What is the multiplicative identity of the complex number $-4+8 i$?
A. 0
B. 1
C. $-4+8 i$
D. $4-8 i$
Answer (2)
The multiplicative identity is a number that, when multiplied by any number, results in that same number.
The multiplicative identity for complex numbers is 1.
Multiplying 1 by the complex number − 4 + 8 i gives − 4 + 8 i .
Therefore, the multiplicative identity of the complex number − 4 + 8 i is 1 .
Explanation
Understanding the Problem The question asks for the multiplicative identity of the complex number − 4 + 8 i . The multiplicative identity is a number that, when multiplied by any number, results in that same number. For real numbers, the multiplicative identity is 1. We need to determine if the same holds true for complex numbers.
Definition of Multiplicative Identity The multiplicative identity is a number x such that for any number y , x "." y = y . In the case of complex numbers, we are looking for a complex number x + y i such that ( x + y i ) ( − 4 + 8 i ) = − 4 + 8 i .
Multiplicative Identity for Complex Numbers The multiplicative identity for complex numbers is 1, which can be written as 1 + 0 i .
Verification To verify, we can multiply 1 by the complex number − 4 + 8 i :
1"." ( − 4 + 8 i ) = − 4 + 8 i
This confirms that 1 is indeed the multiplicative identity.
Conclusion Therefore, the multiplicative identity of the complex number − 4 + 8 i is 1.
Examples
In algebra, the multiplicative identity is crucial for solving equations and simplifying expressions. For example, when working with complex numbers in electrical engineering, the multiplicative identity helps in analyzing AC circuits. The voltage, current, and impedance are often represented as complex numbers, and using the multiplicative identity ensures that the properties of these numbers are maintained during calculations. This is also applicable in signal processing, where complex numbers are used to represent signals, and the multiplicative identity preserves the signal's characteristics.
The multiplicative identity of the complex number − 4 + 8 i is 1, as this is the number that, when multiplied by any number, leaves that number unchanged. In this case, multiplying 1 by − 4 + 8 i results in − 4 + 8 i . Therefore, the answer is option B: 1.
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