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Chapter 7: Problem 2

In Exercises \(1-10,\) plot each complex number and find its absolute value. $$ z=3 i $$

Short Answer

Expert verified
The complex number \(z = 3i\) is represented in the complex plane at point \((0, 3)\). Its absolute value is \(3\)

Step by step solution

01

Identify the real and imaginary parts

To plot a complex number, the real and imaginary parts are needed. The complex number given is \(z = 3i\). Here, the real part is \(0\) and the imaginary part is \(3\). So, it can be represented as \((0, 3)\) in the complex plane.
02

Graph the complex number

Starting from the origin of the graph, move 0 units along the real (x) axis (because the real part is 0), then move vertically 3 units up along the imaginary (y) axis (because the imaginary part is 3). Place a point at this location to denote the complex number \(z = 3i\).
03

Compute the absolute value

The absolute value (or modulus) of a complex number \(z = a + bi\) is \(\sqrt{a^2 + b^2}\). Substituting into this formula with \(a = 0\) and \(b = 3\) gives \(\sqrt{0^2 + 3^2} = \sqrt{9} = 3\).

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Key Concepts

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

Complex Plane
The complex plane, also known as the Argand plane, is a graphical representation of complex numbers. Unlike the regular Cartesian coordinate system, the complex plane has an axis for real numbers and another for imaginary numbers. Any complex number can be represented as a point in this plane.
For example, the complex number \(z = 3i\) can be depicted on this plane by moving along the imaginary axis since its real component is zero. Here is how you visualize this:
  • The x-axis represents the real part of the number.
  • The y-axis represents the imaginary part of the number.
  • Given \(z = 3i\), we start at the origin of the graph (0,0).
  • Since the real part is 0, we do not move horizontally.
  • The imaginary part is 3, so we move 3 units upwards on the imaginary axis.
The location \( 0, 3 \) on the complex plane represents \(z = 3i\). This graphical approach aids in comprehending how complex numbers extend traditional number line concepts.
Absolute Value
The absolute value, or modulus, of a complex number measures its distance from the origin in the complex plane. In simpler terms, it is the length of the line segment from the point (0,0) to the point (a,b) representing the complex number \(z = a + bi\).
To calculate this, we apply the formula:\[|z| = \sqrt{a^2 + b^2}\]For the complex number \(z = 3i\), which translates to \((0, 3)\) in the complex plane, computation would be:
  • Substitute \(a = 0\) and \(b = 3\) into the formula.
  • Calculate: \(\sqrt{0^2 + 3^2} = \sqrt{9} = 3\).
Thus, the absolute value is 3. This signifies that \(z = 3i\) is 3 units away from the origin, effectively a reflection of its magnitude in the complex plane.
Imaginary Numbers
Imaginary numbers are a pivotal concept in complex number theory. They are essential for solving equations that do not have solutions within the realm of real numbers. Typically represented by the letter \(i\), an imaginary unit is defined as \(i^2 = -1\).
When you multiply any real number with \(i\), you get an imaginary number. For instance, \(3i\) is an imaginary number where 3 is the real multiplier.
Characteristics of imaginary numbers include:
  • They lie vertically along the imaginary axis on the complex plane.
  • Imaginary numbers are not real but are used in augmenting real numbers to solve complex problems.
  • Every real number can combine with an imaginary number to form a complex number (e.g., \(a + bi\)).
In the example we used, \(z = 3i\), the absence of a real part emphasizes its position purely on the imaginary axis, underlining how imaginary numbers are distinctly, yet integrally positioned in the complex plane.

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Most popular questions from this chapter

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