is something subtle about proofing that SQRT(5) < (SQRT(5)+1) ?

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Nasser M. Abbasi

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Oct 30, 2021, 3:03:32 AM10/30/21
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since sqrt of number is taken as the positive root, why
then Maple 2021.1 says it can't show this is true or not? Is
there something deep I am overlooking here?

if evalb( 5^(1/2) < (5^(1/2)+1) ) then
"yes, smaller";
fi;

Error, cannot determine if this expression is true or false: 5^(1/2) < 5^(1/2)+1

if 5^(1/2) < (5^(1/2)+1) then
"yes, smaller";
fi;

Error, cannot determine if this expression is true or false: 5^(1/2) < 5^(1/2)+1

But in Mathematica it did not complain

5^(1/2) < (5^(1/2) + 1)
True

This is very strange. Does your CAS have any problem showing
that 5^(1/2) is smaller than 5^(1/2)+1?

In Maple, it can do it if I convert everything to float

if evalf(5^(1/2)) < evalf((5^(1/2)+1)) then
"yes, smaller";
fi;

"yes, smaller"

--Nasser

acer

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Oct 30, 2021, 9:03:43 AM10/30/21
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You are simply using the wrong command, `evalb`, instead of an appropriate command such as `is`.

> evalb( 5^(1/2) < (5^(1/2)+1) );
1/2 1/2
5 < 5 + 1

> is( 5^(1/2) < (5^(1/2)+1) );
true

Your incorrect preconceptions as to the designed functionality of the `evalb` command are getting in your way here.

acer

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Oct 30, 2021, 11:53:50 AM10/30/21
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I could add that the functionality changes according to the `type` of the input, for a clear programmatic distinction.

The `evalb` command tests an inequality if the arguments are of type `numeric`, in Maple's technical sense of the `type` command. And that type does not include exact radicals (though it does include, say, rationals and floats).

For strict equality testing the functionality is also according to that same `numeric` type. But in this case the remaining cases are subjested to a pure address check (which is a structural rather than a mathematical comparison).
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