Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 12: Problem 22
Express the given numbers in exponential form. $$47.3-10.9 j$$
Short Answer
Expert verified
The exponential form is \( 48.537 e^{-0.227i} \).
Step by step solution
01
Understand the Complex Number
A complex number has two parts: a real part and an imaginary part. In this case, the complex number is given as $47.3 - 10.9 j$, where the real part is $47.3$ and the imaginary part is $-10.9$.
02
Calculate Magnitude of the Complex Number
The magnitude (or modulus) of a complex number is found using the formula \( |z| = \sqrt{a^2 + b^2} \), where \( a \) and \( b \) are the real and imaginary parts, respectively. Here, \( a = 47.3 \) and \( b = -10.9 \). Substitute into the formula:\[ |z| = \sqrt{47.3^2 + (-10.9)^2} \].Calculate:\[ |z| = \sqrt{2236.29 + 118.81} = \sqrt{2355.1} \].Upon solving, \( |z| \approx 48.537 \).
03
Determine the Argument
The argument \( \theta \) of a complex number is calculated using \( \theta = \arctan\left(\frac{b}{a}\right) \), where \( a \) and \( b \) are the real and imaginary parts. Here, \( a = 47.3 \) and \( b = -10.9 \).Calculate:\[ \theta = \arctan\left(\frac{-10.9}{47.3}\right) \].This gives \( \theta \approx -0.227 \) radians.
04
Express in Exponential Form
A complex number can be expressed in exponential form as \( z = |z| e^{i \theta} \), where \( |z| \) is the magnitude and \( \theta \) is the argument.Substituting the known values:\[ z = 48.537 e^{-0.227i} \].This is the exponential form of the complex number.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Exponential Form
Exponential form is a way of expressing complex numbers that is particularly useful for simplifying multiplication and division of complex numbers. In this form, a complex number is represented as \( z = |z| e^{i \theta} \),where \( |z| \) is the magnitude of the complex number, and \( \theta \) is the argument.
- The symbol \( e \) represents the base of the natural logarithm, which is approximately equal to 2.718.
- The \( i \theta \) term in the exponent involves the imaginary unit \( i \), which satisfies \( i^2 = -1 \).
- This form provides a compact way to encode the information about the magnitude and angle of a complex number.
Magnitude of Complex Numbers
The magnitude of a complex number, also known as the modulus, measures its size or how far it is from the origin on the complex plane. For a complex number \( z = a + bi \),the magnitude is given by the formula:\[ |z| = \sqrt{a^2 + b^2} \].
- In this equation, \( a \) is the real part and \( b \) is the imaginary part of the complex number.
- The squaring of these components ensures that the magnitude is always a non-negative value, representing the distance from the origin.
- For example, in the original exercise, the complex number \( 47.3 - 10.9j \) had a magnitude of approximately \( 48.537 \).
Argument of Complex Numbers
The argument of a complex number is the angle it makes with the positive real axis on the complex plane. It is usually expressed in radians. To determine it for a complex number \( z = a + bi \), we use:\[ \theta = \arctan\left(\frac{b}{a}\right) \].
- It鈥檚 important to pay attention to the signs of \( a \) and \( b \) to determine which quadrant the angle belongs to, as angles can range from \(-\pi\) to \(\pi\).
- In the exercise, the given complex number had a negative imaginary part, leading to an argument of \(-0.227 \approx -13\) degrees.
- This value of \( \theta \) helps us describe the direction of the complex number relative to the horizontal axis.
Real and Imaginary Parts
A complex number is composed of two parts: the real part and the imaginary part. For a given complex number \( z = a + bi \),
- The **real part** is \( a \), which can be thought of as the horizontal component of the number on the complex plane.
- The **imaginary part** is \( b \), noted by its use of the imaginary unit \( i \), and can be visualized as the vertical component.
- When plotted on a plane, these two parts create a right-angled triangle, with the hypotenuse representing the magnitude of the complex number.
