Roots of Negatives

Now every negative number has square roots. Split the number into a positive part and -1: since -9 is 9 times -1, √(-9) = √(9) × √(-1), which is 3 times i, or 3i.

Check it by squaring: (3i)^2 = 9i^2 = -9. The rule in general: for a positive number a, √(-a) = i × √(a). So √(-81) = 9i and √(-144) = 12i. The number comes first and the i after it.

Note that √(-9) is not -3: (-3)^2 is 9, a positive number. The minus sign under the root is carried by the i.