    # Algebra Of Complex Numbers • 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 = A + iB 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 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 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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