Problem 14 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

Graph the point at \((-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})\) and the modulus is 1.

Step 1: Expand the Expression

Start by writing the complex number in its standard form. We have the complex number \( \frac{-\sqrt{2}+i\sqrt{2}}{2} \), which can be rewritten in standard form as \( -\frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2} \). This separation allows us to clearly see the real and imaginary parts.

Step 2: Identify Real and Imaginary Components

Identify the real part (x-coordinate) and the imaginary part (y-coordinate) from the complex number. In this case, the real part is \( -\frac{\sqrt{2}}{2} \) and the imaginary part is \( \frac{\sqrt{2}}{2} \). These values correspond to the coordinates on the complex plane.

Step 3: Graph the Complex Number

Plot the point \( (-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}) \) on the complex plane. The x-axis represents the real part, and the y-axis represents the imaginary part. It is located in the second quadrant since the real part is negative, and the imaginary part is positive.

Step 4: Calculate the Modulus

Use the formula for the modulus of a complex number \( a + bi \), which is \( \sqrt{a^2 + b^2} \). Substitute \( a = -\frac{\sqrt{2}}{2} \) and \( b = \frac{\sqrt{2}}{2} \), and calculate:\[\text{Modulus} = \sqrt{\left(-\frac{\sqrt{2}}{2}\right)^2 + \left(\frac{\sqrt{2}}{2}\right)^2} \= \sqrt{\frac{2}{4} + \frac{2}{4}} \= \sqrt{1} = 1\]Thus, the modulus of the complex number is 1.

Key Concepts

Modulus of a Complex Number

The modulus of a complex number is like measuring the magnitude or length of the vector representing the number on the complex plane. Think of it as the "distance" from the origin to the complex point. For a complex number in the form of \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part, the modulus \(|z|\) is calculated using the formula: \[|z| = \sqrt{a^2 + b^2}.\] This formula stems from Pythagoras’ theorem, as it finds the hypotenuse of a right triangle formed by the real and imaginary components.
For example, to find the modulus of \(-\frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2}\), you plug these components into the formula:

• \(a = -\frac{\sqrt{2}}{2}\)
• \(b = \frac{\sqrt{2}}{2}\)

This results in: \[\sqrt{\left(-\frac{\sqrt{2}}{2}\right)^2 + \left(\frac{\sqrt{2}}{2}\right)^2} = \sqrt{\frac{2}{4} + \frac{2}{4}} = \sqrt{1} = 1.\] Thus, the modulus provides a concise measure of a complex number’s size without considering its direction on the complex plane.

Graphing Complex Numbers

Complex numbers can be visually understood by graphing them on the complex plane. This plane consists of two axes: the horizontal axis (the real axis) and the vertical axis (the imaginary axis). Each complex number corresponds to a unique point in this plane.
To graph a complex number \(a + bi\):

• Identify the real component \(a\) and mark this on the horizontal axis.
• Identify the imaginary component \(b\) and mark this on the vertical axis.
• Plot the point where these two values intersect.

For example, take the number \(-\frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2}\):

• The real part is \(-\frac{\sqrt{2}}{2}\)
• The imaginary part is \(\frac{\sqrt{2}}{2}\)
• This point lies in the second quadrant because the real part is negative while the imaginary part is positive.

Graphing complex numbers helps intuitively understand their properties and relationships, such as addition or subtraction, by visualizing how these numbers combine in space.

Real and Imaginary Components

Complex numbers consist of two essential parts: the real and the imaginary components. These parts enable us to visualize and manipulate complex numbers effectively.
The real component \(a\) of a complex number \(a + bi\) is the familiar part that you encounter in everyday arithmetic. It represents the position on the horizontal axis of the complex plane. The imaginary component \(bi\) involves the imaginary unit \(i\), which is defined as \(\sqrt{-1}\). It indicates the position on the vertical axis.
Take for example the complex number \(-\frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2}\):

• \(a = -\frac{\sqrt{2}}{2}\,\) is the real component.
• \(b = \frac{\sqrt{2}}{2}\,\) is the coefficient of the imaginary component.

These components are crucial because they allow for operations like addition, subtraction, and graphing. Understanding how to interpret these parts enables one to navigate more advanced concepts in complex number theory.