An irrational number is defined as a number that cannot be expressed as a fraction of two integers, meaning it cannot be written in the form \( \frac{a}{b} \) where both \(a\) and \(b\) are integers and \(b\) is not zero.
Based on your question, let's clarify:
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A fraction with zero in the numerator: This fraction is \(\frac{0}{b}\), which simplifies to \(0\). Zero is a rational number (it can be expressed as \(\frac{0}{1}\)), so this does not qualify as an irrational fraction.
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An improper fraction: An improper fraction is simply a fraction where the numerator is greater than or equal to the denominator, such as \(\frac{5}{3}\) or \(\frac{3}{3}\). Improper fractions can be rational numbers, so this does not inherently make a fraction irrational.
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A fraction with \(\pi\) as a numerator or denominator: Fractions that involve \(\pi\) (like \(\frac{\pi}{2}\) or \(\frac{3}{\pi}\)) are considered irrational. Since \(\pi\) is itself an irrational number, any fraction where \(\pi\) is involved will also be irrational.
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A fraction with a negative integer as the numerator: This would still be a rational number if the denominator is a non-zero integer. For example, \(\frac{-3}{4}\) is a rational number.
In summary, the only kind of fraction that qualifies as irrational is one that includes an irrational number in either the numerator or the denominator (like those involving \(\pi\), \(\sqrt{2}\), etc.).
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