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I'm reading the book, "Annotated Turing" in trying to understand what computation really means. It has a section about algebraic equations and trancendental numbers. I'm quiet not understanding why they are special.

csp2018
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Short (and perhaps too general) answer: Algebraic numbers are special because they are solutions to polynomial equations, and polynomial equations are special because they are essentially the only equations you can make from just addition and multiplication.

The availability of different structures with addition and multiplication leads to a great interest from mathematicians in general to study polynomial equations, their properties and their applications. As an extension of that, we have a lot of theory about their solutions. So we know a lot about algebraic numbers and thus the distinction between algebraic and transcendental becomes important.

Arthur
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  • Thanks! Is it a special property that one side of the equation can be zero? If so, why is that the case? – csp2018 May 31 '19 at 15:09
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    @csp2018 No, that's just a tidy and standard way to write polynomial equations. There is no significant difference between the equations $x^2-5x+6=0$ and $x^2=5x-6$. – Arthur May 31 '19 at 15:11
  • Hmm, my first impression was that transcendental numbers are unique because you can't go from them to zero using those operations. Why can't you do multiplication or addition with transcendental numbers then? – csp2018 May 31 '19 at 15:19
  • @csp2018 Algebraic versus transcendental is all about what you have to start with. Usually that's the rational numbers. So if you have some number $x$, and through multiplication and addition with $x$ and rational numbers can reach a rational result (with some restrictions against "cheating", like $x-x$ not being allowed), then $x$ is algebraic. $0$ is usually set as the ultimate goal of this process, for no significant reason. Most real numbers are transcendental, but we can still add them and multiply them. It's kindof like the distinction between rational and irrational. – Arthur May 31 '19 at 15:24
  • Ah, so I think my confusion then comes from not knowing what the rules are to be considered not cheating. What are these rules? – csp2018 May 31 '19 at 16:51
  • @csp2018 I don't think that that's a detail you should worry too much about. Clearly, for any number $x$, we have that $x+ (-1)\cdot x$ is rational. Does that make $x$ algebraic? No, it does not. At least not by itself. Formally, we require that the final expression describing the operations you do, when maximally simplified, is a polynomial of degree at least $1$. So, for instance, "Take $x$, subtract $5$, then multiply the result by $x$" becomes $(x-5)x$, or $x^2 - 5x$. That's valid. If you do that, and the result is, say, $3$, then the $x$ you started with is algebraic. – Arthur May 31 '19 at 22:44
  • Usually, however, instead of saying that $x^2 - 5x = 3$, we would say $x^2 - 5x - 3 = 0$. In other words, "If you take $x$, subtract $5$, multiply the result by $x$ and then subtract $3$, you get $0$". This describes exactly the same $x$ as I did above, but the "$=0$" on one side (and the $0$ as the result of the algebraic manipulations I described with words) is considered more aesthetically pleasing. – Arthur May 31 '19 at 22:47
  • I had to look at some other sources, and it seems that they suggest trancendental numbers are numbers that you can't get to from rational numbers using polynomial functions. Is that what you meant by "the answer" to the polynomial function? – csp2018 Sep 07 '19 at 10:30