Complex numbers may sound, well, complex, but adding them doesn’t have to be confusing. Today, we’re going to delve into the addition of complex numbers through two different methods. Whether you’re a fan of visual explanations or more algebraic methods, there’s a technique here for you!
The Complex Plane & The Parallelogram Law
Recalling The Complex Plane
Imagine plotting a complex number [latex] z=a+bi [/latex] on this plane.
The number ‘a’ determines its horizontal position, and ‘b’ determines its vertical position. Hence, every point on the complex plane corresponds to a unique complex number.
You can read this detailed blog on Complex Plane and How it Shapes Math –
Parallelogram Law of Addition of Complex Numbers
Imagine two complex numbers, let’s call them W and Z. On the complex plane, these two numbers will be two points. If you were to draw lines from the origin (0,0) to each of these points, you’d form two vectors.
To add these complex numbers, follow these steps:
- Draw the vector for the first complex number, W.
- Starting at the tip of W, draw the vector for the second complex number, Z.
- Now, connect the origin to the tip of the resultant vector (which ends at the tip of Z). This resultant vector represents the sum of W and Z.
In other words, the diagonal of the parallelogram formed by the two complex numbers W and Z gives the resultant sum. This is the parallelogram law for the Addition of Complex Numbers.
Watch this video for a better understanding –
Example of Addition of Complex Numbers using Parallelogram Law
Let’s consider two complex numbers: z = 5 – 8i and w = -5 – i.
When plotted as vectors, these two numbers form two sides of a parallelogram. Using the parallelogram law, the diagonal represents the addition of complex numbers: w + z = -9i.
Here is the video explanation of the above example –
Algebraic Addition of Complex Numbers
This method is perfect for those who prefer a step-by-step approach without the need for graphical representations.
Breaking Down a Complex Number
Every complex number can be expressed in the form a +bi, with ‘a’ as the real part and ‘b’ as the imaginary part. Adding two complex numbers is about separately adding their real and imaginary components.
Example for Algebraically Adding 2 or More Complex Numbers
Consider two complex numbers z = -8 + 3i and w = 2 – 7i.
In order to add the two complex numbers z and w, let’s add their real parts first.
Real part addition : -8 + 2 = -6
And now the imaginary parts –
Imaginary part addition : 3 + (-7) = 3 – 7 = -4
Now combine both the results in order to get z + w = -6 – 4i.
Let’s also confirm the answer that we got by adding the two complex numbers in the previous section by PArallelogram Law.
We had z = 5 – 8i and w = -5 – i.
Real part addition : 5 + (-5) = 0
And now the imaginary parts –
Imaginary part addition : -8 + (-1) = -8 – 1 = -9
Now combine both the results in order to get z + w = 0 + (-9)i = -9i.
Common Mistakes to Avoid While Adding Complex Numbers
When working with the addition of complex numbers, there are a few pitfalls one should be aware of:
- Not Separating Real and Imaginary Parts: Always ensure that you’re adding like terms. Real parts with real parts, and imaginary with imaginary.
- Misplotting on the Complex Plane: Ensure that you’re plotting your initial complex numbers correctly. Double-checking can save you from errors down the road.
- Overthinking: Whether you’re using the parallelogram law or the algebraic method, keep it simple. Both techniques are straightforward, so trust in the process.
Conclusion
Complex numbers might seem tricky at first, but there are simple ways to add them. You can use easy-to-understand pictures like the parallelogram rule or just do regular math. The more you practice and play around with these numbers, the easier adding them will get. So, keep trying, and soon you’ll find it a breeze!
