Rational Numbers: Properties, Operations, and Applications

Number Theory and the Real Number System

Rational Numbers

The set of rational numbers (denoted by Q) extends the set of integers by including all numbers that can be written as a fraction of two integers, with a nonzero denominator. Rational numbers are fundamental in mathematics and are used to represent quantities that are not whole numbers.

• Definition: A rational number is any number of the form $\frac{a}{b}$, where $a$ and $b$ are integers and $b \neq 0$.

• Examples: $0$,$-3$, $-\frac{2}{5}$, and$\frac{4}{-10}$ are all rational numbers.

• Closure Properties: The sum, difference, or product of any two integers is always an integer, but the quotient is not always an integer (e.g., $24 \div 7$ is not an integer).

• Every integer is rational: Since $a = \frac{a}{1}$ for any integer $a$, all integers are rational numbers.

• Equivalent Fractions: $-\frac{2}{5}$ and $\frac{4}{-10}$ are equivalent because they represent the same value.

Fundamental Principle of Rational Numbers

The fundamental principle of rational numbers states that multiplying the numerator and denominator of a rational number by the same nonzero integer yields an equivalent fraction.

• Formula: $\frac{a}{b} = \frac{a \times c}{b \times c}$, where $c \neq 0$.

• Example: Show that $-\frac{2}{5}$ and $\frac{4}{-10}$ are equivalent:

  • $\frac{4}{-10} = \frac{-2 \times -2}{5 \times -2} = -\frac{2}{5}$ (using $c = -2$)

Reducing Fractions

A rational number is in reduced terms or simplified form if the numerator and denominator have no common divisors other than 1 (i.e., they are relatively prime).

• Definition: $\frac{a}{b}$ is reduced if $\gcd(a, b) = 1$.

• Example: $\frac{1890}{2016}$ can be reduced by dividing numerator and denominator by their GCD (126):

  • $\frac{1890}{2016} = \frac{15 \times 126}{16 \times 126} = \frac{15}{16}$

• Uniqueness: The fundamental theorem of arithmetic ensures every rational number has a unique reduced form.

Mixed Numbers and Improper Fractions

Mixed numbers combine an integer and a proper fraction, while improper fractions have numerators greater than or equal to denominators.

• Mixed Number: $3\frac{4}{5} = 3 + \frac{4}{5}$

• Improper Fraction: $\frac{19}{5}$ is improper because $19 > 5$.

• Conversion:

  • To convert $3\frac{4}{5}$ to an improper fraction: $3 = \frac{15}{5}$, so $3 + \frac{4}{5} = \frac{19}{5}$.

  • To convert $\frac{19}{5}$ to a mixed number: $19 \div 5 = 3$ remainder $4$, so$\frac{19}{5} = 3\frac{4}{5}$.

Rational Numbers and Decimals

Every rational number can be written as a decimal that either terminates or repeats. The decimal expansion is found using long division.

• Terminating Decimal: The division ends with a remainder of zero.

• Repeating Decimal: The division results in a repeating pattern of digits.

• Example: $\frac{13}{7}$ as a decimal:

  • Long division yields $1.857142...$ (the sequence 857142 repeats).

Converting Decimals to Fractions

Any terminating or repeating decimal can be written as a fraction of two integers.

• Terminating Decimal: If the decimal terminates after $n$ digits, multiply by $10^n$ to form the numerator, and use $10^n$ as the denominator.

  • Examples:

    • $8.25 = \frac{825}{100}$

    • $3.14159 = \frac{314159}{100000}$

    • $1234.5678 = \frac{12345678}{10000}$

• Repeating Decimal: More complex; use algebraic manipulation.

  • Example: Convert $1.2345$ (where 45 repeats) to a fraction:

    1. Multiply by 100: $123.45$ (repeating part is now to the left of the decimal).

    2. Write as $123 + \frac{45}{99}$ (since two digits repeat, use 99).

    3. Combine: $123 + \frac{45}{99} = \frac{123 \times 99 + 45}{99} = \frac{12222}{99}$

    4. Adjust for the initial multiplication by 100: $\frac{12222}{9900}$

Multiplying Rational Numbers

To multiply two rational numbers, multiply their numerators and denominators.

• Formula: $\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$

• Example: $-3\frac{2}{3} \times -1\frac{1}{4}$

  • Convert to improper fractions: $-3\frac{2}{3} = -\frac{11}{3}$, $-1\frac{1}{4} = -\frac{5}{4}$

  • Product: $-\frac{11}{3} \times -\frac{5}{4} = \frac{55}{12} = 4\frac{7}{12}$

Dividing Rational Numbers

To divide by a rational number, multiply by its reciprocal.

• Reciprocal: The reciprocal of $\frac{a}{b}$ is $\frac{b}{a}$.

• Formula: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{a \times d}{b \times c}$

• Example: $-3\frac{2}{3} \div -1\frac{1}{4}$

  • Convert to improper fractions: $-\frac{11}{3} \div -\frac{5}{4}$

  • Multiply by reciprocal: $\frac{11}{3} \times \frac{4}{5} = \frac{44}{15} = 2\frac{14}{15}$

Adding and Subtracting Rational Numbers

To add or subtract rational numbers:

• Same Denominator: Add or subtract numerators, keep the denominator.

  • Examples: $\frac{2}{7} + \frac{6}{7} = \frac{8}{7}$, $\frac{2}{7} - \frac{6}{7} = -\frac{4}{7}$

• Different Denominators: Find a common denominator (often the least common multiple, LCM), convert, then add or subtract.

  • Example: $\frac{3}{4} + \frac{1}{6}$

    • LCM of 4 and 6 is 12.

    • Convert: $\frac{3}{4} = \frac{9}{12}$, $\frac{1}{6} = \frac{2}{12}$

    • Add: $\frac{9}{12} + \frac{2}{12} = \frac{11}{12}$

Density of Rational Numbers

The rational numbers are dense in the real number line, meaning between any two rational numbers, there is always another rational number.

• Fact: If $r$ and $t$ are rational numbers with $r < t$, there exists a rational $s$ such that $r < s < t$.

• Method: The average $s = \frac{r + t}{2}$ is always a rational number between $r$ and $t$.

• Example: Find a rational number between $\frac{1}{3}$ and $\frac{1}{2}$:

  • Average: $\frac{1}{2} \left( \frac{1}{3} + \frac{1}{2} \right) = \frac{1}{2} \left( \frac{2}{6} + \frac{3}{6} \right) = \frac{1}{2} \left( \frac{5}{6} \right) = \frac{5}{12}$

Problem Solving with Rational Numbers

Rational numbers are used in real-world problem solving, such as scaling recipes or comparing ratios.

• Example: A recipe calls for $\frac{3}{4}$ cup of sugar for 5 dozen cookies. How much sugar is needed for 8 dozen cookies?

  • Set up a proportion: $\frac{3/4 \text{ cup}}{5 \text{ dozen}} = \frac{x \text{ cup}}{8 \text{ dozen}}$

  • Solve for $x$: $x = \frac{3}{4} \times \frac{8}{5} = \frac{24}{20} = \frac{6}{5} = 1\frac{1}{5}$ cups