A complex number comprises of a real number and an imaginary number. It is usually represented in the form of z = a + ib, where a is the real part and b is the imaginary part. Here, i represents an imaginary unit number, whose value is equal to √-1. Thus, i = √-1.
Steps while adding or subtracting the complex numbers
- Adding or subtracting the complex numbers is simply combining the real and imaginary parts of the complex numbers and applying the operations separately on each of the combinations.
- All real numbers are complex numbers with imaginary parts equals to zero, but all complex numbers are not real numbers
- If the complex numbers are in polar form, we first convert them into cartesian form and apply the operations.
Addition of Complex Numbers
Let us consider two complex numbers z1 = a + ib and z2 = c + id. For adding the complex numbers, we simply combine the real and imaginary parts of the two complex numbers and then apply the addition operation. The formula for adding the complex numbers is given by:
z1 + z2 = (a + ib) + (c + id) = (a + c) + i (b + d)
If z = z1 + z2, then
z = (a + c) + i (b + d)
Subtraction of Complex Numbers
Two complex numbers z1 = a + ib and z2 = c + id can be subtracted by combining the real and imaginary parts of both the complex numbers and applying the subtraction operation separately on each of them. The formula for subtracting the complex numbers is given by:
z1 – z2 = (a + ib) – (c + id) = (a – c) + i (b – d)
If z = z1 – z2, then
z = (a – c) + i (b – d)
Properties of Adding or Subtracting Complex Numbers
- Closure property: The complex number formed after adding or subtracting complex numbers are also complex numbers.
- Associative property: This property holds for the addition of complex numbers only. That is for any three complex numbers z1, z2, and z3, we have
(z1 + z2) + z3 = z1 + (z2 + z3)
- Commutative property: This property holds for the addition of two complex numbers. For any two complex numbers z1 and z2, we have
z1 + z2 = z2 + z1
- Additive property: Here, 0 is the additive identity of complex numbers, since
z + 0 = 0 + z
- Additive inverse: For a complex number z, the additive inverse is -z, since z + (-z) = 0
Sample Problems
Question 1. Find the sum of the two complex numbers z = 3 + 5i and w = 6 – 2i.
Solution:
Since the given complex numbers have real and imaginary parts, we can combine them to find the net sum of both the complex numbers.
z + w = (3 + 5i) + (6 – 2i) = (3 + 6) + i (5 – 2)
z + w = 9 + 3i
Thus, the sum of the complex numbers is equal to 9 + 3i.
Question 2. Subtract the complex numbers z = 2 – 3i and w = -4 + 2i.
Solution:
Since, we can combine the real and imaginary terms of the complex numbers and apply our operations, we can write
z – w = (2 – 3i) – (-4 + 2i) = (2 -(-4)) + i (-3 -2)
z – w = 6 – 5i
Thus, the result is 6 – 5i.
Question 3. Given the complex numbers z1 = 3 + 2i, z2 = 5 – 3i and z3 = 1 + 2i, find the value of z1 + z2 – z3.
Solution:
Given the three complex numbers z1 = 3 + 2i, z2 = 5 – 3i and z3 = 1 + 2i, we can apply associative property of complex numbers to find the result.
Thus, we can write,
z1 + z2 – z3 = (z1 + z2) – z3 = ((3 + 2i) + (5 – 3i)) – (1 + 2i)
z1 + z2 – z3 = (8 – i) – (1 + 2i) = (8 – 1) + i(-1 – 2)
z1 + z2 – z3 = 7 – 3i
So, the answer is 7 – 3i.
Question 4. Given the two complex numbers z and v, where z = 6 + 9i. If the sum of the two complex numbers is twice the value when v is subtracted from z, find the value of v.
Solution:
Given, the complex number z = 5 + 2i.
According to the question,
z + v = 2 (z – v)
z + v = 2z – 2v
3v = z
v = z/3
Putting the value of z = 6 + 9i, we get
v = (6 + 9i)/3 = 6/3 + i (9/3) = 2 + 3i
v = 2 + 3i