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[parent] rational and irrational (Result)

The sum, difference, product and quotient of two non-zero real numbers, from which one is rational and the other irrational, is irrational.

Proof. Let $ a$ be a rational and $ \alpha$ irrational number. Here we prove only that $ \frac{a}{\alpha}$ is irrational -- the other cases are similar. If $ \frac{a}{\alpha}$ were a rational number $ r \neq 0$, then also $ \alpha = ar^{-1}$ would be rational as a product of two rationals. This contradiction shows that $ \frac{a}{\alpha}$ is irrational.

Note. In the result, the words real, rational and irrational may be replaced resp. by the words complex, algebraic and transcendental or resp. by the words complex, real and imaginary (the last term here meaning, as commonly in Continental Europe, a complex number having non-zero imaginary part).



"rational and irrational" is owned by pahio.
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See Also: examples of periodic functions, commensurable numbers


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Cross-references: imaginary part, complex number, transcendental, algebraic, complex, contradiction, product, rational number, similar, proof, irrational, rational, real numbers, quotient, difference, sum
There are 5 references to this entry.

This is version 4 of rational and irrational, born on 2005-01-28, modified 2005-01-29.
Object id is 6677, canonical name is RationalAndIrrational.
Accessed 4243 times total.

Classification:
AMS MSC: 11J72 (Number theory :: Diophantine approximation, transcendental number theory :: Irrationality; linear independence over a field)
 11J82 (Number theory :: Diophantine approximation, transcendental number theory :: Measures of irrationality and of transcendence)
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