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anemone
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Prove $\sqrt{2}+\sqrt{3}\gt \pi$.
anemone said:Prove $\sqrt{2}+\sqrt{3}\gt \pi$.
Albert said:$\begin{vmatrix}
\sqrt 3-\dfrac{\pi}{2}
\end{vmatrix}>\begin{vmatrix}
\sqrt 2-\dfrac{\pi}{2}
\end{vmatrix}=\begin{vmatrix}
\dfrac{\pi}{2}-\sqrt 2
\end{vmatrix}$
we have:
$\sqrt 3-\dfrac{\pi}{2}>\dfrac {\pi}{2}-\sqrt 2>0$
$\therefore \sqrt 3+\sqrt 2>\pi$
the proof is done
[sp]I don't understand this argument. Why is $\Bigl|\sqrt 3-\dfrac{\pi}{2}\Bigr| > \Bigl|\dfrac{\pi}{2}-\sqrt 2\Bigr|$?[/sp]Albert said:$\begin{vmatrix}
\sqrt 3-\dfrac{\pi}{2}
\end{vmatrix}>\begin{vmatrix}
\sqrt 2-\dfrac{\pi}{2}
\end{vmatrix}=\begin{vmatrix}
\dfrac{\pi}{2}-\sqrt 2
\end{vmatrix}$
we have:
$\sqrt 3-\dfrac{\pi}{2}>\dfrac {\pi}{2}-\sqrt 2>0$
$\therefore \sqrt 3+\sqrt 2>\pi$
the proof is done
Opalg said:[sp]I don't understand this argument. Why is $\Bigl|\sqrt 3-\dfrac{\pi}{2}\Bigr| > \Bigl|\dfrac{\pi}{2}-\sqrt 2\Bigr|$?[/sp]
Opalg said:[sp]I don't understand this argument. Why is $\Bigl|\sqrt 3-\dfrac{\pi}{2}\Bigr| > \Bigl|\dfrac{\pi}{2}-\sqrt 2\Bigr|$?[/sp]
Albert said:if $\Bigl|A\Bigr| > \Bigl|B\Bigr|$
and $\Bigl|B\Bigr| = \Bigl|C\Bigr|$
then
$\Bigl|A\Bigr| > \Bigl|C\Bigr|$
√2 + √3 > π
? The mathematical challenge of proving √2 + √3 > π
is to show that the sum of the irrational numbers √2
and √3
is greater than the irrational number π
. This challenge requires the use of advanced mathematical techniques and concepts, such as real analysis and the properties of irrational numbers.
√2
and √3
? Yes, this statement is true for all values of √2
and √3
. The inequality √2 + √3 > π
holds for any values of √2
and √3
, as long as they are both irrational numbers.
Some techniques that can be used to prove √2 + √3 > π
include real analysis, the properties of irrational numbers, and the use of inequalities and limits. Other advanced mathematical concepts, such as calculus and trigonometry, may also be used in the proof.
This statement is important in mathematics because it demonstrates the relationship between different types of irrational numbers. It also highlights the complexity of irrational numbers and the need for advanced mathematical techniques to understand and prove their properties. Additionally, this statement has applications in various fields such as physics, engineering, and finance.
One real-life example that illustrates this statement is the construction of a regular pentagon using only a ruler and compass. The length of one side of a regular pentagon is √5
, which can be expressed as √2 + √3
. This shows that the sum of two irrational numbers, √2
and √3
, is greater than the irrational number π
, which is the ratio of the circumference of a circle to its diameter.