what type of number is $12i + 360$?
choose all answers that apply:
a real
b imaginary
c complex

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Accepted Answer

# Brief Explanations:
1. A complex number is in the form $a + bi$, where $a$ is the real part and $bi$ is the imaginary part. For $12i + 360$ (or $360+12i$), $a = 360$ (real) and $b = 12$ (so $12i$ is imaginary).
2. - **Option A (Real)**: The real part $360$ is a real number, but the entire number $360 + 12i$ is not just a real number (since it has an imaginary part). However, complex numbers include real numbers (when $b = 0$) and imaginary numbers (when $a = 0$). But here, the number has both real and imaginary parts, so it is a complex number. Also, the imaginary part exists, so it is related to imaginary numbers in the complex system.
 - **Option B (Imaginary)**: An imaginary number is typically in the form $bi$ (where $a = 0$ in $a + bi$). But $360+12i$ has $a
eq0$, so it's not a pure imaginary number. However, it contains an imaginary component ($12i$), and in the context of complex numbers, numbers with an imaginary part (even with a real part) are part of the complex number system which encompasses imaginary - related numbers.
 - **Option C (Complex)**: By definition, any number of the form $a+bi$ (where $a,b\in\mathbb{R}$) is a complex number. Here $a = 360$ and $b = 12$, so $360 + 12i$ is a complex number. Also, since it has an imaginary part ($12i$), it is a complex number that is not purely real. And as a complex number with a non - zero imaginary part, it is associated with imaginary numbers in the complex system. So both B (in the sense of having an imaginary component) and C apply, and also, since the real part is real, but the key is the form. Wait, more accurately:
    - A real number is a complex number with $b = 0$ (e.g., $360=360 + 0i$). Our number has $b = 12
eq0$, so it's not a real number (so A is incorrect).
    - A pure imaginary number is a complex number with $a = 0$ (e.g., $12i=0 + 12i$). Our number has $a = 360
eq0$, so it's not a pure imaginary number. But complex numbers include all numbers of the form $a + bi$, so $360+12i$ is a complex number (C is correct). Also, since it has an imaginary part ($12i$), it is a complex number that involves an imaginary component, so in the set of complex numbers, it is a number that has an imaginary part, so B (imaginary - related, as part of complex) and C (complex) apply. Wait, maybe a better way:
      - Real numbers: Numbers without an imaginary part ($bi = 0$). Our number has $12i$, so not real (A is out).
      - Imaginary numbers: Pure imaginary numbers are $bi$ ($a = 0$), but our number is $a+bi$ with $a
eq0$. However, the term "imaginary" can be used in the context of complex numbers to refer to numbers with a non - zero imaginary part. But strictly, pure imaginary is $a = 0$. But the options: the number $360 + 12i$ is a complex number (C). Also, since it has an imaginary part, it is a complex number that is not real, and in the classification, complex numbers include real (when $b = 0$) and imaginary (when $a = 0$) and numbers with both. But the question is "choose all that apply". Let's re - evaluate:
        - The number $360+12i$ is a complex number (because it is in the form $a + bi$, $a = 360$, $b = 12$) – so C is correct.
        - It has an imaginary part ($12i$), so it is a complex number that is not real, and in the sense of the options, "Imaginary" might be considered as a number that has an imaginary component (even with a real part) in the complex system. Also, the real part $360$ is real, but the entire number is complex. Wait, maybe the intended answers are B and C? Wait no:
          - Real number: A real number is a complex number with $b = 0$. Since our number has $b = 12
eq0$, it's not a real number (A is wrong).
          - Imaginary number: A pure imaginary number is $bi$ with $a = 0$. Our number has $a = 360
eq0$, so it's not a pure imaginary number. But the number contains an imaginary term ($12i$), and complex numbers are the set of all $a + bi$. So the number is complex (C). Also, since it has an imaginary part, it is a complex number that is not real, and in the context of the options, maybe "Imaginary" is meant to include numbers with an imaginary component in the complex system. Alternatively, maybe the question considers that any number with $i$ is imaginary - related and complex. So the correct options are B and C? Wait, no:
            - Let's recall definitions:
              - Real number: A number that can be represented on the real number line, no imaginary unit $i$. So $360 + 12i$ has $i$, so not real (A is out).
              - Imaginary number: A number of the form $bi$ where $b$ is a non - zero real number and $a = 0$ in $a+bi$. So $12i$ is imaginary, but $360 + 12i$ is not a pure imaginary number. But the term "imaginary" can be used more loosely to refer to numbers involving $i$, so in that case, since $360+12i$ involves $i$, it is imaginary - related.
              - Complex number: Any number of the form $a + bi$, $a,b\in\mathbb{R}$. So $360+12i$ is complex (C).
            - So the correct answers are B and C? Wait, maybe the question has a bit of a loose definition for "Imaginary" (not just pure imaginary) and "Complex". So:
              - A: No, because it has an imaginary part.
              - B: Yes, because it has $i$ (imaginary component).
              - C: Yes, because it is in the form $a + bi$.

# Answer:
B. Imaginary, C. Complex

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