Comprehensive Notes on Rational Numbers and Arithmetic Operations

Course Logistics and Academic Context

• Timeline and Expectations: The schedule for the upcoming three weeks will be demanding, particularly with homecoming week at Benedict College occurring during this period. Freshman students who are new to the university environment must stay focused and on target.
• Classroom Participation: Active engagement is mandatory to earn participation points. Students are called directly to the board to demonstrate problem-solving methods in front of the class throughout the semester.
• Calculator Rules: Calculators are strictly prohibited for foundational arithmetic modules (such as Module 4). Computations, including multi-digit long division, must be performed manually.

Fundamentals of Rational Numbers

• Etymology and Concept: The word "rational" contains the root word "ratio". Consequently, rational numbers are numbers that can be expressed as a ratio of two integers.
• Formal Definition: A rational number is any number that can be written in the fractional form:

$\frac{a}{b}$

where $a$ and $b$ are integers, and $b \neq 0$ (since division by zero is undefined).

• Fraction Anatomy:

  • Numerator: The top integer $a$ in the ratio $\frac{a}{b}$.
  • Denominator: The bottom integer $b$ in the ratio $\frac{a}{b}$.

• Classification Hierarchy of Numbers:

  • Whole Numbers: Non-negative counting numbers beginning at zero:

$0, 1, 2, 3, 4, 5, \dots$

• Integers: Includes all whole numbers along with their negative whole counterparts:

$\dots, -4, -3, -2, -1, 0, 1, 2, 3, 4, \dots$

• Rational Numbers: The overarching category that encompasses all whole numbers, all integers, plus fractions and terminating or repeating decimals (e.g., $0.5$, $\frac{3}{4}$, $0.\bar{3}$).

Simplifying and Reducing Rational Numbers

• Cancellation Principle: A rational number can be reduced if there is a common nonzero integer factor $c$ in both the numerator and the denominator:

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

provided $c \neq 0$. The common terms $\frac{c}{c}$ evaluate to $1$ and cancel each other out.

• Factor Tree and Greatest Common Divisor (GCD) Method:
  • To reduce unfamiliar fractions, systematically decompose the numerator and denominator into prime factor trees.
  • Example: Reduce the rational number $\frac{130}{455}$.
  • Factor $130$: Ends in $0$, indicating divisibility by $5$.

$130 = 2 \times 5 \times 13$

- Factor $455$: Ends in $5$, indicating divisibility by $5$. Performing $455 \div 5$ yields $91$. $91$ is an odd number divisible by $7$ ($7 \times 13 = 91$).

$455 = 5 \times 7 \times 13$

- Determine Greatest Common Divisor (GCD): Identify all common prime factors present in both factorizations ($5$ and $13$).

$\text{GCD} = 5 \times 13 = 65$

- Cancel Common Factors:

$\frac{130}{455} = \frac{2 \times 5 \times 13}{5 \times 7 \times 13}$

- Canceling the $5$s and $13$s from numerator and denominator reduces the fraction completely to:

$\frac{2}{7}$

Mixed Numbers and Improper Fractions

• Improper Fraction: A rational number where the numerator is greater than or equal to the denominator ($\text{numerator} \ge \text{denominator}$).

• Converting a Mixed Number to an Improper Fraction:

  • Procedure:
  1. Multiply the denominator of the fractional part by the whole integer.
  2. Add the numerator to the resulting product.
  3. Place the total sum over the original denominator.
  • Example: Convert $3\frac{4}{5}$ to an improper fraction.
  • Step 1: Multiply denominator and whole number: $3 \times 5 = 15$
  • Step 2: Add numerator: $15 + 4 = 19$
  • Step 3: Write over original denominator: $\frac{19}{5}$

• Converting an Improper Fraction to a Mixed Number:

  • Procedure:
  1. Divide the numerator (dividend inside) by the denominator (divisor outside) using long division.
  2. The whole number quotient becomes the integer portion of the mixed number.
  3. The remainder is placed over the original denominator as the fractional portion.
  • Example: Convert $\frac{42}{5}$ to a mixed number form.
  • Set up division with $42$ on the inside and $5$ on the outside.
  • $5$ goes into $42$ a total of $8$ times ($8 \times 5 = 40$).
  • Subtract $40$ from $42$ to find the remainder: $42 - 40 = 2$.
  • Express result as quotient plus remainder over original denominator: $8\frac{2}{5}$.

Expressing Rational Numbers as Decimals via Long Division

• Manual Long Division Method: When converting a rational number $\frac{a}{b}$ into a decimal without a calculator, place $a$ inside the division bracket ($a.00\dots$) and $b$ outside.

• Example 1: Step-by-Step Long Division continuation ($50 \div 8$ context):

  • Subtract $48$ from $50$ to get $2$.
  • Check if $8$ goes into $2$: No. Bring down a zero to form $20$
  • Determine how many times $8$ goes into $20$: $2$ times ($8 \times 2 = 16$
  • Subtract $16$ from $20$ to get $4$
  • Check if $8$ goes into $4$: No. Bring down another zero to form $40$
  • Determine how many times $8$ goes into $40$: $5$ times ($8 \times 5 = 40$
  • The decimal expansion terminates with trailing digits $.25$

• Example 2: Converting $\frac{7}{11}$ to Decimal Form:

  • Set up division $7.000\dots \div 11$.
  • $11$ does not go into $7$. Place decimal point directly above in quotient.
  • $11$ goes into $70$ six ($6$) times ($6 \times 11 = 66$).
  • Subtract $66$ from $70$ to get $4$.
  • Bring down zero to form $40$.
  • $11$ goes into $40$ three ($3$) times ($3 \times 11 = 33$).
  • Subtract $33$ from $40$ to get $7$.\n - Bring down zero to form $70$.\n - 11$ goes into $70$ six (6$) times ($6 \times 11 = 66).\n - Subtract 66$from$70$to get$4$.
  • Bring down zero to form $40$.
  • $11$ goes into $40$ three ($3$) times ($3 \times 11 = 33$).
  • Repeating Decimal: The sequence $63$ repeats indefinitely ($0.636363\dots$). Indicate infinite repetition by placing a horizontal bar over the repeating digits:

$0.\overline{63}$

Place Value and Converting Decimals to Fractions

• Place Value Chart Structure:

  • Left of Decimal: Ones place, Tens place.
  • Right of Decimal: Tenths place ($\frac{1}{10}$), Hundredths place ($\frac{1}{100}$), Thousandths place ($\frac{1}{1000}$).
  • Monetary Analogy: Align decimal place values with currency (e.g., tenths and hundredths represent dimes and cents).

• Converting Terminating Decimals to $\frac{a}{b}$ Form:

  • Example A: $0.7$
  • Represents $0.70$ ($70\text{ cents}$).
  • The digit $7$ resides in the tenths position.
  • Expressed as rational quotient: $\frac{7}{10}$ ("seven tenths").
  • Example B: $0.49$
  • The rightmost digit $9$ resides in the hundredths position.
  • Expressed as rational quotient: $\frac{49}{100}$ ("forty-nine hundredths").
  • Example C: Decimal with rightmost digit in thousandths position
  • Expressed as a quotient over $1000$.

• Converting Repeating Decimals to $\frac{a}{b}$ Form:

  • For a repeating decimal like $0.\bar{6} = 0.6666\dots$, define a variable $n = 0.\bar{6}$ and use algebraic subtraction to express as an integer fraction.

Comparing Quantities of Rational Numbers

• Comparison Problem: Determine the largest quantity among $\frac{3}{4}$ of a cookie, $0.7$ of a cookie, and $\frac{2}{3}$ of a cookie.

• Decimal Conversions:

  • $\frac{3}{4} = 0.75$ (equivalent to $75\text{ hundredths}$ or $75\text{ cents}$).
  • $0.7 = 0.70$ (equivalent to $70\text{ hundredths}$ or $70\text{ cents}$).
  • $\frac{1}{3} = 0.\bar{3} = 0.3333\dots$
  • $\frac{2}{3} = 0.\bar{6} = 0.6666\dots$

• Comparing Place Values:

  • Compare Tenths Place: Both $0.75$ and $0.70$ have $7$ in the tenths place, whereas $0.666\dots$ has $6$. Because $7 > 6$, $\frac{2}{3}$ is the smallest quantity.
  • Compare Hundredths Place: Between $0.75$ and $0.70$, $5 > 0$, making $0.75$ greater than $0.70$

• Final Order (Largest to Smallest):

$\frac{3}{4} > 0.7 > \frac{2}{3}$

$0.75 > 0.70 > 0.666\dots$

Selecting $\frac{3}{4}$ provides the maximum amount.

Operations on Rational Numbers

• Multiplication of Rational Numbers:
  • Terminology: The result of multiplying numbers is called the product.
  • Rule: Multiply the numerators together and multiply the denominators together:

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

• Reduce the final product to lowest terms.

• Example Problem: Multiply $-\frac{2}{3} \times -\frac{9}{4}$

  • Step 1: Multiply numerators: $-2 \times -9 = 18$

  • Step 2: Multiply denominators: $3 \times 4 = 12$

  • Step 3: Resulting improper fraction: $\frac{18}{12}$

  • Step 4: Reduce to lowest terms:

    • Method 1 (Cancel common factor $6$): $\frac{18 \div 6}{12 \div 6} = \frac{3}{2} = 1\frac{1}{2}$
    • Method 2 (Long division): $18 \div 12 = 1$ with a remainder of $6$, yielding $1\frac{6}{12} = 1\frac{1}{2}$

  • Division of Rational Numbers:

• Terminology:

  • The result of dividing numbers is called the quotient.
  • The reciprocal (multiplicative inverse) is obtained by flipping the numerator and denominator ($\frac{c}{d} \rightarrow \frac{d}{c}$).

• Rule: To divide by a rational number, multiply the first fraction by the reciprocal of the second fraction:

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

• Example Problem: Divide $-\frac{3}{5} \div \frac{7}{11}$
  • Step 1: Find reciprocal of $\frac{7}{11}$, which is $\frac{11}{7}$.
  • Step 2: Change division sign to multiplication:

$-\frac{3}{5} \times \frac{11}{7}$

- Step 3: Multiply numerators across: $-3 \times 11 = -33$
- Step 4: Multiply denominators across: $5 \times 7 = 35$
- Final quotient: $-\frac{33}{35}$

Guided Classroom Exercises and Dialogue

• Long Division Step Dialogue:

  • Question: What is $50 - 48$?
  • Answer: $2$
  • Question: Does $8$ go into $2$?
  • Answer: No. Move over and bring down a zero to make $20$
  • Question: How many times does $8$ go into $20$?
  • Answer: $2$ times ($8 \times 2 = 16$
  • Question: What is $20 - 16$?
  • Answer: $4$. Bring down another zero to make $40$
  • Question: How many times does $8$ go into $40$?
  • Answer: $5$ times ($8 \times 5 = 40$

• Repeating Decimal Division Dialogue ($\frac{7}{11}$):

  • Question: How many times does $11$ go into $70$?
  • Answer: $6$ times ($6 \times 11 = 66$
  • Question: What is $70 - 66$?
  • Answer: $4$. Bring down zero to make $40$
  • Question: How many times does $11$ go into $40$?
  • Answer: $3$ times ($3 \times 11 = 33$
  • Question: What is $40 - 33$?
  • Answer: $7$. Bring down zero to make $70$
  • Observation: The sequence repeats infinitely as $0.6363\dots$, written as $0.\overline{63}$.