Subjects complex numbers

Modulus Ratio 643040

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1. **Stating the problem:** We are asked to find the value of the expression $$\frac{|z_2|}{|z_1|}$$ where $z_1$ and $z_2$ are complex numbers. 2. **Formula and rules:** The modulus (or absolute value) of a complex number $z = a + bi$ is given by $$|z| = \sqrt{a^2 + b^2}$$ where $a$ is the real part and $b$ is the imaginary part. 3. **Intermediate work:** - Calculate $|z_2|$ by finding the square root of the sum of squares of its real and imaginary parts. - Calculate $|z_1|$ similarly. - Then compute the fraction $$\frac{|z_2|}{|z_1|}$$. 4. **Simplification:** If possible, simplify the fraction by canceling common factors. 5. **Explanation:** The modulus represents the distance of the complex number from the origin in the complex plane. Dividing the moduli gives the ratio of these distances. Since no specific values for $z_1$ and $z_2$ are given, the expression remains $$\frac{|z_2|}{|z_1|}$$ as the final answer.