Question
Rational numbers cannot
be negative
go on forever and ever
• repeat digits in the decimal
• be square roots of non-perfect squares
_- (1 point)
be negative
go on forever and ever
• repeat digits in the decimal
• be square roots of non-perfect squares
_- (1 point)
Answers
GPT-4o mini
Rational numbers are defined as numbers that can be expressed as the quotient or fraction \( \frac{p}{q} \) of two integers, where \( p \) is the numerator, \( q \) is the non-zero denominator. Let's go through the statements provided:
1. **Cannot be negative**: This is false. Rational numbers can be negative (e.g., \( -\frac{1}{2} \)) as well as positive (e.g., \( \frac{1}{2} \)) and zero (e.g., \( 0 \)).
2. **Go on forever and ever**: In the context of decimal representations, this can refer to the fact that some rational numbers have repeating decimal expansions (e.g., \( \frac{1}{3} = 0.333... \)). However, not all rational numbers have decimals that "go on forever"; finite decimal representations also exist (e.g., \( \frac{1}{2} = 0.5 \)).
3. **Repeat digits in the decimal**: This is true. Rational numbers have either terminating or repeating decimal expansions.
4. **Be square roots of non-perfect squares**: This is false. The square root of a non-perfect square (e.g., \( \sqrt{2} \)) is an irrational number, not a rational number.
In summary:
- Rational numbers **can** be negative.
- They can have decimal expansions that are either terminating or repeating.
- They **do not** include square roots of non-perfect squares, which are irrational.
So the correct understanding is that rational numbers can be either positive or negative, can have repeating decimals, and are not square roots of non-perfect squares.
1. **Cannot be negative**: This is false. Rational numbers can be negative (e.g., \( -\frac{1}{2} \)) as well as positive (e.g., \( \frac{1}{2} \)) and zero (e.g., \( 0 \)).
2. **Go on forever and ever**: In the context of decimal representations, this can refer to the fact that some rational numbers have repeating decimal expansions (e.g., \( \frac{1}{3} = 0.333... \)). However, not all rational numbers have decimals that "go on forever"; finite decimal representations also exist (e.g., \( \frac{1}{2} = 0.5 \)).
3. **Repeat digits in the decimal**: This is true. Rational numbers have either terminating or repeating decimal expansions.
4. **Be square roots of non-perfect squares**: This is false. The square root of a non-perfect square (e.g., \( \sqrt{2} \)) is an irrational number, not a rational number.
In summary:
- Rational numbers **can** be negative.
- They can have decimal expansions that are either terminating or repeating.
- They **do not** include square roots of non-perfect squares, which are irrational.
So the correct understanding is that rational numbers can be either positive or negative, can have repeating decimals, and are not square roots of non-perfect squares.