Here's a list of irrational numbers which almost fulfill your criteria.
Each were chosen to be accurate to within ±0.1, so that no two of them implicitly express the same mathematical approximation, and so that none of them "cheats" in order to fudge an exact result involving integers to obtain a slightly inexact, irrational result. Only the last number fails to meet your criteria, as it is slightly larger than 12.
- $\ln(3)$
- $7\pi/11$
- $\sqrt 2 + \frac\pi2$
- $7/{\sqrt[3]5}$
- $\mathrm e^\phi$
- $\sec^2(20)$
- $4\sqrt3$
- $5\phi$
- $2\pi+\mathrm e$
- $\sinh(3)$
- $\pi^3-20$
- $\csc^2(16)$
I would prefer not to use cosecant, integers greater than 12, or more than two additions/subtractions — it is too easy to get results if you rely on these — but I think I've spent enough time on this diversion for now. :-)
[EDIT: revised the formula for 4 two times now: this first to change the formula for 4 from √3 + √5 — which is too close to √4 + √4 — and the second time to correct the formula as I somehow copied a result which was not approximately 4.]
Added: Since it seems that I can't sleep tonight, here is a list of approximate values:
$\begin{align*}
\ln(3) & \approx 1.0986 \\
7\pi/11 & \approx 1.9992 \\
\sqrt 2 + \tfrac\pi2 & \approx 2.9850 \\
7 / \sqrt[3]5 & \approx 4.0936 \\
\mathrm e^\phi & \approx 5.0432 \\
\sec^2(20) & \approx 6.0049 \\
4\sqrt3 & \approx 6.9282 \\
5\phi & \approx 8.0902 \\
2\pi+\mathrm e & \approx 9.0015 \\
\sinh(3) & \approx 10.018 \\
\pi^3-20 & \approx 11.006 \\
\csc^2(16) & \approx 12.064
\end{align*}$
I have an analogy that I use with my students, and it is applicable here.
Mathematics is like literature. Things like elementary arithmetics (grade school $+$,$-$,$\cdot$, etc. on real numbers) are like the "abc"s. Things like algebraic manipulation (but not Algebra) (e.g. $log_2 (4 x)=9$, solve for $x$) are like words. (A lot of Americans reach only this stage and proclaim they hate math; it's as preposterous as saying you hate books when you can't even read the word the). Things like elementary calculus and real analysis are like sentences, and by the time you have an undergraduate degree, you can probably read a board book or two (without help, gasp!).
The whole rest of mathematics lies beyond.
The application is this: your "computations" are as fundamental to mathematics itself as the printed word is to literacy. It's great that you're excited about math, but I think it's extremely important to not just understand, but grok the basics before you try to build anything on it.
A lot of the stuff grade school kids learn is crap--like about not-dividing-by-zero, like about fractions, like about conics. And a lot of it is a dumbed-down-version or otherwise irrelevant once you get to higher math. But, a lot of it provides useful intuition. The analogy is: do you need to know what the letters' names are to read a book? No. Do you need to know that "?" is called "question mark"? No. As long as you understand that a is different from b and that ? is interrogative, then you're in good shape for understanding--but knowing those facts is useful too. If you know that a and e represent vowels, which all words in English phonology require, for example, then you automatically know why sdfslkjhrwfbv is an ill-formed English word.
The main points here are:
- You do not have to understand useless dogma educators teach kids to simplify lessons. I'm of the opinion that such lies should not be taught in the first place, since they screw up peoples' ideas later. I know. I've fixed a lot of it.
- However, by not being familiar with basic methods, you are missing part of the fundamental essence of math. If "just don't think about math like that", then you're missing part of what math actually is.
- You do not technically need to be competent at executing computations, so long as you do thoroughly understand them. It is difficult, however, to have the latter without the former--and, as one commenter noted, it is easy to delude yourself.
- Being incompetent at computation will harm your ability to interact with other mathematicians.
- Being incompetent at computation will minorly harm your ability to understand higher math.
Under these circumstances, I recommend becoming competent. If you really do understand computations, this should be relatively straightforward. It will be well, well worth your increased understanding and increased ability to interact. You don't have to aim for complete fluency at first, but I'd still hit the grammar monograph before I tackle Tolstoy.
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