The Correct Answer and Explanation is:
The rational numbers in the image are:
- 0.3 (with a bar over the 3)
- -2
- 22/6
- 11.23654 (with a bar over the 4)
- 3 1/2
A rational number is any number that can be expressed as a fraction, or ratio, in the form p/q, where p and q are both integers and the denominator, q, is not zero. There are several clear indicators to help identify them.
The numbers from the list that fit this definition are rational for the following reasons. The number 0.3 with a bar over the 3 represents the repeating decimal 0.333…, which is equivalent to the fraction 1/3. All repeating decimals are rational. The number -2 is an integer, and all integers are rational because they can be written as a fraction with a denominator of 1, in this case, -2/1. The number 22/6 is already presented as a fraction of two integers, so it is rational by definition. The mixed number 3 1/2 is also rational because it can be converted into the improper fraction 7/2. Finally, 11.23654 with a bar over the 4 represents 11.2365444…, which is a repeating decimal and therefore a rational number.
The remaining numbers are irrational. An irrational number cannot be expressed as a simple fraction, and its decimal representation continues infinitely without any repeating pattern. Pi (π) is the most famous example of an irrational number. The numbers 90.790170… and 82.208701… are also irrational. The ellipsis (…) indicates that their decimals continue forever, and since there is no repeating bar, we assume the digits do not form a repeating pattern.
