Problem 14 - Precalculus Mathematics for Calculus

Question

Graph the complex number and find its modulus. $$\frac{-\sqrt{2}+i \sqrt{2}}{2}$$

Step-by-Step Solution

Answer

The modulus of the complex number is 1.

Step 1: Express the Complex Number

First, express the complex number in standard form as \(a + bi\). The complex number given is \(-\sqrt{2}+i \sqrt{2}\) divided by 2. Simplifying this expression, we write: \[ z = \frac{-\sqrt{2}}{2} + i \frac{\sqrt{2}}{2} \] where \(a = \frac{-\sqrt{2}}{2}\) and \(b = \frac{\sqrt{2}}{2}\).

Step 2: Plot the Complex Number on the Argand Plane

To graph a complex number, plot the real part \(a\) on the x-axis and the imaginary part \(b\) on the y-axis. Plot the point \(\left( \frac{-\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right)\) on the Argand plane. This involves moving \(\frac{-\sqrt{2}}{2}\) units along the x-axis (left) and \(\frac{\sqrt{2}}{2}\) units up along the y-axis.

Step 3: Calculate the Modulus of the Complex Number

The modulus of a complex number \(a + bi\) is given by \(|z| = \sqrt{a^2 + b^2}\). Substitute in the values of \(a\) and \(b\): \[ |z| = \sqrt{\left(\frac{-\sqrt{2}}{2}\right)^2 + \left(\frac{\sqrt{2}}{2}\right)^2} \]This simplifies to: \[ |z| = \sqrt{\frac{2}{4} + \frac{2}{4}} = \sqrt{1} = 1 \]

Key Concepts

Modulus of a Complex Number

The modulus of a complex number is a crucial concept that helps in understanding its magnitude. When we talk about the modulus, we're referring to the distance of the complex number from the origin in the complex plane. For a complex number expressed as \(a + bi\), its modulus \(|z|\) is calculated using the formula: \[|z| = \sqrt{a^2 + b^2}\]Breaking it down:

• \(a\) is the real part.
• \(b\) is the imaginary part.

The formula provides a measure of the "size" of the complex number, much like finding the length of the hypotenuse of a right triangle. In simpler terms, it gives the straight-line distance from the origin \((0,0)\) to the point \((a,b)\) on the Argand plane. This is similar to how you would calculate the distance in normal 2-dimensional geometry.
In our exercise, with \(a = \frac{-\sqrt{2}}{2}\) and \(b = \frac{\sqrt{2}}{2}\), the modulus indeed turns out to be 1. This tells us that the point \(\left(\frac{-\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)\) is exactly 1 unit away from the origin.

Argand Plane

The Argand plane is a clever way to represent complex numbers visually. Instead of living in the clouds as mysterious mathematical objects, complex numbers find their home on this plane. It's very similar to the regular Cartesian coordinate plane but tailored specifically for complex numbers. Here's how it works:

• The horizontal axis (x-axis) represents the real part of the complex number.
• The vertical axis (y-axis) represents the imaginary part of the complex number.

So if you have a complex number \(a + bi\), you'd plot it at the point \((a, b)\) on the Argand plane. This visual representation is incredibly helpful because it gives you a tangible way to see and analyze the complex number, making abstract concepts more approachable.
The Argand plane also allows for straightforward operations like adding and subtracting complex numbers because you can visualize these actions as vector movements. For instance, in our exercise, the complex number \(\frac{-\sqrt{2}}{2} + i \frac{\sqrt{2}}{2}\) is placed on the Argand plane, making it easy to see its relationship with other numbers and the origin.

Graphing Complex Numbers

Graphing complex numbers might seem daunting at first, but it's a straightforward process once you understand the Argand plane. Here’s how you go about it:

• Identify the real part \(a\) to position along the horizontal axis.
• Identify the imaginary part \(b\) for the vertical axis position.

Now, simply mark the point \((a, b)\) on the plane. For our exercise, with \(a = \frac{-\sqrt{2}}{2}\) and \(b = \frac{\sqrt{2}}{2}\), you move left from the origin \(\frac{-\sqrt{2}}{2}\) units on the x-axis, and up \(\frac{\sqrt{2}}{2}\) units on the y-axis.
This point represents the complex number. By graphing complex numbers, you gain insights into their magnitude (via modulus) and direction (argument), which can be incredibly handy for various applications in mathematics and engineering. This graphical representation not only brings clarity but also helps in performing operations like addition, subtraction, and finding the conjugate visually. It's a gateway to visualizing the beauty of complex numbers in a structured manner.