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.
- 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.
- 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.
- 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.
- 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.
- 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_{1} = a + ib and z_{2} = c + id be two complex numbers where a, b, c and d are real numbers. Then the sum of the complex numbers = z_{1} + z_{2} = 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 z_{1} = 3 + 2i and z_{2} = 4 - 3i are two complex numbers. Show that their sum is also a complex number.
Solution:
The sum of the complex numbers = z_{1} + z_{2} = 3 + 2i + 4 - 3i
= 7 - i which is also a complex number. (Proved)
Let z_{1} = a + ib and z_{2} = c + id be two complex numbers where a, b, c and d are real numbers. Then the difference of the complex numbers = z_{1} - z_{2} = 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 z_{1} = 5 + 3i and z_{2} = 3 + 2i are two complex numbers. Show that their difference is also a complex number.
Solution:
The difference of the complex numbers = z_{1} - z_{2} = (5 + 3i) - (3 + 2i)
= 2 + i which is also a complex number.
(Proved)
Let z_{1} = a + ib and z_{2} = c + id be two complex numbers where a, b, c and d are real numbers. Then the product of the complex numbers = z_{1}z_{2} = (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 z_{1} = 2 + i and z_{2} = 3 + 2i are two complex numbers. Show that their product is also a complex number.
Solution:
The product of the complex numbers = z_{1}z_{2} = (2 + i)(3 + 2i)
= 6 + 4i + 3i + 2i^{2}
= 6 + 7i - 2 (putting i^{2} = -1)
= 4 + 7i which is also a complex number. (Proved)
Let z_{1} = a + ib and z_{2} = 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 z_{1} = z_{1} = 1 + i and z_{2} = 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)
If n is a positive integer then, z^{n} = 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:
z^{n} = z^{2} = (1 + i)(1 + i) = (1 + i)^{2} = 1 + 2i + i^{2}
= 1 + 2i - 1 = 2i which is also a complex number. (Proved)
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 = m^{n} (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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