Question

What is the square root of the negative one?

Answer 1

Hint: In this question, we have to find the square root of an expression. Thus, we will use the complex numbers to get the solution. Since, we have to find the value of a square root, so we know that the number inside the square root must be positive, but if the number is negative, then we have to use the complex number. As we know, complex terms include one real part and another as an imaginary part, where the imaginary part is the square of a negative number. Thus, we will use the same logic to get an accurate solution.

Complete step by step solution:
According to the problem, we have to find the value of a square root of a number.
Thus, we will first convert the language statement into mathematical statements and then apply the complex number term to get the solution.
The statement given to us is ‘the square root of the negative one.’ Thus, its mathematical term is equal to $\sqrt{-1}$
Now, we know that the $\sqrt{x}$ is always a real number, if $x\ge 0$ , but if $x<0$ , then its square root $\sqrt{x}$ is always an imaginary part of a complex number.
Also, complex numbers are the extension of Real numbers from a line to a plane. The unit in the $x$ direction is the number 1 and the unit in the $y$ direction is the number $i$ , which is called iota. Thus, $i$ is called the imaginary unit.
Now, iota has a property that its square is equal to minus 1, that is
$\Rightarrow {{i}^{2}}=-1$
So, we will take the square root on both sides in the above equation, we get
$\Rightarrow \sqrt{{{i}^{2}}}=\sqrt{-1}$
On further simplify the above equation, we get
$\Rightarrow \pm i=\sqrt{-1}$
Therefore, the square root of negative one is equal to $\pm i$ , where $+i$ is the principal square root and $-i$ is the other root.

Note: While solving this problem, do not forget the definition of complex numbers. Always remember that when we remove the square root sign, we get both the plus and minus sign.