The set of numbers that are real numbers but cannot be expressed as one integer divided by another negative integer or as a proper irrational or rational number includes all numbers that do not fit within these definitions.
To break this down:
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Rational Numbers: These are numbers that can be expressed as the quotient of two integers \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b \neq 0 \). A number is proper if \( |a| < |b| \). However, since we are focusing on the condition where the denominator is a negative integer, a proper rational number would not meet this criterion because \(|a| < |b|\) cannot hold if one of the numbers is negative.
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Irrational Numbers: These are numbers that cannot be expressed as a fraction of two integers. They include numbers such as \( \sqrt{2} \), \( \pi \), and \( e \).
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Numbers expressed as "one integer divided by another negative integer": This typically includes numbers like \(-\frac{1}{k}\) where \( k \) is a positive integer.
Given these definitions, the set you're describing can be interpreted as follows:
- It contains any real number that cannot be expressed in forms that are either rational (derivable from integers in the way stated) or proper irrational (since proper irrational numbers do not fit into the format of one integer divided by another negative integer).
Thus, the specific set you are referring to could include:
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Positive irrational numbers such as \( \sqrt{2} \), \( \pi \), etc., as these cannot be expressed as the result of dividing integers in the specified form.
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Negative numbers that are not of the form \(\frac{a}{b}\) where \(b\) is a negative integer, such as \(-\sqrt{2}\), which cannot be expressed as \(\frac{a}{b}\) with both \(a\) and \(b\) fitting the criteria given.
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Any real number that is not specifically a proper rational form as designated or negated variants thereof.
In summary, the entire set of real numbers (including all real irrational numbers and all rational forms outside of the prescribed negative integer division) aside from the rational numbers expressed in the specific form you described qualifies.