What does "$-2,25$ in $\mathbb{C}$" mean?

Question

Which of the following numbers is one of the square roots of $-2,25$ in $\mathbb{C}$?

a. $-1,5$

b. $1,5$

c. $1,5i$

d. $2,25i$

I can calculate the square roots of a complex number without problems, what I don't understand here is what this problem is asking me exactly. What does "$-2,25$ in $\mathbb{C}$" mean?

My book is in portuguese so commas are usually used instead of periods to mark the decimal part of a number (i.e $9,5=9.5$). However, commas and colons are also used to separate 2 different numbers (i.e $9,5$ is sometimes equal to $9;5$).

So I am pretty confused as to what "$-2,25$ in $\mathbb{C}$" might exactly be.

My book says the solution is c. Anyone knows?

Answer 1

Sounds like you are already interpreting the comma correctly: $-2,25$ is just another way of saying $-2.25$, and it is indeed a cultural thing.

The "in $\Bbb C$" part is just clarifying that you should be looking for a complex number (i.e., a number in $\Bbb C$) as your answer. The directions are specifying that it's the square root that's "in $\Bbb C$", not the $-2,25$ that's "in $\Bbb C$". ($-2,25$ actually is in $\Bbb C$ but that's not really relevant - see my digression below, which is also not relevant.)

Technically the "in $\Bbb C$" is redundant because the square root of any negative number is necessarily in $\Bbb C$.

(And even more technically, "in $\Bbb C$" is redundant because every real number [including rationals, irrationals, integers, etc.]) is a complex number with imaginary part zero. But I digress.)

Basically you want to ask yourself, "Which of these 4 answer choices will give me $-2.25$ when I square it?"

Answer 2

The comma here is just the decimal mark: $-2,25$ means $-\frac{225}{100}$, or $-2.25$ as you may be more accustomed to. It's not saying "square roots of ($-2,25$ in $\mathbb{C}$)" but rather "(square roots of $-2,25$) in $\mathbb{C}$". That is, it is asking which of the options is a complex number whose square is $-2.25$.

(The same notation is used in the options, so for instance, $2,25i$ means $\frac{225}{100}i$.)