Until now, we have been dealing with real numbers. We have not been able to take the square root of a negative number because the square root of a negative number is not a real number. Instead, the square root of a negative number is an imaginary number--a number of the form , where k < 0 . Imaginary numbers are represented as ki , where i = . For example, = 5i and = i .
We can simplify square roots of negative numbers by factoring out = i and simplifying the resulting root.
Examples:
= | · | ||
= | i· | ||
= | i·4· | ||
= | 4i . |
= | · | ||
= | i·10 | ||
= | 10i. |
= | · | ||
= | i· | ||
= | i·5· | ||
= | 5i . |
Observe the following:
i ^{1} | = | i | |
i ^{2} | = | ()^{2} = - 1 | |
i ^{3} | = | i ^{2} i = - 1(i) = - i | |
i ^{4} | = | i ^{3} i = - i(i) = - i ^{2} = - (- 1) = 1 | |
i ^{5} | = | i ^{4} i = 1(i) = i | |
i ^{6} | = | i ^{5} i = - 1 | |
i ^{7} | = | i ^{6} i = - i | |
i ^{8} | = | i ^{7} i = 1 | |
i ^{9} | = | i | |
^{ ... } |
Examples: