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Imagine the complex plane as a two-dimensional graph. It is kind of like your usual coordinate system, but for complex numbers. Here's how it works:

• The horizontal axis is the real axis, representing the real parts of complex numbers.
• The vertical axis is the imaginary axis, for the imaginary parts of complex numbers.

When you have a complex number, say \(-6 + 4i\), you plot it on this plane by:- Moving \(-6\) units to the left on the real axis.- Then, going \(+4\) units up on the imaginary axis.This creates a point on the plane, which visually represents your complex number. It helps in understanding complex numbers by making it easier to see their relationships and operations like addition, subtraction, and more geometrically.

Every complex number has two parts: real and imaginary. Think of them like different ingredients in a cooking recipe.- The **real part** contributes to the 'standard' value of the number. It's the amount you can directly understand without any 'twist.' For example, in \(-6 + 4i\), the real part is \(-6\).- The **imaginary part** is a bit more abstract. It involves the imaginary unit \(i\), which is defined as the square root of \(-1\). Here, the imaginary part is \(+4i\).These parts act independently in mathematical operations but together form a complex number.If you represent \(-6 + 4i\) geometrically on the complex plane:- \-6\ is plotted on the real axis.- \4i\ is represented on the imaginary axis.Understanding these parts separately can simplify operations involving complex numbers.

Multiplying complex numbers involves some interesting features that differ from regular multiplication.When multiplying \((2i)\times(2 + 3i)\):

• Use the distributive law: multiply each part separately, like \(2i imes 2\) and \(2i \times 3i\).
• Simplify by remembering that \(i^2 = -1\):\ (i) \ 4i \ from \(2i \times 2\)\ (ii) \ 6i^2 \ becomes \-6\ because multiplying \i^2\ results in \-1\, changing the sign.

Therefore, the result from \((2i)\times(2 + 3i)\) is \(-6 + 4i\).This shows that multiplication can change both the position and orientation of a number on the complex plane by altering its real and imaginary parts.