Some of you may have noticed that the two complex numbers whose square roots we took using our derived formula shared a specific mathematical property: the two coefficients are the two smaller numbers of a Pythagorean Triple. Clearly, I chose them to get only rational numbers under the radical symbol, so we could be certain of getting a result that we could express in a familiar form. Whether this form will always have rational coefficients, I’ll leave for another day.
For today, I am interested in what happens to a number of the form, , where b is not a perfect square. Can this be simplified to a form in which no radicals contain non-monomial expressions containing radicals? Again assuming that we don’t have to invent a new kind of number, let , where a, b, p and q are rational. Then . Since, if a and c are rational and b and d are not perfect squares, , we can say (so ) and , so .
Since I know where this is going and what leads to extraneous roots (you didn’t think I was making this up as I was typing, did you?), we will use only the add-the-radical branch of the quadratic formula to get . And . So . Similarly, .
The right side of these identities is simpler than the left only if is a perfect square and b is not, e.g., where or or or , etc. If , then . Example: , which simplifies to .
Is there no hope for numbers that do not meet this criterion? Are we doomed to an infinite regress?
TO BE CONTINUED