Question
What does the Closure property tell you about the rational and irrational numbers under subtraction?
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GPT-4o mini
The Closure Property in mathematics refers to a set being closed under a particular operation if performing that operation on members of the set always produces a member of the same set.
1. **Rational Numbers under Subtraction**:
- The set of rational numbers is closed under subtraction. This means that if you take any two rational numbers \(a\) and \(b\) (where \(a\) and \(b\) can be expressed as fractions \(\frac{p}{q}\) and \(\frac{r}{s}\), respectively), their difference \(a - b\) is also rational. Specifically, the result can be expressed as \(\frac{p}{q} - \frac{r}{s} = \frac{ps - qr}{qs}\), which is again a fraction and therefore rational.
2. **Irrational Numbers under Subtraction**:
- The set of irrational numbers is **not** closed under subtraction. This means that if you take two irrational numbers, their difference might not be irrational. For example, consider the irrational numbers \(\sqrt{2}\) and \(-\sqrt{2}\). Their difference is \(\sqrt{2} - (-\sqrt{2}) = \sqrt{2} + \sqrt{2} = 2\), which is a rational number. Hence, it is possible for the difference of two irrational numbers to be a rational number.
In summary:
- The set of rational numbers is closed under subtraction.
- The set of irrational numbers is not closed under subtraction.
1. **Rational Numbers under Subtraction**:
- The set of rational numbers is closed under subtraction. This means that if you take any two rational numbers \(a\) and \(b\) (where \(a\) and \(b\) can be expressed as fractions \(\frac{p}{q}\) and \(\frac{r}{s}\), respectively), their difference \(a - b\) is also rational. Specifically, the result can be expressed as \(\frac{p}{q} - \frac{r}{s} = \frac{ps - qr}{qs}\), which is again a fraction and therefore rational.
2. **Irrational Numbers under Subtraction**:
- The set of irrational numbers is **not** closed under subtraction. This means that if you take two irrational numbers, their difference might not be irrational. For example, consider the irrational numbers \(\sqrt{2}\) and \(-\sqrt{2}\). Their difference is \(\sqrt{2} - (-\sqrt{2}) = \sqrt{2} + \sqrt{2} = 2\), which is a rational number. Hence, it is possible for the difference of two irrational numbers to be a rational number.
In summary:
- The set of rational numbers is closed under subtraction.
- The set of irrational numbers is not closed under subtraction.