We will discuss here about the usual mathematical operation  subtraction of two complex numbers.
How do you subtract Complex Numbers?
Let z\(_{1}\) = p + iq and z\(_{2}\) = r + is be any two complex numbers, then the subtraction of z\(_{2}\) from z\(_{1}\) is defined as
z\(_{1}\)  z\(_{2}\) = z\(_{1}\) + (z\(_{2}\))
= (p + iq) + (r  is)
= (p  r) + i(q  s)
The following steps of subtraction of complex numbers are given below:
Step I: Distribute the negative
Step II: Group the real part of the complex number and the imaginary part of the complex number.
Step III: Combine the like terms and simplify
For example, let z\(_{1}\) = 6 + 4i and z\(_{2}\) = 7 + 5i, then
z\(_{1}\)  z\(_{2}\) = (6 + 4i)  (7 + 5i)
= (6 + 4i) + (7  5i), [Distributing the negative sign]
= (6 + 7) + (4  5)i, [Grouping the real part of the complex number and the imaginary part of the complex number.]
= 13  i, [Combining the like terms and simplify]
and z2  z1 = (7 + 5i)  (6 + 4i)
= (7 + 5i) + (6  4i), [Distributing the negative sign]
= (7  6) + (5  4)i, [Grouping the real part of the complex number and the imaginary part of the complex number.]
= 13 + i
Solved examples on subtraction of complex numbers:
1. Find the difference between the complex numbers (2 + 3i) from (9  2i).
Solution:
(9  2i)  (2 + 3i)
= (9  2i) + (2  3i), [Distributing the negative sign]
= ( 9  2) + (2  3)i, [Grouping the real part of the complex number and the imaginary part of the complex number.]
= 11  5i
2. Evaluate: (7√5 + 3i)  (√5  2i)
Solution:
(7√5 + 3i)  (√5  2i)
= (7√5 + 3i) + (√5 + 2i), [Distributing the negative sign]
= (7√5  √5) + (3 + 2)i, [Grouping the real part of the complex number and the imaginary part of the complex number.]
= 6√5 + 5i
3. Express the complex number (8  3i)  (6 + 2i) in the standard form a + ib.
Solution:
(8  3i)  (6 + 2i)
= (8  3i) + (6  2i), [Distributing the negative sign]
= (8 + 6) + (3 – 2)i, [Grouping the real part of the complex number and the imaginary part of the complex number.]
= 14  5i, which is the required form.
Note: The final answer of Subtraction of complex numbers must be in simplest or standard form a + ib.
`11 and 12 Grade Math
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