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Magazine Chapter 8: Problem 43
Rewrite each complex number into polar \(r e^{i \theta}\) form. $$ 5+3 i $$
Short Answer
Expert verified
\(5 + 3i\) in polar form is \(\sqrt{34} e^{i \tan^{-1}(\frac{3}{5})}\).
Step by step solution
01
Identify the Components
The given complex number is \(5 + 3i\). Here, the real part \(a = 5\) and the imaginary part \(b = 3\).
02
Calculate the Magnitude
The magnitude \(r\) of a complex number \(a + bi\) is calculated using the formula: \[r = \sqrt{a^2 + b^2}\]Substitute the values:\[r = \sqrt{5^2 + 3^2} = \sqrt{25 + 9} = \sqrt{34}\]
03
Determine the Argument
The argument \(\theta\) is given by the formula: \[\theta = \tan^{-1}\left(\frac{b}{a}\right)\]Substitute the values:\[\theta = \tan^{-1}\left(\frac{3}{5}\right)\]This needs to be calculated using a calculator to find \(\theta\) in radians.
04
Express in Polar Form
Using the magnitude \(r\) and the argument \(\theta\), the polar form is \[r e^{i \theta}\]So the polar form of \(5 + 3i\) is \[\sqrt{34} e^{i \tan^{-1}(\frac{3}{5})}\].
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding the Magnitude of Complex Numbers
The magnitude of a complex number is akin to the distance from the origin in the complex plane. If we have a complex number in the form of \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part, the magnitude \( r \) is given by the formula: \[ r = \sqrt{a^2 + b^2} \] This formula is derived from the Pythagorean theorem, where \( a \) and \( b \) are akin to the legs of a right triangle and \( r \) is the hypotenuse.
- The magnitude helps in determining how far away the complex number is from the origin (0, 0).
- For example, with the number \( 5+3i \), the calculation \( r = \sqrt{25 + 9} = \sqrt{34} \) gives us an idea of this distance.
Grasping the Argument of Complex Numbers
The argument of a complex number gives us the angle that the complex number makes with the positive real axis in the complex plane. Calculating the argument involves using the formula: \[ \theta = \tan^{-1}\left(\frac{b}{a}\right) \] In this formula, \( a \) and \( b \) are the real and imaginary parts of the complex number, respectively.
- The argument is measured in radians and can be positive or negative depending on the quadrant in which the point \((a, b)\) lies.
- For the complex number \( 5+3i \), \( \theta = \tan^{-1}\left(\frac{3}{5}\right) \), which needs a calculator for precise evaluation.
Exploring Complex Number Transformation
Transforming complex numbers into polar form is a way to represent them using magnitude and argument, which is expressed as \( r e^{i\theta} \). This transformation is particularly useful in simplifying the multiplication and division of complex numbers. Below are key reasons for using this transformation:
- Easier Multiplication and Division: When in polar form, multiply the magnitudes and add the arguments for quick calculations.
- Simplification: With expressions like \( e^{i\theta} \), Euler's formula connects exponential functions with trigonometry.
Decoding Imaginary Numbers
Imaginary numbers are a key concept in understanding complex numbers. Represented with the symbol \( i \), the imaginary unit is defined as \( i^2 = -1 \). These numbers allow for the extension of the real number system to include solutions to equations that don't have real solutions, such as \( x^2 = -1 \).
- An imaginary number is of the form \( bi \), where \( b \) is a real number.
- They arise naturally when dealing with roots of negative numbers and are essential for complex number construction (e.g., \( a + bi \)).
