Complex numbers can be thought of as points on a special map called the complex plane.

Each complex number looks like this:

z = a + bi

Here, a is the real part, and b is the imaginary part.

When we want to add two complex numbers, let’s say:

z1 = a1 + b1i
z2 = a2 + b2i

We can break this addition down into parts:

• Add the real parts: a1 + a2
• Add the imaginary parts: b1 + b2

So, when we add them together, we get:

z1 + z2 = (a1 + a2) + (b1 + b2)i

Looking at it Geometrically:

• Each complex number represents a point on the complex plane. This plane has an x-axis (the real part) and a y-axis (the imaginary part).

• When you add complex numbers, it’s like doing vector addition.

  • The real part moves you left or right (x-axis).
  • The imaginary part moves you up or down (y-axis).

So, to add two complex numbers, you place the tail of the second number (z2) at the head of the first number (z1). Then, you draw a line from the starting point (the origin) to the new endpoint of the combined vector.

In Summary:

• This way of visualizing complex numbers helps us understand how to add them.

• It also shows us important properties like commutativity (the order doesn't matter) and associativity (how we group them doesn’t matter either).

• Knowing this relationship makes it easier to work with complex numbers in different math areas.