<story><title>Quantum physics falls apart without imaginary numbers</title><url>https://www.scientificamerican.com/article/quantum-physics-falls-apart-without-imaginary-numbers/</url></story><parent_chain><item><author>andrew_eu</author><text>There is much more to the history of complex numbers, and that is also worth a read [0]. In particular, Gauss was very against the term &amp;quot;imaginary numbers&amp;quot; because it implies some mystery around them. I vaguely remember reading that he preferred the term &amp;quot;lateral&amp;quot; numbers, but that may be a mistake. Euler&amp;#x27;s formula connects them very plainly with rotations in a complex number plane.&lt;p&gt;The intuition I developed with them while studying physics was that, unlike &amp;quot;real&amp;quot; numbers which interact by stacking, complex numbers interact by stacking and rotating. This is bizarre to think about with single numbers in a 1D world, but we don&amp;#x27;t live in a 1D world. In higher dimensions they rotate and sheer rather than just scale.&lt;p&gt;And indeed, QM (at least as it was thought to me) would fall apart without complex numbers. Whether a physical theory can be consistent without them is an interesting question, but not because a physical theory with them creates some kind of metaphysical paradox.&lt;p&gt;[0] &lt;a href=&quot;https:&amp;#x2F;&amp;#x2F;en.m.wikipedia.org&amp;#x2F;wiki&amp;#x2F;Complex_number#History&quot; rel=&quot;nofollow&quot;&gt;https:&amp;#x2F;&amp;#x2F;en.m.wikipedia.org&amp;#x2F;wiki&amp;#x2F;Complex_number#History&lt;/a&gt;</text></item></parent_chain><comment><author>lacker</author><text>&lt;i&gt;Whether a physical theory can be consistent without them is an interesting question&lt;/i&gt;&lt;p&gt;I agree, it is interesting, and the answer is yes, a physical theory can be completely consistent while never using complex numbers.&lt;p&gt;If you don&amp;#x27;t want to use imaginary numbers, you can avoid them for almost anything by using matrices instead. Use 2x2 matrices instead of complex numbers, and instead of 1, use the identity matrix:&lt;p&gt;1 0&lt;p&gt;0 1&lt;p&gt;Instead of i, use a matrix that corresponds to a 90 degree rotation:&lt;p&gt;0 -1&lt;p&gt;1 0&lt;p&gt;This means that i * i = -1, and so pretty much everything else will just work the same way as it does with complex numbers. Adding, multiplying, calculus, all the same. No need for complex numbers if you don&amp;#x27;t like them.&lt;p&gt;I do like complex numbers though. They are just a bit more concise and convenient than using matrices.</text></comment>
<story><title>Quantum physics falls apart without imaginary numbers</title><url>https://www.scientificamerican.com/article/quantum-physics-falls-apart-without-imaginary-numbers/</url></story><parent_chain><item><author>andrew_eu</author><text>There is much more to the history of complex numbers, and that is also worth a read [0]. In particular, Gauss was very against the term &amp;quot;imaginary numbers&amp;quot; because it implies some mystery around them. I vaguely remember reading that he preferred the term &amp;quot;lateral&amp;quot; numbers, but that may be a mistake. Euler&amp;#x27;s formula connects them very plainly with rotations in a complex number plane.&lt;p&gt;The intuition I developed with them while studying physics was that, unlike &amp;quot;real&amp;quot; numbers which interact by stacking, complex numbers interact by stacking and rotating. This is bizarre to think about with single numbers in a 1D world, but we don&amp;#x27;t live in a 1D world. In higher dimensions they rotate and sheer rather than just scale.&lt;p&gt;And indeed, QM (at least as it was thought to me) would fall apart without complex numbers. Whether a physical theory can be consistent without them is an interesting question, but not because a physical theory with them creates some kind of metaphysical paradox.&lt;p&gt;[0] &lt;a href=&quot;https:&amp;#x2F;&amp;#x2F;en.m.wikipedia.org&amp;#x2F;wiki&amp;#x2F;Complex_number#History&quot; rel=&quot;nofollow&quot;&gt;https:&amp;#x2F;&amp;#x2F;en.m.wikipedia.org&amp;#x2F;wiki&amp;#x2F;Complex_number#History&lt;/a&gt;</text></item></parent_chain><comment><author>TRiG_Ireland</author><text>Or, as someone I know (Gnomon on h2g2) remarked, they&amp;#x27;re just as real as real numbers, which is to say, not real at all.</text></comment>