🔑 Key Concepts
- A rational number can be written as \(\displaystyle \frac{a}{b}\), where \(a\) and \(b\) are integers and \(b\ne0\).
- Every integer is rational because it can be written with a denominator of \(1\).
- Every fraction made from integers with a nonzero denominator is rational.
- A terminating decimal is rational.
- A repeating decimal is rational because it follows a repeating pattern and can be written as a fraction.
- The square root of a perfect square is rational.
- The square root of a positive integer that is not a perfect square is irrational.
- When a question asks which set is all rational, every number in the selected set must be rational.
✏️ Worked Example
🧠 Math Vocabulary
- Rational number: A number that can be written as \(\displaystyle \frac{a}{b}\), where \(a\) and \(b\) are integers and \(b\ne0\).
- Irrational number: A real number that cannot be written as a ratio of two integers.
- Integer: A positive whole number, a negative whole number, or zero.
- Fraction: A number written as a quotient of a numerator and denominator.
- Numerator: The number above the fraction bar.
- Denominator: The number below the fraction bar.
- Terminating decimal: A decimal that ends after a finite number of digits.
- Repeating decimal: A decimal in which one digit or group of digits repeats indefinitely.
- Nonterminating decimal: A decimal that continues without ending.
- Nonrepeating decimal: A decimal that continues without developing a repeating pattern.
- Square root: A value that produces a given number when multiplied by itself.
- Perfect square: A number that is the square of an integer.
- Radical: An expression containing a root symbol.
- Radicand: The number or expression written inside a radical symbol.
- Real numbers: The set containing all rational and irrational numbers.
💡 Main Idea
A set contains all rational numbers only when every value in the set can be written as a ratio of two integers. Fractions, integers, terminating decimals, repeating decimals, and square roots of perfect squares are rational. Square roots of positive integers that are not perfect squares are irrational, so finding even one such value eliminates the entire choice.
📚 What You Should Already Know
Students should know how to simplify fractions, evaluate common square roots, recognize perfect squares, distinguish integers from non-integers, and interpret terminating and repeating decimal representations.
🚀 What Comes Next
Students can extend this understanding by placing rational and irrational numbers within the real number system, estimating non-perfect square roots, locating irrational numbers between consecutive integers, and comparing approximate values on a number line.
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