Subtracting complex numbers

Join us for today's lesson where we will demystify the process of subtracting complex numbers. If you've ever wondered how to tackle this, then you're in the right place.

Here's the gist: when you need to calculate the difference between two complex numbers, let's say (a,b)=a+bi and (c,d)=c+di, all you need to do is subtract the corresponding elements from each pair. In mathematical terms $$ (a,b)-(c,d)=(a-c,b-d) $$

Let's simplify this. Picture the first complex number (minuend), with its real part (illustrated in red) and the imaginary part (in blue). The subtraction process involves taking these parts and subtracting from them their counterparts in the second complex number (subtrahend).

Let's take a deep dive into an example to make it crystal clear.

Let's work with two arbitrary complex numbers:

$$ z_1 = (2,3) = 2+3i $$

$$ z_2 = (4,2) = 4+2i $$

Now, let's visualize these on the complex plane:

To perform the subtraction, extract the real and imaginary parts of the first complex number (z₁) and deduct the corresponding parts from the second complex number (z₂).

$$ z_1 - z_2 = (2,3) - (4,2) = (2-4,3-2) = (-2,1) $$

Voila! The resulting difference between the two complex numbers is another complex number (-2,1), which can be expressed as -2+i.

We can also bring to life the subtraction of complex numbers graphically, using the principles of vector addition.

Step one is to identify the opposite vector -v of the second complex number (subtrahend).

Next, you add the vector u of the first complex number (minuend) to this opposite vector -v of the subtrahend. You can use either the parallelogram method or the head-to-tail method.

The resulting vector is none other than the difference z₁-z₂ between the two complex numbers.

It's also worth noting that the subtraction of complex numbers isn't exclusive to their coordinate form. You can also perform this operation when the complex numbers are in algebraic form.

Let's say we have:

$$ z_1-z_2 = (2,3i)-(4+2i) $$

The method remains the same - associate and subtract the real and imaginary parts of both complex numbers:

$$ z_1+z_2 = (2+3i)-(4+2i) = (2-4)+(3i-2i) = -2+i $$

And just like that, you arrive at the same outcome.

We hope you've enjoyed this journey into the world of complex numbers with Nigiara. We encourage you to stay with us for more insightful lessons.