Math Problem Statement
In the picture it is Complex Variables and Transforms, now slove the equation as given in picture
Solution
The problem in the image asks to define a complex number and its modulus and argument, then find the modulus and principal argument of the expression:
Step 1: Simplify the expression
We will first simplify the expression by multiplying the numerators and expanding the denominator.
Numerator:
= 6 - 2i - 15i + 5i^2$$ Since $$i^2 = -1$$, the above becomes: $$6 - 17i + 5(-1) = 6 - 17i - 5 = 1 - 17i$$ So, the numerator simplifies to $$1 - 17i$$. #### Denominator: $$(1 - 2i)^2 = (1 - 2i)(1 - 2i) = 1 \times 1 + 1 \times (-2i) + (-2i) \times 1 + (-2i) \times (-2i) = 1 - 2i - 2i + 4i^2$$ Again, using $$i^2 = -1$$, we get: $$1 - 4i - 4 = -3 - 4i$$ So, the expression simplifies to: $$\frac{1 - 17i}{-3 - 4i}$$ ### Step 2: Convert to polar form To find the modulus and argument, we need to convert the expression to polar form. #### Modulus: The modulus of a complex number $$z = a + bi$$ is given by: $$|z| = \sqrt{a^2 + b^2}$$ For the numerator $$1 - 17i$$: $$|1 - 17i| = \sqrt{1^2 + (-17)^2} = \sqrt{1 + 289} = \sqrt{290}$$ For the denominator $$-3 - 4i$$: $$|-3 - 4i| = \sqrt{(-3)^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5$$ Thus, the modulus of the whole expression is: $$\frac{\sqrt{290}}{5}$$ #### Argument: The argument of a complex number $$z = a + bi$$ is given by: $$\arg(z) = \tan^{-1}\left(\frac{b}{a}\right)$$ For the numerator $$1 - 17i$$, we get: $$\arg(1 - 17i) = \tan^{-1}\left(\frac{-17}{1}\right) = \tan^{-1}(-17)$$ For the denominator $$-3 - 4i$$, we get: $$\arg(-3 - 4i) = \tan^{-1}\left(\frac{-4}{-3}\right) = \tan^{-1}\left(\frac{4}{3}\right)$$ The total argument is: $$\arg\left(\frac{1 - 17i}{-3 - 4i}\right) = \arg(1 - 17i) - \arg(-3 - 4i)$$ Now, let's calculate these values numerically.The results for the expression $$\frac{(2 - 5i)(3 - i)}{(1 - 2i)^2}$$ are: - **Modulus**: $$\approx 3.41$$ - **Principal argument**: $$\approx 0.702$$ radians (or $$\approx 40.24^\circ$$) Would you like more details on any of the steps? Here are some related questions you might explore: 1. How can we use polar form to multiply and divide complex numbers? 2. What are the geometric interpretations of modulus and argument? 3. How can we find the argument of a complex number when the real part is negative? 4. What role does the principal value of the argument play in complex number operations? 5. Can the modulus of a complex number ever be negative? **Tip:** Always make sure to express the argument in radians or degrees based on the context, and use the principal value between $$-\pi$$ and $$\pi$$ radians.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Complex Numbers
Modulus
Argument
Polar Form
Formulas
Modulus: |z| = √(a^2 + b^2)
Argument: arg(z) = tan^(-1)(b/a)
Theorems
Properties of Complex Numbers
Suitable Grade Level
Grades 11-12
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