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I am working through some problems in Axler's Linear Algebra Done Right textbook, and I noticed that I haven't really developed an intuitive feel for how to approach existence in the proofs.

The idea of existence just seems very vague to me. For instance, in one of the problems, the book asks you to find real numbers c and d given real numbers a and b in the following expression:

1/(a+bi) = c+di

From what it looks like, if I can express c and d in terms of a and b, that means that c and d exist because a and b are assumed to exist? Is there are more solid way for me to approach existence in proofs?

Richard K Yu
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  • Note that $$\frac1{a+bi}=\frac{a-bi}{(a+bi)(a-bi)} = \frac{a-bi}{a^2+b^2}=\frac a{a^2+b^2}-i\frac b{a^2+b^2}$$So, if $c=\frac a{a^2+b^2},;d=-\frac b{a^2+b^2}$ then $$\frac1{a+bi}=c+di$$ – Rushabh Mehta May 14 '19 at 02:17
  • I get this part, but I mean is it proof that something exists if it can be expressed in variables that we assume to exist already? – Richard K Yu May 14 '19 at 02:24
  • I don't understand what you mean by "assume to exist already" The idea of existence is that given some complex number expressed generally as $a+bi$, if I can come up with a construction for the inverse also in the form $c+di$, then I've shown that one exists for all complex numbers. Not all existence proofs are constructive, but many are. – Rushabh Mehta May 14 '19 at 02:24
  • Welcome to Math Stack Exchange. If you can construct $c$ and $d$ using mathematical operations valid for real numbers (e.g., not dividing by zero, not taking a square root of a negative number), that is a solid proof of existence. In some situations (not this one) you might have only a non-constructive proof, which proves something exists without showing how to compute it – J. W. Tanner May 14 '19 at 02:26
  • I think I got it! Is there an example of a non-constructive proof of existence I can look at? – Richard K Yu May 14 '19 at 02:29
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    I'd suggest you read this wikipedia article – J. W. Tanner May 14 '19 at 02:35
  • Note that when you say "express $c$ and $d$ in terms of..." it sounds like you are already assuming they exist, which may have been what was troubling you. But as other commenters have noted, that act of expressing is actually a proof of existence of a $c$ and $d$ with the required properties. If you like, you are "defining $c$ and $d$ in terms of..." and then showing these defined objects have the required properties, which has less of a connotation that they are some prior existing objects. – spaceisdarkgreen May 14 '19 at 02:45
  • Also, a lot of non-constructive existences proofs have the form of a proof by contradiction. The prototype (which is oddly not mentioned in the wikipedia article) is probably Cantor's proof of the existence of transcendental numbers. The algebraic numbers are countable, so the nonexistence of transcendental numbers implies there is a bijection of the reals with the integers, which Cantor's diagonalization argument shows leads to a contradiction. However this does not give you any example of a transcendental number. (Although Liouville had already demonstrated transcendentals constructively.) – spaceisdarkgreen May 14 '19 at 02:52

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