Question

What is the square root of -2?

Answer 1

Hint: We use the basic definitions of square root to calculate the answer to this question. We need to note an important point here that we cannot obtain the square root of negative numbers. Therefore, we use the concept of complex numbers here which states that the $\sqrt{-1}$ is considered $i.$ This is an imaginary number and we use this to calculate the value of $\sqrt{-2}.$

Complete step by step solution:
In order to solve this question, we need to use the concept of complex numbers and imaginary numbers. We know that the square root of negative numbers does not exist. Therefore, we introduce the concept of square root of -1 which is an imaginary number represented by $i.$ The exact value of these imaginary numbers is not known to us but it is just a concept to deal with the square root of negative numbers.
We also need to note an important property that,
$\Rightarrow {{i}^{2}}=\sqrt{-1}\times \sqrt{-1}=-1$
For the given problem, we can rewrite $\sqrt{-2}$ as,
$\Rightarrow \sqrt{-2}=\sqrt{2\times -1}$
We can split the two terms inside the root as follows,
$\Rightarrow \sqrt{-2}=\sqrt{2}\times \sqrt{-1}$
We know the standard value of $\sqrt{2}$ is given as 1.414 and we know that $\sqrt{-1}$ is represented by an imaginary number $i.$ Using these,
$\Rightarrow \sqrt{-2}=1.414i$

Hence, the square root of -2 is \[1.414i.\]

Note: We need to know the concept of imaginary numbers in order to understand this question better. The reason there does not exist a square root value for a negative number is because we cannot multiply any two same numbers to get a number with a negative sign. Even if we multiply two negative numbers, we get a positive number. This is the reason we introduced the concept of imaginary numbers so we can obtain the square root of negative numbers too.