Comparing expressions exactly

Question

Suppose I have two expressions; call them $A$ and $B$. The following values of $A$ and $B$ are good examples for my question...

$$ A = \pi e^2\\ B = \pi^2 e $$

Is there a method to determine the truth of comparisons such as $A \ge B$, $A \lt B$, etc? Especially, I want to ensure that exactness is preserved if possible, so I'd like to avoid converting $A$ and $B$ to numbers.

In this particular case, we can at least factorize: $$A\ge B \iff A-B\ge0 \iff \pi e(e-\pi)\ge0 \iff e-\pi\ge0$$ — Librecoin, Apr 19 '13 at 20:41
@gh403 Use the definitions of the quantities $e$ and $\pi$ to replace them by finite or infinite sums, products, and the like, and then try to simplify once more. Continue (if possible) until the order of the remaining quantities is known. — Librecoin, Apr 19 '13 at 21:19

score 3 · Accepted Answer · answered Apr 19 '13 at 20:51

3

Actually, I believe I can be a bit more specific. Even if the number is the result of using integers, addition, subtraction, multiplication, division, and the placement of square root signs anywhere, there is no canonical way of writing this number. Meaning there is no evident way of telling whether it is positive or negative: a certain amount of cleverness in squaring sides, gathering on one side, etc. will be required. Thus, if we take the square root of what we have so far, we cannot entirely be sure the result is real.

answered Apr 19 '13 at 20:51

Will Jagy

139,541

Suppose my program (for it is a program I'm writing ;) has its own notion of "canonical form." Is it possible once I have both $A$ and $B$ in canonical form? – thirtythreeforty Apr 19 '13 at 21:01
@gh403, first, tell me whether $$ 18991028052879952877717003280980106896077255885649556410 - 317799749249667142896491448615083010692333748314716137 \cdot \sqrt{3571} $$ is positive or negative, then we can discuss it further. – Will Jagy Apr 19 '13 at 22:48
Then $$ 3026807861414232601576476947093931340883926646932707209616924 - 27015380505739903416049453225901131788373049465835213058397 \cdot \sqrt{12553}
$$ – Will Jagy Apr 19 '13 at 23:18
Gotcha, makes sense. – thirtythreeforty Apr 21 '13 at 16:01

Comparing expressions exactly

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