Question
Are negative numbers always rational numbers?
Answers
GPT-4o mini
Yes, negative numbers are always rational numbers if they can be expressed as a ratio of two integers. A rational number is defined as any number that can be written in the form \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \) is not zero.
For example:
- The negative number \(-2\) can be expressed as \(-2 = \frac{-2}{1}\), which is a ratio of two integers.
- The negative number \(-\frac{1}{3}\) is also a rational number because it is already in the form \( \frac{p}{q} \) with \( p = -1 \) and \( q = 3 \).
Thus, all negative numbers that can be expressed in this manner are indeed rational numbers. However, it is worth noting that irrational numbers can also be negative; for example, \(-\sqrt{2}\) is a negative number but not a rational one. So, while all rational negative numbers are rational, not all negative numbers are rational.
For example:
- The negative number \(-2\) can be expressed as \(-2 = \frac{-2}{1}\), which is a ratio of two integers.
- The negative number \(-\frac{1}{3}\) is also a rational number because it is already in the form \( \frac{p}{q} \) with \( p = -1 \) and \( q = 3 \).
Thus, all negative numbers that can be expressed in this manner are indeed rational numbers. However, it is worth noting that irrational numbers can also be negative; for example, \(-\sqrt{2}\) is a negative number but not a rational one. So, while all rational negative numbers are rational, not all negative numbers are rational.