Generate 2 questions on modulus of complex numbers.
Understand the Problem
The question is asking for the creation of two specific questions related to the modulus of complex numbers. The modulus of a complex number is a measure of its distance from the origin in the complex plane, and generating questions would typically involve calculations or properties related to that concept.
Answer
1. What is the modulus of \( z = 3 + 4i \)? Answer: \( |z| = 5 \) 2. Are the moduli of \( z_1 = 2 - 3i \) and \( z_2 = 2 + 3i \) equal? Answer: Yes, both are \( |z| = \sqrt{13} \)
Answer for screen readers
- The modulus of the complex number ( z = 3 + 4i ) is
$$ |z| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 $$
- The moduli of ( z_1 ) and ( z_2 ) are both
$$ |z_1| = |z_2| = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13} $$
Thus, they are equal.
Steps to Solve
- Identify the properties of the modulus of complex numbers
Recall that the modulus ( |z| ) of a complex number ( z = a + bi ) is given by the formula
$$ |z| = \sqrt{a^2 + b^2} $$
This property can be used to create questions that involve numerical calculations or comparisons.
- Create a question involving calculation
For instance, create a question that asks for the modulus of a specific complex number.
Example question: "What is the modulus of the complex number ( z = 3 + 4i )?"
- Create a question involving properties
Create a question that involves understanding the properties of moduli.
Example question: "If ( z_1 = 2 - 3i ) and ( z_2 = 2 + 3i ), are the moduli of ( z_1 ) and ( z_2 ) equal? What are they?"
- The modulus of the complex number ( z = 3 + 4i ) is
$$ |z| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 $$
- The moduli of ( z_1 ) and ( z_2 ) are both
$$ |z_1| = |z_2| = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13} $$
Thus, they are equal.
More Information
The modulus of a complex number provides an important geometric interpretation, as it corresponds to the distance from the origin in the complex plane. Additionally, the equality of moduli between conjugate pairs emphasizes the symmetrical properties of complex numbers.
Tips
- Miscalculating the squares when applying the modulus formula. Remember to square both the real part and the imaginary part properly.
- Forgetting the square root when finding modulus. Ensure to always take the square root after calculating the sum of squares.
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