How To Plot A Complex Number

Plotting complex numbers might seem daunting at first, but it unlocks a fascinating visual understanding of these mathematical entities, extending the familiar number line into a two-dimensional plane where both real and imaginary components coexist. Learning how to plot complex numbers is fundamental for anyone delving into electrical engineering, quantum mechanics, or pure mathematics, offering a geometric interpretation that simplifies calculations and reveals deeper connections between numbers and geometry.

Understanding the Complex Plane

Before diving into the plotting process, grasping the concept of the complex plane is crucial. Unlike the real number line, which represents numbers with only a magnitude, the complex plane introduces a second dimension to represent imaginary numbers.

• Real Axis: This is the horizontal axis, just like the number line you're familiar with. It represents the real part of a complex number.
• Imaginary Axis: This is the vertical axis, perpendicular to the real axis. It represents the imaginary part of a complex number.

A complex number is generally expressed in the form a + bi, where:

• a is the real part.
• b is the imaginary part.
• i is the imaginary unit, defined as the square root of -1.

Therefore, the complex number a + bi can be represented as a point (a, b) on the complex plane, similar to how you plot points on a Cartesian coordinate system.

Steps to Plot a Complex Number

Plotting a complex number on the complex plane is straightforward. Here’s a step-by-step guide:

1. Identify the Real and Imaginary Parts: Given a complex number in the form a + bi, identify the values of a (the real part) and b (the imaginary part).
2. Locate the Real Part on the Real Axis: Find the value of a on the horizontal (real) axis. This point represents the real component of your complex number.
3. Locate the Imaginary Part on the Imaginary Axis: Find the value of b on the vertical (imaginary) axis. This point represents the imaginary component of your complex number.
4. Plot the Point: Draw a vertical line from the point you located on the real axis and a horizontal line from the point you located on the imaginary axis. The point where these two lines intersect is the graphical representation of your complex number a + bi on the complex plane.
5. Label the Point: Label the point with the complex number it represents, such as a + bi or simply z.

Examples of Plotting Complex Numbers

Let's illustrate this process with a few examples:

Example 1: Plotting z = 3 + 2i

1. Identify Real and Imaginary Parts: a = 3 (real part), b = 2 (imaginary part).
2. Locate Real Part: Find 3 on the real axis.
3. Locate Imaginary Part: Find 2 on the imaginary axis.
4. Plot the Point: Draw a vertical line from 3 on the real axis and a horizontal line from 2 on the imaginary axis. The intersection point is (3, 2).
5. Label the Point: Label the point as 3 + 2i or z.

Example 2: Plotting z = -1 - i

1. Identify Real and Imaginary Parts: a = -1 (real part), b = -1 (imaginary part). Note that –i is the same as -1i.
2. Locate Real Part: Find -1 on the real axis.
3. Locate Imaginary Part: Find -1 on the imaginary axis.
4. Plot the Point: Draw a vertical line from -1 on the real axis and a horizontal line from -1 on the imaginary axis. The intersection point is (-1, -1).
5. Label the Point: Label the point as -1 - i or z.

Example 3: Plotting z = 4i

1. Identify Real and Imaginary Parts: a = 0 (real part), b = 4 (imaginary part). Notice that there's no real part explicitly written, so it's 0.
2. Locate Real Part: Find 0 on the real axis (the origin).
3. Locate Imaginary Part: Find 4 on the imaginary axis.
4. Plot the Point: The point is directly on the imaginary axis at (0, 4).
5. Label the Point: Label the point as 4i or z.

Example 4: Plotting z = -2

1. Identify Real and Imaginary Parts: a = -2 (real part), b = 0 (imaginary part). Notice that there's no imaginary part explicitly written, so it's 0.
2. Locate Real Part: Find -2 on the real axis.
3. Locate Imaginary Part: Find 0 on the imaginary axis (the origin).
4. Plot the Point: The point is directly on the real axis at (-2, 0).
5. Label the Point: Label the point as -2 or z.

Polar Form of Complex Numbers and Plotting

Complex numbers can also be represented in polar form, which uses the magnitude (or modulus) and argument (or angle) to define the number’s position on the complex plane.

• Modulus (r): The distance from the origin (0, 0) to the point representing the complex number on the complex plane. It is calculated as r = √(a² + b²), where a is the real part and b is the imaginary part.
• Argument (θ): The angle between the positive real axis and the line connecting the origin to the point representing the complex number, measured counterclockwise. It is calculated as θ = arctan(b/a). You need to be mindful of the quadrant in which the complex number lies to get the correct angle.

The polar form of a complex number is expressed as z = r(cos θ + i sin θ), often abbreviated as z = r cis θ.

Plotting from Polar Form:

1. Determine the Modulus (r): Calculate the distance from the origin.
2. Determine the Argument (θ): Calculate the angle from the positive real axis.
3. Draw a Circle: Imagine a circle centered at the origin with a radius equal to the modulus r.
4. Locate the Angle: Starting from the positive real axis, measure the angle θ counterclockwise.
5. Plot the Point: The point where the line representing the angle θ intersects the circle is the graphical representation of the complex number in polar form.

Example: Plotting z = 2(cos(π/3) + i sin(π/3))

1. Modulus: r = 2
2. Argument: θ = π/3 (which is 60 degrees)
3. Draw a Circle: Imagine a circle with a radius of 2 centered at the origin.
4. Locate the Angle: Measure 60 degrees counterclockwise from the positive real axis.
5. Plot the Point: The point where the 60-degree line intersects the circle is the point representing the complex number.

Why Plotting Complex Numbers Matters

Visualizing complex numbers through plotting isn't just a theoretical exercise; it has practical applications in various fields:

• Electrical Engineering: In AC circuit analysis, complex numbers are used to represent impedances (resistance, capacitance, and inductance). Plotting these impedances on the complex plane (often called an impedance diagram) helps engineers understand the behavior of circuits and design filters.
• Quantum Mechanics: Complex numbers are fundamental in quantum mechanics, where wave functions describing the state of a particle are complex-valued. Visualizing these wave functions on the complex plane can help understand the probability amplitudes and phases of quantum states.
• Signal Processing: Complex numbers are used to represent signals in the frequency domain using Fourier transforms. Plotting these complex numbers helps analyze the frequency components of a signal.
• Mathematics: Understanding the geometry of complex numbers is crucial in complex analysis, a branch of mathematics dealing with functions of complex variables. Plotting helps visualize concepts like complex mappings and transformations.
• Control Systems: Complex numbers are used to analyze the stability of control systems. Plotting the poles and zeros of a system's transfer function on the complex plane (the s-plane) provides insights into the system's stability and performance.

Transformations and Operations on the Complex Plane

Plotting complex numbers also provides a visual way to understand how operations on complex numbers affect their positions on the complex plane.

• Addition: Adding two complex numbers z1 = a + bi and z2 = c + di results in a new complex number z3 = (a+c) + (b+d)i. Geometrically, this corresponds to vector addition on the complex plane. If you draw vectors from the origin to the points representing z1 and z2, the vector sum will point to the location of z3.
• Subtraction: Subtracting two complex numbers z1 = a + bi and z2 = c + di results in a new complex number z3 = (a-c) + (b-d)i. Geometrically, this is similar to vector subtraction.
• Multiplication: Multiplying two complex numbers in polar form, z1 = r1(cos θ1 + i sin θ1) and z2 = r2(cos θ2 + i sin θ2), results in a new complex number z3 = r1r2(cos(θ1+θ2) + i sin(θ1+θ2)). Geometrically, the modulus of the product is the product of the moduli (r1r2), and the argument of the product is the sum of the arguments (θ1 + θ2). This means that multiplication involves scaling (changing the distance from the origin) and rotation (changing the angle).
• Division: Dividing two complex numbers in polar form, z1 = r1(cos θ1 + i sin θ1) and z2 = r2(cos θ2 + i sin θ2), results in a new complex number z3 = (r1/r2)(cos(θ1-θ2) + i sin(θ1-θ2)). Geometrically, the modulus of the quotient is the quotient of the moduli (r1/r2), and the argument of the quotient is the difference of the arguments (θ1 - θ2). This means that division involves scaling and rotation in the opposite direction compared to multiplication.
• Complex Conjugate: The complex conjugate of a complex number z = a + bi is z̄ = a - bi. Geometrically, the complex conjugate is a reflection of the original point across the real axis.
• Rotation: Multiplying a complex number by i corresponds to a 90-degree counterclockwise rotation around the origin. Multiplying by -i corresponds to a 90-degree clockwise rotation.

Common Mistakes and How to Avoid Them

While plotting complex numbers is generally straightforward, here are some common mistakes and how to avoid them:

• Confusing Real and Imaginary Axes: Always remember that the horizontal axis is the real axis and the vertical axis is the imaginary axis. Double-check which value corresponds to which axis.
• Incorrectly Calculating the Argument: When finding the argument (angle) using arctan(b/a), remember to consider the quadrant in which the complex number lies. The arctan function typically returns values between -π/2 and π/2, so you might need to add π or 2π to get the correct angle in other quadrants. For example, if both a and b are negative, the complex number is in the third quadrant, and you need to add π to the result of arctan(b/a).
• Misinterpreting Polar Form: Make sure you understand the relationship between the modulus, argument, and the rectangular form (a + bi) of a complex number. If you're given a complex number in polar form, correctly convert it to rectangular form before plotting if you find it easier.
• Forgetting the Imaginary Unit: Always remember to include the imaginary unit i when writing the imaginary part of a complex number. For example, if the imaginary part is 3, write it as 3i, not just 3.
• Incorrectly Applying Operations: When performing operations like multiplication or division in polar form, double-check your calculations for the modulus and argument. Remember that multiplication involves multiplying the moduli and adding the arguments, while division involves dividing the moduli and subtracting the arguments.
• Not Labeling the Points: Always label the points you plot with the corresponding complex number. This helps avoid confusion and makes it easier to refer to specific complex numbers on the plane.
• Assuming all numbers are real: Remember the real part can be zero, or the imaginary part can be zero. Don't assume a complex number must have both a real and imaginary component.

Advanced Concepts: Riemann Sphere

Beyond the basic complex plane lies a more advanced concept: the Riemann sphere. This provides a way to visualize complex infinity. The Riemann sphere is a sphere where each point on the sphere corresponds to a complex number, and a single point at the "north pole" of the sphere represents complex infinity. This representation elegantly handles the behavior of complex functions as their arguments approach infinity. While plotting directly on a Riemann sphere is beyond the scope of introductory plotting, understanding its existence highlights the richer and more sophisticated geometric interpretations possible with complex numbers.

Conclusion

Plotting complex numbers is a fundamental skill that unlocks a visual understanding of these essential mathematical entities. By mastering the process of representing complex numbers on the complex plane, you gain a powerful tool for solving problems and understanding concepts in various fields, from electrical engineering to quantum mechanics. The ability to visualize complex numbers provides a deeper intuition and simplifies complex calculations, revealing the beautiful interplay between algebra and geometry. Embrace the complex plane, and you'll unlock a new dimension in your mathematical understanding. Remember to practice, pay attention to detail, and explore the fascinating world of complex numbers!