Question

what type of number is $52i + 7.5$?
choose all answers that apply:
a real
b imaginary
c complex

Explanation:

1. Recall the definitions:

• A real number is a number that can be represented on the number line, like \(7.5\), but the term with \(i\) (imaginary unit, \(i = \sqrt{-1}\)) makes it not purely real.
• An imaginary number is of the form \(bi\) where \(b\) is real and \(b

eq0\), but here we have a real part (\(7.5\)) and an imaginary part (\(52i\)), so it's not purely imaginary.

• A complex number is of the form \(a + bi\), where \(a\) and \(b\) are real numbers, \(a\) is the real part, and \(bi\) is the imaginary part. For \(52i+7.5\) (or \(7.5 + 52i\)), \(a = 7.5\) (real) and \(b = 52\) (real), so it fits the complex number form. Also, all real and imaginary numbers are complex numbers (real numbers can be written as \(a+0i\), imaginary as \(0 + bi\)).

1. Analyze each option:

• Option A: Real numbers have no imaginary part (\(bi = 0\)). Since \(52i

eq0\) (because \(52
eq0\)), \(52i + 7.5\) is not a real number (it has an imaginary component), so A does not apply.

• Option B: Imaginary numbers are of the form \(bi\) with \(a = 0\). Here \(a = 7.5

eq0\), so it's not a purely imaginary number (it has a real part), so B does not apply (note: the term "imaginary" here is for purely imaginary, but the number has both real and imaginary parts, so it's complex).

• Option C: Complex numbers are defined as \(a + bi\), \(a,b\in\mathbb{R}\). Here \(a = 7.5\) and \(b = 52\), so \(52i+7.5\) is a complex number. Also, real numbers (\(a+0i\)) and imaginary numbers (\(0 + bi\)) are subsets of complex numbers, so this number is complex.

Answer:

C. Complex, and also note that while it's not purely real or purely imaginary, it is a complex number (and since all real/imaginary are complex, but the question is about the type of \(52i + 7.5\) which has both parts, so the correct applicable answer is C. Complex. Wait, but also, technically, complex numbers include real (when \(b = 0\)) and imaginary (when \(a = 0\)) as special cases. But in terms of the options:

• A: No, because it has an imaginary part.
• B: No, because it has a real part (not purely imaginary, which is \(0 + bi\)).
• C: Yes, because it's \(a + bi\) with \(a = 7.5\), \(b = 52\), so it's complex.

So the correct answer is C. Complex. But also, since the question says "choose all that apply", let's re - check:

• Real: No, because of the \(52i\) term.
• Imaginary: No, because there is a real part (\(7.5\)) (purely imaginary would have \(a = 0\)).
• Complex: Yes, as it's in the form \(a+bi\), \(a,b\in\mathbb{R}\).

So the answer is C. Complex.