Addition of Two Complex Numbers

We will discuss here about the usual mathematical operation - addition of two complex numbers.

How do you add Complex Numbers?

Let z\(_{1}\) = p + iq and z\(_{2}\) = r + is be any two complex numbers, then their sum z\(_{1}\) + z\(_{2}\) is defined as

z\(_{1}\) + z\(_{2}\) = (p + r) + i(q + s).

For example, let z\(_{1}\) = 2 + 8i and z\(_{2}\) = -7 + 5i, then

z\(_{1}\) + z\(_{2}\) = (2 + (-7)) + (8 + 5)i = -5 + 13i.


If z\(_{1}\), z\(_{2}\), z\(_{3}\) are any complex numbers, then it is easy to see that

(i) z\(_{1}\) + z\(_{2}\) = z\(_{2}\) + z\(_{1}\)             (Commutative law)

(ii) (z\(_{1}\) + z2) + z\(_{3}\) = z\(_{1}\) + (z\(_{2}\) + z\(_{3}\)), (Associative law)

(iii) z + 0 = z = 0 + z, so o acts as the additive identity for the set of complex numbers.


Negative of a complex number:

For a complex number, z = x + iy, the negative is defined as -z = (-x) + i(-y) = -x - iy.

Note that z + (-z) = (x - x) + i(y - y) = 0 + i0 = 0.

Thus, -z acts as the additive inverse of z.


Solved examples on addition of two complex numbers:

1. Find the addition of two complex numbers (2 + 3i) and (-9 - 2i).

Solution:

(2 + 3i) + (-9 - 2i)

= 2 + 3i - 9 - 2i

= 2 - 9 + 3i - 2i

= -7 + i

 

2. Evaluate: (2√3 + 5i) + (√3 - 7i)

Solution:

2√3 + 5i + √3 - 7i

= 2√3 + √3 + 5i - 7i

= 3√3 - 2i

 

3. Express the complex number (1 - i) + (-1 + 6i) in the standard form a + ib.

Solution:

(1 - i) + (-1 + 6i)

= 1 - i -1 + 6i

= 1 - 1 - i + 6i

= 0 + 5i, which is the required form.


Note: The final answer of addition of two complex numbers must be in simplest or standard form a + ib.







11 and 12 Grade Math 

From Addition of Two Complex Numbers to HOME PAGE




Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.