 #1
Grasshopper
Gold Member
 210
 111
 TL;DR Summary

Some clarification on what the positive 1 solution of the root of the products means, and more
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So presumably ##x = \sqrt{1}## can be derived as a solution to the equation
##x^2 + 1 = 0##, thus,
##x = ± \sqrt{1}##
(1) We always use the positive root. What is the reason for that? Is it simply a convention? Or are there dire consequences for using the negative root, such as causing the universe to spontaneously collapse in an illconceived division by zero operation? ;)
(2) Regarding the argument that ##i^2 = (\sqrt{1})(\sqrt{1}) = \sqrt{(1)(1)} = \sqrt{1} = 1##, is that simply an uninteresting and useless result, and that's why it's never taught? Or is it not allowed mathematically to do this? It seems almost like transforming from the complex axis to the real axis. What does it mean? Is there anything interesting regarding this result?
Thanks!
##x^2 + 1 = 0##, thus,
##x = ± \sqrt{1}##
(1) We always use the positive root. What is the reason for that? Is it simply a convention? Or are there dire consequences for using the negative root, such as causing the universe to spontaneously collapse in an illconceived division by zero operation? ;)
(2) Regarding the argument that ##i^2 = (\sqrt{1})(\sqrt{1}) = \sqrt{(1)(1)} = \sqrt{1} = 1##, is that simply an uninteresting and useless result, and that's why it's never taught? Or is it not allowed mathematically to do this? It seems almost like transforming from the complex axis to the real axis. What does it mean? Is there anything interesting regarding this result?
Thanks!