If in lowest terms, then makes even. Substituting then makes even, contradicting that the integers are coprime. More quantitatively, for every nonzero integer , and factoring the difference of squares gives . To verify the constant, for the other factor is less than four; outside this interval the approximation error is at least .
Every nonempty open real interval contains an irrational number. Choose a rational number strictly inside the interval using density of the rational numbers; then remains inside for sufficiently large positive integers and is irrational. Thus the irrational numbers form a dense subset of the real line. Together with rational density, this gives approaching sequences of both types at every real point, useful for testing continuity of functions defined differently on the two sets.
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An irrational number is a type of real number that cannot be expressed as a simple fraction or ratio of two integers. This means that if a number is irrational, it cannot be written in the form \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \neq 0 \). Irrational numbers have non-repeating, non-terminating decimal expansions. This means their decimal representations go on forever without repeating a pattern.