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Chapter 6: Problem 18

Graph each complex number, and find its absolute value. \(\frac{\sqrt{3}}{2}+\frac{i}{2}\)

Short Answer

Expert verified
Plot the point \( \frac{\root{3}}{2}, \frac{1}{2} \) and its absolute value is 1.

Step by step solution

01

- Identify Real and Imaginary Components

The given complex number is \(\frac{\root{3}}{2} + \frac{i}{2}\). Identify the real part and the imaginary part. Here, the real part is \(\frac{\root{3}}{2}\) and the imaginary part is \(\frac{1}{2}\).
02

- Plot on the Complex Plane

To graph the complex number, plot the point \(\frac{\root{3}}{2} \) on the x-axis (real axis) and \(\frac{1}{2} \) on the y-axis (imaginary axis). The point \(\frac{\root{3}}{2}, \frac{1}{2} \) represents the complex number on the complex plane.
03

- Find the Absolute Value

The absolute value (or modulus) of a complex number \(a + bi\) is given by \(\root{a^2 + b^2}\). Here, \(a = \frac{\root{3}}{2}\) and \(\frac{1}{2}\). Calculate as follows: \(\root{\big(\frac{\root{3}}{2}\big)^2 + \big(\frac{1}{2}\big)^2} = \root{\frac{3}{4} + \frac{1}{4}} = \root{1} = 1\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

graphing complex numbers
Graphing complex numbers can be quite intuitive once you understand the basics. Imagine the complex plane as a two-dimensional space where each point represents a complex number. The horizontal axis is called the real axis, and the vertical axis is known as the imaginary axis.

For example, consider the complex number \(\frac{\root{3}}{2} + \frac{i}{2}\). To graph this number:
  • Identify the real part \(\frac{\root{3}}{2}\) which can be plotted along the x-axis.
  • Identify the imaginary part \(\frac{1}{2}\) which can be plotted along the y-axis.
Plotting these, you get the point (\(\frac{\root{3}}{2}, \frac{1}{2}\)) on the complex plane. It's like plotting any other point in a regular 2D Cartesian coordinate system, only here the axes are representing real and imaginary parts.
absolute value of complex numbers
The absolute value of a complex number, also known as the modulus, gives us an idea of its 'distance' from the origin (0,0) in the complex plane. For a complex number represented as \(a + bi\), the formula for the absolute value is: \[ \root{a^2 + b^2} \]

Let's take our example \(\frac{\root{3}}{2} + \frac{i}{2}\):
  • Identify the real part \(\frac{\root{3}}{2}\) and the imaginary part \(\frac{1}{2}\).
  • Square both parts: (\(\frac{\root{3}}{2}\))^2 = \(\frac{3}{4}\) and (\(\frac{1}{2}\))^2 = \(\frac{1}{4}\).
  • Add the results: \(\frac{3}{4} + \frac{1}{4} = 1\).
  • Finally, take the square root of the sum to get the absolute value: \(\root{1} = 1\)
So, the absolute value of \(\frac{\root{3}}{2} + \frac{i}{2}\) is 1. It tells us that the complex number is a distance of 1 unit away from the origin on the complex plane.
complex plane
The complex plane is a fundamental concept when working with complex numbers. It鈥檚 essentially a fancy way of saying a 2D coordinate system where complex numbers are visualized.

Here's a quick breakdown of the complex plane:
  • Real Axis: The horizontal axis where you plot the real part of the complex number.
  • Imaginary Axis: The vertical axis where you plot the imaginary part of the complex number.
This setup helps in visualizing complex numbers as coordinates. For instance, if you have the complex number \(\frac{\root{3}}{2} + \frac{i}{2}\), you would plot it at the point (\(\frac{\root{3}}{2}, \frac{1}{2}\)) in this plane.

Understanding the complex plane simplifies many operations involving complex numbers, such as addition, subtraction, and even multiplication and division, as each number can be represented geometrically.

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Most popular questions from this chapter

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