To find the sum [tex]\((4 + 5i) + (-3 + 7i)\)[/tex], let's carefully follow the correct mathematical steps.
1. Breaking down the terms:
- The first complex number is [tex]\(4 + 5i\)[/tex], where [tex]\(4\)[/tex] is the real part and [tex]\(5i\)[/tex] is the imaginary part.
- The second complex number is [tex]\(-3 + 7i\)[/tex], where [tex]\(-3\)[/tex] is the real part and [tex]\(7i\)[/tex] is the imaginary part.
2. Adding the real parts and the imaginary parts separately:
[tex]\[
(4 + (-3)) + (5i + 7i)
\][/tex]
3. Simplifying the real and imaginary sums:
[tex]\[
(4 - 3) + (5i + 7i) = 1 + 12i
\][/tex]
Now, let's look at what Aiko did. She rewrote the sum as [tex]\((-3 + 7)i + (4 + 5)i\)[/tex].
- Here, Aiko incorrectly placed the terms inside the imaginary unit [tex]\(i\)[/tex], instead of keeping the real and imaginary parts separate.
To explain the specific error Aiko made:
- Mathematical properties:
- Commutative Property: This property states that changing the order of the numbers does not change the sum. This property was not misused by Aiko since she didn't merely switch the order.
- Associative Property: This property allows for grouping of numbers in any order. This property was also not relevant to her error since she didn't change grouping of terms.
- Identity Property: This property refers to adding zero to a number or multiplying by one to get the number itself. This property wasn't part of the error here.
- Aiko's specific error involves improperly combining the real part with the coefficient of the imaginary part, in essence, rewriting and combining terms incorrectly.
Thus, the correct explanation for Aiko's error is:
"Aiko incorrectly used the distributive property by combining the real number and the coefficient of the imaginary part."
