form A + iB where A ≠ 0 and B ≠ 0 are real.

    Hence any root of a complex numbers is a complex number.

    Example:

    If z = 2 + i and n = 3 then show that n√z is also a complex number.

    Solution:

    Let n√z = m ⇒ 3√z = m

                        ⇒ 3√2+ i = m

                       ⇒ m = 3√2+ i

    Hence m is also an imaginary number of the form A + iB.

    Hence any root of z is also a complex number. (Proved)


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