Questions: Which of the following is true? The square root of 4 is a rational number and an integer. -6.133... is an irrational number. The square root of 3 is a rational number. 0 is neither a rational number nor an irrational number.

Solution Steps

To determine which statements are true, we need to evaluate each one based on the definitions of rational and irrational numbers, as well as integers.

1. $\sqrt{4}$ is a rational number and an integer: Since $\sqrt{4} = 2$, which is both a rational number (can be expressed as a fraction) and an integer, this statement is true.
2. $-6.1 \overline{33}$ is an irrational number: This number is a repeating decimal, which can be expressed as a fraction, making it a rational number. Therefore, this statement is false.
3. $\sqrt{3}$ is a rational number: The square root of 3 is not a perfect square and cannot be expressed as a fraction, making it an irrational number. Therefore, this statement is false.
4. 0 is neither a rational number nor an irrational number: 0 can be expressed as a fraction (0/1), making it a rational number. Therefore, this statement is false.

We calculate \( \sqrt{4} = 2 \). Since \( 2 \) is an integer and can be expressed as a fraction \( \frac{2}{1} \), we conclude that \( \sqrt{4} \) is a rational number and an integer. Thus, the statement is False.

The number \( -6.1\overline{33} \) can be expressed as a fraction. Specifically, it simplifies to \( -\frac{92}{15} \). Since this fraction has a finite denominator, \( -6.1\overline{33} \) is a rational number. Therefore, the statement is True.

The value of \( \sqrt{3} \) is approximately \( 1.7321 \). Since \( \sqrt{3} \) cannot be expressed as a fraction, it is classified as an irrational number. Thus, the statement is False.

The number \( 0 \) can be expressed as \( \frac{0}{1} \), which confirms that it is a rational number. Therefore, the statement that \( 0 \) is neither a rational number nor an irrational number is False.

Final Answer