Aiko is finding the sum $(4+5 i)+(-3+7 i)$. She rewrites the sum as $(-3+7) i+(4+5) i$. Which statement explains the error Aiko made by using a mathematical property incorrectly?
A. Aiko incorrectly used the commutative property by changing the order of the two complex numbers.
B. Aiko incorrectly used the distributive property by combining the real number and the coefficient of the imaginary part.
C. Aiko incorrectly used the associative property by changing the order of the two complex numbers.
D. Aiko incorrectly used the identity property by combining the real number and the coefficient of the imaginary part.
Answer (2)
Aiko incorrectly rewrites the sum of complex numbers ( 4 + 5 i ) + ( − 3 + 7 i ) as ( − 3 + 7 ) i + ( 4 + 5 ) i .
The correct method involves grouping real and imaginary parts separately: ( 4 − 3 ) + ( 5 + 7 ) i = 1 + 12 i .
Aiko's mistake is incorrectly applying the distributive property by improperly combining real and imaginary parts.
The correct answer is that Aiko incorrectly used the distributive property by combining the real number and the coefficient of the imaginary part.
Explanation
Understanding the Problem Aiko is trying to add two complex numbers, ( 4 + 5 i ) and ( − 3 + 7 i ) . However, she made a mistake in how she rearranged the terms. Let's break down what she did and why it's incorrect.
Identifying the Error Aiko rewrites the sum as ( − 3 + 7 ) i + ( 4 + 5 ) i . This is where the error lies. In complex number addition, you should group the real parts together and the imaginary parts together. The correct way to add these complex numbers is to combine the real parts ( 4 + ( − 3 )) and the imaginary parts ( 5 i + 7 i ) .
Correct Calculation The correct sum is calculated as follows: ( 4 + 5 i ) + ( − 3 + 7 i ) = ( 4 + ( − 3 )) + ( 5 i + 7 i ) = ( 4 − 3 ) + ( 5 + 7 ) i = 1 + 12 i Aiko's method leads to: ( − 3 + 7 ) i + ( 4 + 5 ) i = 4 i + 9 i = 13 i This is not the correct sum.
Identifying the Incorrect Property Aiko incorrectly applied the distributive property. The distributive property applies when you're multiplying a single term by a sum or difference inside parentheses, like a ( b + c ) = ab + a c . In this case, Aiko seems to be trying to rearrange and group terms, but she's doing it in a way that mixes real and imaginary parts incorrectly. She is not distributing anything; instead, she is incorrectly grouping terms as if both -3 and 7 were coefficients of i in the original expression, which is not the case.
Conclusion The correct statement that explains Aiko's error is:
Aiko incorrectly used the distributive property by combining the real number and the coefficient of the imaginary part.
Examples
Complex numbers are used in electrical engineering to analyze AC circuits. The voltage and current in an AC circuit can be represented as complex numbers, and operations with complex numbers (like addition and multiplication) are used to calculate the impedance and power in the circuit. For example, if you have two voltage sources in series represented as complex numbers V 1 = 3 + 4 i and V 2 = 1 − 2 i , the total voltage V T is the sum of these complex numbers: V T = V 1 + V 2 = ( 3 + 1 ) + ( 4 − 2 ) i = 4 + 2 i .
Aiko incorrectly rewrote the sum of complex numbers by mixing real and imaginary parts, leading to the wrong answer. She misunderstood the addition rules for complex numbers, which requires grouping real parts and imaginary parts separately. The chosen explanation for her error is that she incorrectly used the distributive property by combining the real number and the coefficient of the imaginary part.
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