Question

Simplify the following complex numbers and diagram: show them on an Argand diagram: (a) \(3+3 \mathrm{j}\) (b) \(2=4\) (e) \(-\mathrm{i}\) (f) \(-5=1 \mathrm{j}\) (c) \(-0.5\) (d) \(6 \mathrm{j}\) (a) \(j^{2}\) (b) \(-\mathrm{j}^{2}\) (c) \((-\mathrm{j})^{2}\) (d) \(j^{3}\left(\right.\) e) \(j^{4}\)

Short answer

Question: Simplify the given complex numbers, compute and simplify the expressions with j raised to different powers, and plot the results on an Argand diagram. Answer: The complex numbers are already in their simplest form. The simplified expressions for j raised to different powers are: (a) j^2 = -1, (b) -j^2 = 1, (c) (-j)^2 = -1, (d) j^3 = -j, (e) j^4 = 1. The corresponding Argand diagram represents these complex numbers as points or vectors in a 2-dimensional plane, with real parts along the x-axis and imaginary parts along the y-axis.

Step-by-step solution

Convert the given complex numbers into the form a + bj

All the given complex numbers are already in the simplest form, where a and b are real numbers and j is the imaginary unit.

Plot the complex numbers on an Argand diagram

An Argand diagram is a plot of complex numbers in a 2-dimensional plane showing the real part on the x-axis and imaginary part on the y-axis. Identify the real and imaginary parts of each complex number and plot them as points or vectors on the plane.

Compute and simplify j raised to different powers

Calculate j raised to various powers based on the properties and cyclicality of imaginary unit: - j^2 = -1 - j^3 = j * j^2 = j * -1 = -j - j^4 = j^2 * j^2 = -1 * -1 = 1 - (-j)^2 = (-1 * j)^2 = (-1)^2 * j^2 = 1 * -1 = -1 The simplified expressions are: (a) j^2 = -1 (b) -j^2 = 1 (c) (-j)^2 = -1 (d) j^3 = -j (e) j^4 = 1

Plot results of Step 3 on Argand diagram

For the results found in Step 3, plot the points on Argand diagram using their real and imaginary parts as coordinates: (a) -1 + 0j (b) 1 + 0j (c) -1 + 0j (d) 0 - 1j (e) 1 + 0j

Key concepts

Argand Diagram

The Argand diagram, also known as the complex plane, is a method of visualizing complex numbers. Each complex number corresponds to a point in this two-dimensional plane. Here, the horizontal axis (x-axis) represents the real part of the complex number, and the vertical axis (y-axis) represents the imaginary part.

This means for a number like \(a + bj\), \(a\) would be your real coordinate on the x-axis, and \(b\) is your imaginary coordinate on the y-axis. To plot a complex number on an Argand diagram, you:

• Identify the real and imaginary components.
• Mark the real part on the x-axis and the imaginary part on the y-axis.
• Draw a point where these two meet.

For example, for \(3 + 3j\), you'd plot the point (3, 3) on the plane. As a result, complex numbers are effectively graphically represented, helping us understand their properties visually.

Imaginary Unit

In the world of complex numbers, the imaginary unit, represented by \(j\) or \(i\), is a fundamental concept. It is defined as the square root of -1, which cannot be represented by any real number. In mathematical terms, we express this as \(j^2 = -1\).

The imaginary unit helps us express numbers that cannot be solved with just real numbers, especially those involving the square roots of negative numbers. Here’s how the imaginary unit is used:

• Given a negative square root, such as \(\sqrt{-4}\), it is expressed as \(2j\), where \(2\) is real and \(j\) the imaginary part.
• This allows for the creation of complex numbers, expressed generally as \(a + bj\), with \(a\) being the real part and \(bj\) the imaginary part.

Understanding \(j\) is vital in operating with complex numbers since it enables us to simplify and solve more complex mathematical problems.

Simplification of Complex Numbers

Simplifying complex numbers involves expressing them in the standard form \(a + bj\), where both \(a\) and \(b\) are real numbers. When given a complex expression or equation, the goal is to arrange it so that all the real parts and imaginary parts are clearly defined.

During simplification, follow these steps:

• Combine like terms, which means adding up the real parts separately from the imaginary parts.
• Simplify expressions involving \(j\) using known powers (like \(j^2 = -1\)).
• Rewrite the number in the form \(a + bj\).

For example, simplifying an expression that results in \(2+3j-4j+j^2\) would involve calculating \(j^2 = -1\), then combining the real and imaginary components to find the simplest form, like \(1-j\) here.

Powers of Imaginary Unit

Understanding the pattern of the powers of the imaginary unit \(j\) is key to working with complex numbers.

The imaginary unit follows a cyclical pattern:

• \(j^1 = j\)
• \(j^2 = -1\)
• \(j^3 = j \times j^2 = -j\)
• \(j^4 = j^2 \times j^2 = 1\)

This cycle repeats every four powers. So, for \(j^5\), the pattern starts over, \(j^5 = j^1\), and so on.

Having this understanding allows you to simplify expressions featuring powers of \(j\) quickly. For instance, when dealing with \(j^7\), knowing the cycle tells us immediately that \(j^7 = j^3 = -j\). Recognizing this repeating pattern saves time and reduces complexity in calculating expressions involving \(j\).