Pure Imaginary Number

Definition

A pure imaginary number is a complex number that has a real part of zero and only a non-zero imaginary part. It is represented in the form $bi$, where $b$ is a real number and $i$ represents the imaginary unit, defined as the square root of -1.

5 Must Know Facts For Your Next Test

1. Pure imaginary numbers are a subset of complex numbers, where the real part is zero.
2. Arithmetic operations, such as addition, subtraction, multiplication, and division, can be performed on pure imaginary numbers using the properties of complex numbers.
3. Pure imaginary numbers have important applications in various fields, including electrical engineering, quantum mechanics, and the study of complex-valued functions.
4. The absolute value (or modulus) of a pure imaginary number $bi$ is simply $|b|$, the absolute value of the coefficient $b$.
5. Pure imaginary numbers can be plotted on the imaginary axis of the complex plane, which is perpendicular to the real axis.

Review Questions

• Explain the relationship between pure imaginary numbers and complex numbers.
  • Pure imaginary numbers are a specific type of complex number, where the real part is zero, and the number is represented solely by the imaginary part. Complex numbers in general are composed of a real part and an imaginary part, with pure imaginary numbers being a special case where the real part is absent. This means that pure imaginary numbers are a subset of the broader set of complex numbers and share many of the same properties and operations, such as addition, subtraction, multiplication, and division.
• Describe the geometric representation of pure imaginary numbers on the complex plane.
  • Pure imaginary numbers are represented on the complex plane by points that lie solely on the imaginary axis, which is perpendicular to the real axis. The imaginary axis is the vertical axis of the complex plane, and pure imaginary numbers can be plotted as points along this axis, with their distance from the origin representing the absolute value of the imaginary coefficient. This geometric representation helps visualize the relationship between pure imaginary numbers and the broader set of complex numbers, which can be plotted anywhere within the complex plane.
• Analyze the importance of pure imaginary numbers in various fields of study.
  • Pure imaginary numbers have significant applications in numerous fields, including electrical engineering, quantum mechanics, and the study of complex-valued functions. In electrical engineering, pure imaginary numbers are used to represent reactive components, such as inductors and capacitors, which play a crucial role in the analysis and design of electrical circuits. In quantum mechanics, pure imaginary numbers are essential in the mathematical formulation of quantum states and the description of wave functions. Additionally, the study of complex-valued functions, which incorporate both real and imaginary components, relies heavily on the properties of pure imaginary numbers to analyze and understand the behavior of these functions.

Related terms

Complex Number: A complex number is a number that can be expressed in the form $a + bi$, where $a$ is the real part and $b$ is the imaginary part.

Imaginary Unit ($i$): The imaginary unit, denoted as $i$, is defined as the square root of -1, meaning $i^2 = -1$.

Real Part: The real part of a complex number is the coefficient of the real component, represented by the variable $a$ in the form $a + bi$.