1. The problem asks to identify properties of irrational and rational numbers. 2. An irrational number is defined as a number that cannot be expressed as a ratio of two integers $\frac{a}{b}$ where $b \neq 0$. 3. Important rules: - Irrational numbers have decimal expansions that are infinite and non-repeating. - Rational numbers can be expressed as $\frac{a}{b}$ with integers $a,b$ and $b \neq 0$. - Rational numbers include integers, finite decimals, and repeating decimals. 4. Checking each statement for irrational numbers: - "A repeating decimal": No, repeating decimals are rational. - "An integer": No, integers are rational. - "A decimal": Yes, irrational numbers can be decimals but specifically infinite non-repeating decimals. - "A number that has a decimal expansion that is infinite and does not repeat": Yes, this is true for irrationals. - "A number that CAN NOT be written as a ratio of two integers $\frac{a}{b}$, $b \neq 0$": Yes, this is the defining property. - "A number that CAN be written as a ratio of two integers $\frac{a}{b}$, $b \neq 0$": No, this is false for irrationals. 5. Checking each statement for rational numbers: - "A repeating decimal": Yes, repeating decimals are rational. - "An integer": Yes, integers are rational numbers. - "A decimal": Yes, rational numbers can be decimals (finite or repeating). - "A number that has a decimal expansion that is infinite and does not repeat": No, this describes irrationals. - "A number that can not be written as a ratio of two integers $\frac{a}{b}$, $b \neq 0$": No, this is false for rationals. - "A number that can be written as a ratio of two integers $\frac{a}{b}$, $b \neq 0$": Yes, this is the defining property. Final answers: - Irrational numbers: check "A decimal", "A number that has a decimal expansion that is infinite and does not repeat", "A number that CAN NOT be written as a ratio of two integers $\frac{a}{b}$, $b \neq 0$". - Rational numbers: check "A repeating decimal", "An integer", "A decimal", "A number that can be written as a ratio of two integers $\frac{a}{b}$, $b \neq 0$".