Math Problem Statement
Convert 1 - 3i and 5 + i into exponential form and hence calculate the modulus and argument of: (1 - 3i)/(5 + i)
Solution
Let's begin by converting the two complex numbers and into their exponential forms. The general form for expressing a complex number in exponential form is:
where:
- is the modulus (magnitude) of the complex number, given by ,
- is the argument (angle) of the complex number, given by .
Step 1: Convert into exponential form.
-
Modulus:
-
Argument: Since the real part is positive and the imaginary part is negative, the complex number is in the fourth quadrant. The argument is:
Thus, in exponential form is:
Step 2: Convert into exponential form.
-
Modulus:
-
Argument: Since the real part is positive and the imaginary part is positive, the complex number is in the first quadrant. The argument is:
Thus, in exponential form is:
Step 3: Calculate .
Now, divide by : This simplifies to:
Step 4: Modulus of .
The modulus is simply the ratio of the moduli of and :
Step 5: Argument of .
The argument is the difference of the arguments of and :
Thus, the modulus and argument of are:
- Modulus:
- Argument:
Would you like further clarification or details on any part of this process?
Here are some follow-up questions you might find interesting:
- How do you calculate the real and imaginary parts from an exponential form?
- What is the significance of the modulus and argument in complex numbers?
- Can you express any complex number in polar form and vice versa?
- How can you geometrically interpret the division of two complex numbers?
- What are the applications of complex number operations in electrical engineering?
Tip: When working with arguments of complex numbers, make sure to adjust the angle depending on the quadrant where the number lies.
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Math Problem Analysis
Mathematical Concepts
Complex Numbers
Exponential Form
Modulus and Argument
Formulas
Modulus: r = √(x^2 + y^2)
Argument: θ = tan⁻¹(y/x)
Exponential Form: z = r e^{iθ}
Theorems
Euler's Formula: z = r e^{iθ}
Properties of Moduli and Arguments in Complex Division
Suitable Grade Level
Grades 10-12