Subtraction of Complex Numbers

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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