In geometry, the cube's edge length \( a \) is rational, but some measurements related to the cube may require the use of irrational numbers. The most common examples are:
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Diagonal of a face: The diagonal \( d \) of a face of the cube can be found using the Pythagorean theorem. For a square face of the cube with sides of length \( a \):
\[ d = \sqrt{a^2 + a^2} = \sqrt{2a^2} = a\sqrt{2} \]
Since \( \sqrt{2} \) is an irrational number, \( d \) becomes irrational when \( a \) is rational.
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Space diagonal: The space diagonal \( D \), which runs from one corner of the cube to the opposite corner, can also be calculated using the Pythagorean theorem in three dimensions:
\[ D = \sqrt{a^2 + a^2 + a^2} = \sqrt{3a^2} = a\sqrt{3} \]
Again, \( \sqrt{3} \) is an irrational number, so the space diagonal \( D \) is also irrational when \( a \) is rational.
In summary, the measurements that require the use of irrational numbers when the edge length \( a \) is a rational number are the lengths of the diagonal of a face and the space diagonal of the cube.