What is wrong with the following expression?
4=√16=√−4×−4=√−4×√−4=−4.
It seems easy. √−4 doesn't make sense. Right? Well, yes and no.
If we just consider the usual real numbers, then square of any number is positive and √−4 simply does not exists.
However, there is something called complex numbers. For complex numbers, square can be negative. In this case, there are two ways of interpreting √−4.
1. √−4=2i, where i is the imaginary unit. In this case, the equality √−4×−4=√−4×√−4 does not hold.
2. √−4={±2i}, i.e. it is a set. We would also reinterpreting √16={±4}. In this case, we have
{±4}=√16=√−4×−4=√−4×√−4={±2}×{±2}={±4},
here we use element-wise multiplication of two sets.
In general, there are two different meanings of √x.
1. √x is the unique positive square root of x, if x>0; √0=0; √x=√−xi if x<0; any one of the square root of x if x is not a real number. Under this meaning, we have √ab=√a×√b only if at least one of a and b are nonnegative.
Under this meaning, the formula for the quadratic equation ax2+bx+c=0 is −b±√b2−4ac2a. As we have extended the definition of √⋅, this formula now makes senses for any complex numbers a≠0, b and c.
2. √x is the set consists of the two roots of x. Indeed, we can generalize it to n√x being the set of all n-th roots of x. For this meaning, we have n√ab=n√a×n√b for all complex numbers a and b.
The formula for the quadratic equation is now simply −b+√b2−4ac2a, which is indeed the set of the two solutions.
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