Imaginary Unit - Proper Use

Proper Use

The imaginary unit is sometimes written √−1 in advanced mathematics contexts (as well as in less advanced popular texts). However, great care needs to be taken when manipulating formulas involving radicals. The notation is reserved either for the principal square root function, which is only defined for real x ≥ 0, or for the principal branch of the complex square root function. Attempting to apply the calculation rules of the principal (real) square root function to manipulate the principal branch of the complex square root function will produce false results:

(incorrect).

Attempting to correct the calculation by specifying both the positive and negative roots only produces ambiguous results:

(ambiguous).

Similarly:

(incorrect).

The calculation rules

and

are only valid for real, non-negative values of a and b.

These problems are avoided by writing and manipulating i√7, rather than expressions like √−7. For a more thorough discussion, see Square root and Branch point.