Problem 17 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 11-20: Represent the complex number geometrically. $$ 2 i(2+3 i) $$

Step-by-Step Solution

Answer

The complex number \(-6 + 4i\) is represented by the point \((-6, 4)\) on the complex plane.

Step 1: Calculate the Product

To find the product of the two complex numbers, distribute the multiplication: \(2i(2+3i) = 2i \times 2 + 2i \times 3i\). Simplifying, we get \(4i + 6i^2\). Since \(i^2 = -1\), the expression becomes \(4i - 6\). Thus, the final result is \(-6 + 4i\).

Step 2: Understand the Components

The complex number \(-6 + 4i\) can be decomposed into real and imaginary parts:- Real part: \(-6\)- Imaginary part: \(4\)

Step 3: Plot the Complex Number

Complex numbers are represented on the complex plane with the real part on the x-axis and the imaginary part on the y-axis. - Plot the point corresponding to the real part \(-6\) on the x-axis.- Plot the point corresponding to the imaginary part \(4\) on the y-axis.- The point \((-6, 4)\) represents the complex number geometrically.

Key Concepts

Complex Plane

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.

Real and Imaginary Parts

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.

Multiplication of 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.