Math Problem Statement
Solution
The problem in the image asks:
"Čemu je jednako |w| ako je i ?"
This translates to:
"What is the modulus if and ?"
Let's solve this step-by-step.
-
Determine :
We are given . This is a complex number, so we write it as . -
Find the conjugate of :
The conjugate of , denoted as , is obtained by changing the sign of the imaginary part. Thus,
-
Calculate :
Now, calculate : Using the formula , we get: Since , this simplifies to: -
Substitute into :
Now, substitute and into the expression for : Simplifying the denominator: Since , the denominator becomes: Now simplify the numerator: Thus, we have: -
Find :
The modulus of is given by:
Thus, the modulus of , , is .
Would you like a more detailed explanation, or do you have any further questions?
Here are 5 related questions you might find interesting:
- How do you find the conjugate of a complex number?
- What is the geometric interpretation of the modulus of a complex number?
- How can you simplify complex fractions involving imaginary units?
- What are the properties of the modulus function in complex numbers?
- How do you square a complex number using the standard form ?
Tip: Always remember that multiplying the numerator and denominator of a fraction by the conjugate is a useful method to simplify expressions with complex numbers.
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Math Problem Analysis
Mathematical Concepts
Complex Numbers
Conjugates
Modulus of Complex Numbers
Algebraic Manipulation
Formulas
Conjugate of z: \(\bar{z}\)
Modulus formula: \(|w| = \sqrt{(\text{Re}(w))^2 + (\text{Im}(w))^2}\)
Square of complex numbers: \((a + bi)^2 = a^2 + 2abi + b^2i^2\)
Theorems
Modulus of a Complex Number
Conjugate Property in Complex Numbers
Suitable Grade Level
Grades 11-12 (Pre-Calculus, Algebra II)