Operations with rational numbers Knowledge-Based Study

Study notes

1. Addition and Subtraction of Rational Numbers

   A rational number is any number that can be expressed as a quotient or fraction $\frac{a}{b}$, where $a$ and $b$ are integers and $b \neq 0$. Rational numbers include positive and negative fractions, integers, terminating decimals, and repeating decimals.

   Like Denominators

   When adding or subtracting rational numbers with identical denominators, combine the numerators while keeping the denominator constant:

   $$\frac{a}{c} + \frac{b}{c} = \frac{a + b}{c}$$

   Remember that the rules of integer signs apply directly to numerators:

   • Example: $-\frac{5}{9} + \frac{2}{9} = \frac{-5 + 2}{9} = -\frac{3}{9} = -\frac{1}{3}$
   • Example: $-\frac{7}{12} - \left(-\frac{5}{12}\right) = \frac{-7 - (-5)}{12} = \frac{-7 + 5}{12} = -\frac{2}{12} = -\frac{1}{6}$

   Unlike Denominators

   When denominators differ, determine the least common denominator (LCD) to rewrite each fraction as an equivalent fraction before operating:

   To calculate $\frac{3}{4} - \frac{5}{6}$:

   1. The least common multiple of 4 and 6 is 12.
   2. Rewrite each fraction: $\frac{3 \times 3}{4 \times 3} = \frac{9}{12}$ and $\frac{5 \times 2}{6 \times 2} = \frac{10}{12}$.
   3. Subtract: $\frac{9}{12} - \frac{10}{12} = \frac{9 - 10}{12} = -\frac{1}{12}$.

   Multiplication of Rational Numbers

   To multiply rational numbers, multiply the numerators together and multiply the denominators together. You do not need a common denominator.

   $$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$$

   Sign Rules for Multiplication

   The standard sign rules for integers apply to rational numbers:

   • The product of two numbers with the same sign is positive: $(+) \times (+) = (+)$ and $(-) \times (-) = (+)$.
   • The product of two numbers with different signs is negative: $(+) \times (-) = (-)$ and $(-) \times (+) = (-)$.

   Simplifying Before Multiplying

   It is best practice to divide out common factors from any numerator and any denominator before calculating products. This keeps computations manageable.

   Example: Calculate $-\frac{8}{15} \times \frac{5}{12}$

   1. Identify common factors across terms: 8 and 12 share a factor of 4; 5 and 15 share a factor of 5.
   2. Simplify: $-\frac{8 \div 4}{15 \div 5} \times \frac{5 \div 5}{12 \div 4} = -\frac{2}{3} \times \frac{1}{3}$.
   3. Multiply: $-\frac{2 \times 1}{3 \times 3} = -\frac{2}{9}$.

   If working with mixed numbers, always convert them to improper fractions before multiplying.

   Division of Rational Numbers

   Division by a rational number is equivalent to multiplying by its multiplicative inverse (or reciprocal). Two numbers are reciprocals if their product equals 1.

   Finding the Reciprocal

   The reciprocal of a non-zero rational number $\frac{a}{b}$ is $\frac{b}{a}$. Notice that the sign remains the same:

   • The reciprocal of $\frac{3}{7}$ is $\frac{7}{3}$.
   • The reciprocal of $-\frac{4}{5}$ is $-\frac{5}{4}$.
   • The reciprocal of $-6$ (which is $-\frac{6}{1}$) is $-\frac{1}{6}$.
   • The number 0 has no reciprocal, which confirms that division by zero is undefined.

   The Division Rule

   $$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{a \times d}{b \times c} \quad (b, c, d \neq 0)$$

   Example: Calculate $-\frac{9}{14} \div \left(-\frac{3}{7}\right)$

   1. Rewrite as multiplication by the reciprocal: $-\frac{9}{14} \times \left(-\frac{7}{3}\right)$.
   2. Determine the sign: A negative multiplied by a negative yields a positive result.
   3. Simplify common factors: 9 and 3 divide by 3; 7 and 14 divide by 7.
   4. Evaluate: $\frac{3}{2} \times \frac{1}{1} = \frac{3}{2}$ or $1\frac{1}{2}$.

   Order of Operations

   When an expression contains multiple operations involving rational numbers, evaluate it using the standard order of operations (PEMDAS / BODMAS):

   1. Parentheses / Grouping Symbols: Evaluate operations inside innermost brackets or parentheses first.
   2. Exponents / Powers: Compute powers and roots. Remember that $\left(-\frac{1}{2}\right)^2 = \frac{1}{4}$, whereas $-\left(\frac{1}{2}\right)^2 = -\frac{1}{4}$.
   3. Multiplication and Division: Compute from left to right in the order they appear.
   4. Addition and Subtraction: Compute from left to right in the order they appear.

   Worked Multi-Step Example

   Evaluate the expression: $$\frac{1}{2} - \frac{3}{4} \times \left(-\frac{2}{5}\right) + \frac{1}{10}$$

   1. First, carry out the multiplication: $\frac{3}{4} \times \left(-\frac{2}{5}\right) = -\frac{6}{20} = -\frac{3}{10}$.
   2. Substitute back into the expression: $\frac{1}{2} - \left(-\frac{3}{10}\right) + \frac{1}{10}$.
   3. Convert subtraction of a negative into addition: $\frac{1}{2} + \frac{3}{10} + \frac{1}{10}$.
   4. Express all terms with a common denominator of 10: $\frac{5}{10} + \frac{3}{10} + \frac{1}{10} = \frac{9}{10}$.