What is the absolute value of the complex number $-4-\sqrt{2} i $?
Answer (2)
The absolute value of the complex number − 4 − 2 i is 3 2 . This is calculated using the formula ∣ z ∣ = a 2 + b 2 and simplifying accordingly. The final result shows the magnitude of the complex number in a clear form.
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Identify the complex number as z = − 4 − 2 i , where a = − 4 and b = − 2 .
Apply the absolute value formula: a 2 + b 2 = ( − 4 ) 2 + ( − 2 ) 2 .
Simplify the expression: 16 + 2 = 18 .
Further simplify to get the final answer: 3 2 .
3 2
Explanation
Understanding the Problem We are asked to find the absolute value of the complex number z = − 4 − 2 i . The absolute value of a complex number a + bi is given by a 2 + b 2 . In this case, a = − 4 and b = − 2 .
Applying the Formula We need to calculate the absolute value using the formula. The absolute value is given by: ( − 4 ) 2 + ( − 2 ) 2
Simplifying the Expression Now, we simplify the expression: ( − 4 ) 2 + ( − 2 ) 2 = 16 + 2 = 18
Further Simplification We can further simplify 18 as 9 × 2 = 9 × 2 = 3 2 .
Final Answer Therefore, the absolute value of the complex number − 4 − 2 i is 3 2 .
Examples
Complex numbers are used in electrical engineering to analyze alternating current circuits. The absolute value of a complex impedance represents the magnitude of the impedance, which is crucial for calculating current and voltage in the circuit. For example, if the impedance of a circuit is given by Z = − 4 − 2 i ohms, the magnitude of the impedance is ∣ Z ∣ = 3 2 ohms, which helps engineers determine the circuit's response to different frequencies.
