I'm having a debate with my friend. If real numbers a>b>c>d, can we say that a+bi > c+di? I think the answer is yes, and my argument seems to be confirmed by this post Order relation of complex numbers but even though I understand it, admittedly I'm struggling to explain in English why this relation is true. Can someone help me?
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$\begingroup$ No you cannot compare two complex numbers generally.. $\endgroup$– King TutMar 31, 2018 at 16:59
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$\begingroup$ The complex numbers are not an ordered field like the real numbers, so it does not make sense to compare them like this. See en.m.wikipedia.org/wiki/Ordered_field $\endgroup$– aidangallagher4Mar 31, 2018 at 17:00
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$\begingroup$ Ok, thanks so much $\endgroup$– zaxiomsMar 31, 2018 at 17:01
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1 Answer
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As said in the comments, the complex numbers under addition/multiplication do not form an ordered field, so it does not make sense to compare elements. See this post for more details.
A bit of intuition about this is that when relating two real numbers, we have a 1D number line with which we visualise them, meaning it is very obvious whether one is greater than or less than (or equal to) another - ie it is bigger if it is 'to the right' of the other number, less than if 'to the left', and equal if its position coincides.
With complex numbers we work with the 2D complex plane, on which there is no similar concept of 'greater than' or 'less than', since there are two different directions in which we can compare numbers ('above' and 'below' as well as the previous 'left' and 'right').
The only way we can compare complex numbers is through comparing their moduli (which is a function from the complex numbers to the real numbers), however since this loses information about the complex numbers (since $|1+i|=|1-i|=|\sqrt{2}i|=\cdots$), it is not so useful to us.
Hope this helps :)
answered Mar 31, 2018 at 17:09