Compare decimals and fractions

Learning Objectives Construct a logical argument to justify the ordering of a fraction and a decimal. Apply the Trichotomy and Density properties of rational numbers to compare and order fractions and decimals. Prove the relationship between two rational numbers using the cross-multiplication property of inequality. Analyze the structure of terminating and repeating decimals to determine their exact fractional equivalents for precise comparison. Evaluate the validity of mathematical statements involving inequalities between decimals and fractions. Formulate a counterexample to disprove a false claim about the comparison of two numbers. Which is larger: 0.999... or 1? 🤔 The answer is a classic logical puzzle that reveals the true nature of infinite series and number represen...

TermDefinitionExample Rational NumberAny number that can be expressed as a quotient or fraction p/q of two integers, where p is the numerator and q is a non-zero denominator. As a decimal, a rational number will either terminate or repeat.The number 3/8 is a rational number. It can be expressed as the terminating decimal 0.375. Trichotomy PropertyA fundamental axiom of ordering real numbers. For any two real numbers, 'a' and 'b', exactly one of the following three statements is true: a < b, a = b, or a > b.When comparing 1/2 and 0.51, the Trichotomy Property guarantees that only one relationship is possible. Since 1/2 = 0.5, we can conclude that 1/2 < 0.51 is the only true statement. Density Property of Rational NumbersThis property states that between any two...

Cross-Multiplication Property of Inequality For fractions \( \frac{a}{b} \) and \( \frac{c}{d} \), with \( b > 0 \) and \( d > 0 \): \( \frac{a}{b} > \frac{c}{d} \) if and only if \( ad > bc \). This is a powerful logical tool for comparing two fractions without finding a common denominator or converting to decimals. It transforms the comparison of fractions into a comparison of integers. Conversion of a Purely Repeating Decimal to a Fraction For a repeating decimal \( x = 0.\overline{d_1d_2...d_n} \) with \( n \) repeating digits, the fractional equivalent is \( x = \frac{d_1d_2...d_n}{10^n - 1} \). This formula provides a direct method for converting an infinite repeating decimal into its exact rational form, which is essential for precise comparisons.