3.1 The Imaginary Unit
No real number squares to a negative, so \(x^{2}=-1\) has no real solution. We therefore define a new number \(i\) with the property \[i^{2}=-1,\] and build a system containing it. This is the same move made when negatives were invented to solve \(x+3=1\): the equation had no answer, so the number system was extended until it did.
With \(i\) available, square roots of negatives can be written down: \[\sqrt {-25}=\sqrt {25}\cdot i=5i .\]
Note 3.1. The rule \(\sqrt {ab}=\sqrt {a}\sqrt {b}\) holds for non-negative \(a\) and \(b\) and fails once negatives are allowed. Applying it blindly gives \[-1=i^{2}=\sqrt {-1}\cdot \sqrt {-1}=\sqrt {(-1)(-1)}=\sqrt {1}=1,\] which is false. The safe practice is to convert to \(i\) first — write \(\sqrt {-25}\) as \(5i\) before doing anything else — and never to multiply two square roots of negatives as though the rule applied. Writing \(i=\sqrt {-1}\) is convenient shorthand; the defining property is \(i^{2}=-1\), and that is what to rely on.
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