Comprehensive Study Guide: Integers, Fractions, Order of Operations, Ratios, Sets, and Limits

Integer Operations

• Integers comprise the set of all positive and negative whole numbers, as well as zero: $\mathbb{Z} = \{\dots, -4, -3, -2, -1, 0, 1, 2, 3, 4, \dots\}$.

• Integers are used to represent location, direction, magnitude, and changes in real-world contexts.

Addition and Subtraction of Integers

• Number Line Representation:

  • Addition and subtraction can be visually represented on a horizontal number line.

  • Begin at the position of the first integer.

  • To add a positive integer (or subtract a negative integer), move to the right.

  • To add a negative integer (or subtract a positive integer), move to the left.

• Subtracting Integers Rule: To subtract an integer, add its additive opposite:

$a - b = a + (-b)$

$a - (-b) = a + b$

• Simplifying Expressions (Removing Brackets and Double Signs):

  • Standard integer notation eliminates extraneous brackets and signs:

  • $\text{+(+)} \rightarrow +$

  • $\text{+(-)} \rightarrow -$

  • $\text{-(+)} \rightarrow -$

  • $\text{-(-)} \rightarrow +$

• Worked Examples:

  • Simplify $(+1) + (+3)$:

$1 + 3 = 4$

• Simplify $(+1) + (-3)$:

$1 - 3 = -2$

• Simplify $(+6) - (-2)$:

$6 + 2 = 8$

• Simplify $(-7) - (+3)$:

$-7 - 3 = -10$

• Simplify $(+3) + (-5) + (+6) + (-7)$:

$3 - 5 + 6 - 7 = -3$

• Simplify $(+5) - (+9) + (+2) - (-8)$:

$5 - 9 + 2 + 8 = 6$

• Simplify $-3 + 6 - 5 + 4$:

$(-3 - 5) + (6 + 4) = -8 + 10 = 2$

• Real-World Application (Atmospheric Lapse Rate):

  • In Mathville (altitude $350\,\text{m}$), the temperature is $8^\circ\text{C}$. The temperature decreases by $6.5^\circ\text{C}$ for every $1000\,\text{m}$ increase in altitude. Integer-town is located at an altitude of $2350\,\text{m}$. Determine if precipitation in Integer-town will occur as rain or snow.

  • Calculate the change in altitude:

$\Delta h = 2350\,\text{m} - 350\,\text{m} = 2000\,\text{m}$

• Calculate the number of $1000\,\text{m}$ increments:

$\frac{2000\,\text{m}}{1000\,\text{m}} = 2$

• Calculate total temperature decrease:

$\Delta T = 2 \times 6.5^\circ\text{C} = 13^\circ\text{C}$

• Determine temperature at Integer-town:

$T = 8^\circ\text{C} - 13^\circ\text{C} = -5^\circ\text{C}$

• Since the temperature is below the freezing threshold of $0^\circ\text{C}$, the precipitation will fall as snow.

Multiplication and Division of Integers

• Sign Rules for Multiplication and Division:

  • The product or quotient of two integers with identical signs is positive:

$\text{positive} \times \text{positive} = \text{positive}$

$\text{negative} \times \text{negative} = \text{positive}$

• The product or quotient of two integers with opposing signs is negative:

$\text{positive} \times \text{negative} = \text{negative}$

$\text{negative} \times \text{positive} = \text{negative}$

• Worked Examples:

  • Calculate $(-2)(+6)$:

$(-2) \times 6 = -12$

• Calculate $(-7)(-8)$:

$(-7) \times (-8) = 56$

• Calculate $(+2)(-3)(-3)$:

$2 \times (-3) \times (-3) = -6 \times (-3) = 18$

• Calculate $(-24) \div (-6)$:

$\frac{-24}{-6} = 4$

• Calculate $(-15) \div (+5)$:

$\frac{-15}{5} = -3$

• Calculate $\frac{-42}{7}$:

$\frac{-42}{7} = -6$

• Calculate $\frac{5(-7)}{(-3)(+2)}$:

$\frac{-35}{-6} = \frac{35}{6} = 5\frac{5}{6}$

Fraction Arithmetic and Mixed Numbers

• Classifications of Fractions:

  • Proper Fraction: A fraction where the absolute value of the numerator is strictly less than the denominator ($\left|\text{numerator}\right| < \text{denominator}$).

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

  • Mixed Number: An expression combining a non-zero integer part and a proper fraction.

• Conversion Algorithms:

  • Mixed Number to Improper Fraction:

$a\frac{b}{c} = \frac{(a \times c) + b}{c}$

• Improper Fraction to Mixed Number:

$\frac{n}{d} = q\frac{r}{d}$

    where $q = n \div d$ (integer quotient) and $r = n \bmod d$ (remainder).

• Addition and Subtraction Procedures:

  1. Convert all mixed numbers into improper fractions.

  2. Determine a common denominator across all terms (preferably the least common multiple).

  3. Convert each fraction to an equivalent fraction with the common denominator.

  4. Add or subtract the numerators while keeping the denominator constant.

  5. Simplify the final fraction to lowest terms.

• Worked Examples:

  • Evaluate $3\frac{2}{3} + \left(-1\frac{3}{4}\right)$:

$3\frac{2}{3} = \frac{11}{3}$

$-1\frac{3}{4} = -\frac{7}{4}$

$\frac{11}{3} - \frac{7}{4} = \frac{44}{12} - \frac{21}{12} = \frac{23}{12} = 1\frac{11}{12}$

• Evaluate $\frac{2}{5} - \left(-\frac{1}{10}\right)$:

$\frac{2}{5} + \frac{1}{10} = \frac{4}{10} + \frac{1}{10} = \frac{5}{10} = \frac{1}{2}$

• Multiplication and Division Procedures:

  • Multiplication: Convert terms to improper fractions, then multiply numerators directly and denominators directly:

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

• Division: Convert terms to improper fractions, replace division with multiplication by the reciprocal (flip the second fraction):

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

• Worked Multiplication & Division Examples:

  • Evaluate $\frac{3}{4} \times \left(-\frac{2}{3}\right)$:

$\frac{3 \times (-2)}{4 \times 3} = \frac{-6}{12} = -\frac{1}{2}$

• Evaluate $-1\frac{1}{2} \div \left(-2\frac{3}{4}\right)$:

$-1\frac{1}{2} = -\frac{3}{2}$

$-2\frac{3}{4} = -\frac{11}{4}$

$\left(-\frac{3}{2}\right) \div \left(-\frac{11}{4}\right) = \left(-\frac{3}{2}\right) \times \left(-\frac{4}{11}\right) = \frac{12}{22} = \frac{6}{11}$

Fraction Applications and Geometry

• Road Trip Fraction Problem:

  • Four friends share driving duties on a trip. Kiley drives $\frac{1}{3}$ of the trip, Rami drives $\frac{3}{8}$, and Ryan drives $\frac{1}{12}$. Melanie drives the remainder.

  • Part A: Determine the total fraction driven by Kiley, Rami, and Ryan:

$\frac{1}{3} + \frac{3}{8} + \frac{1}{12}$

    Common denominator for 3, 8, 12 is 24:

$\frac{8}{24} + \frac{9}{24} + \frac{2}{24} = \frac{19}{24}$

• Part B: Calculate Melanie\'s fraction:

$1 - \frac{19}{24} = \frac{24}{24} - \frac{19}{24} = \frac{5}{24}$

• Part C: Total trip driving time is 11 hours and 12 minutes ($672\,\text{minutes}$). Calculate each driver\'s time:

  • Kiley: $\frac{1}{3} \times 672\,\text{mins} = 224\,\text{mins} = 3\,\text{hours } 44\,\text{minutes}$

  • Rami: $\frac{3}{8} \times 672\,\text{mins} = 252\,\text{mins} = 4\,\text{hours } 12\,\text{minutes}$

  • Ryan: $\frac{1}{12} \times 672\,\text{mins} = 56\,\text{mins} = 0\,\text{hours } 56\,\text{minutes}$

  • Melanie: $\frac{5}{24} \times 672\,\text{mins} = 140\,\text{mins} = 2\,\text{hours } 20\,\text{minutes}$

• Geometric Area Example:

  • Determine the area of a right-angled triangle with height $\frac{2}{7}$ and base $\frac{14}{5}$:

$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$

$\text{Area} = \frac{1}{2} \times \frac{14}{5} \times \frac{2}{7} = \frac{28}{70} = \frac{2}{5}\text{ units}^2$

• Market Share Application:

  • UFO Industries produces $\frac{1}{3}$ of world widgets, IOU Corporation produces $\frac{2}{5}$, and DDT Enterprises produces $\frac{1}{6}$. Determine the fraction produced by other companies:

$\text{Combined Share} = \frac{1}{3} + \frac{2}{5} + \frac{1}{6} = \frac{10}{30} + \frac{12}{30} + \frac{5}{30} = \frac{27}{30} = \frac{9}{10}$

$\text{Other Companies} = 1 - \frac{9}{10} = \frac{1}{10}$

• Capacity Application:

  • A $3\frac{1}{2}$ gallon gas can contains $\frac{9}{10}$ gallon. How much more gas can be added?

$3\frac{1}{2} - \frac{9}{10} = \frac{7}{2} - \frac{9}{10} = \frac{35}{10} - \frac{9}{10} = \frac{26}{10} = \frac{13}{5} = 2\frac{3}{5}\,\text{gallons}$

• Unit Conversion Application:

  • A kilometer is approximately $\frac{5}{8}$ mile. Find how many kilometers are in $2\frac{3}{4}$ miles:

$2\frac{3}{4} \div \frac{5}{8} = \frac{11}{4} \div \frac{5}{8} = \frac{11}{4} \times \frac{8}{5} = \frac{88}{20} = \frac{22}{5} = 4\frac{2}{5}\,\text{km}$

• Sports Statistics Application:

  • A sports team played 42 games and won $\frac{5}{14}$ of them. Calculate the number of losses:

$\text{Fraction Lost} = 1 - \frac{5}{14} = \frac{9}{14}$

$\text{Games Lost} = 42 \times \frac{9}{14} = 3 \times 9 = 27\,\text{games}$

Order of Operations (BEDMAS)

• The BEDMAS Hierarchy:

  1. Brackets: Evaluate all operations inside parentheses (), brackets [], or braces {} from innermost to outermost.

  2. Exponents: Evaluate powers and exponential expressions ($x^n$).

  3. Division and Multiplication: Evaluate in order of appearance from left to right.

  4. Addition and Subtraction: Evaluate in order of appearance from left to right.

• Left-to-Right Equivalence Rule:

  • Division does not strictly take precedence over multiplication; they hold equal priority and are performed left-to-right.

  • Subtraction does not strictly take precedence behind addition; they hold equal priority and are performed left-to-right.

  • Example: $9 \div 3 \times 2 \div 6 = 3 \times 2 \div 6 = 6 \div 6 = 1$.

  • Example: $8 - 5 + 7 - 1 = 3 + 7 - 1 = 10 - 1 = 9$.

• Problem-Solving Activity (Restoring Brackets):

  • Insert brackets into $1 + 3 \times 5 + 7$ to yield $27$:

$(1 + 3) \times 5 + 7 = 4 \times 5 + 7 = 20 + 7 = 27$

• Insert brackets into $4 + 4 + 4 \times 4$ to yield $48$:

$(4 + 4 + 4) \times 4 = 12 \times 4 = 48$

• Insert brackets into $6 + 8 - 2 \times 3$ to yield $24$:

$6 + (8 - 2) \times 3 = 6 + 6 \times 3 = 6 + 18 = 24$

• Comprehensive BEDMAS Worked Examples:

  • Simplify $3 - 2(8 - (-5))$:

$3 - 2(8 + 5) = 3 - 2(13) = 3 - 26 = -23$

• Simplify $3[-6 - (-2)]$:

$3[-6 + 2] = 3[-4] = -12$

• Simplify $\frac{5}{2} \div \left(-\frac{4}{5}\right) - \left(-\frac{3}{4}\right)\left(\frac{-8}{9}\right)$:

$\text{Division step: } \frac{5}{2} \times \left(-\frac{5}{4}\right) = -\frac{25}{8}$

$\text{Multiplication step: } \left(-\frac{3}{4}\right) \times \left(-\frac{8}{9}\right) = \frac{24}{36} = \frac{2}{3}$

$\text{Subtraction step: } -\frac{25}{8} - \frac{2}{3} = -\frac{75}{24} - \frac{16}{24} = -\frac{91}{24}$

• Simplify $\frac{((-2 + (-1))^2 \div 3}{((-4)^2 - 17)}$:

$\text{Numerator numerator bracket: } -2 + (-1) = -3$

$\text{Numerator exponent: } (-3)^2 = 9$

$\text{Numerator division: } 9 \div 3 = 3$

$\text{Denominator exponent: } (-4)^2 = 16$

$\text{Denominator subtraction: } 16 - 17 = -1$

$\text{Final quotient: } \frac{3}{-1} = -3$

Ratios, Rates, Proportions, and Percentages

• Ratios:

  • A ratio compares quantities belonging to the same unit or type.

• In a group of 2 circles and 4 squares, the ratio of circles to squares is $2 : 4$. Total parts = 6.

• Ratios are simplified by dividing terms by their greatest common factor. A ratio $2 : 4$ simplifies to $1 : 2$

• Recipe Scaling Problem: A pancake recipe calls for 3 cups of flour, 4 cups of milk, and 2 eggs ($3 : 4 : 2$). If only 1 egg is available, adjust proportions proportionally by dividing all quantities by 2:

$\text{Flour} = \frac{3}{2} = 1.5\,\text{cups}$

$\text{Milk} = \frac{4}{2} = 2\,\text{cups}$

$\text{Eggs} = 1$

• Rates and Unit Rates:

  • A rate compares two quantities with different units/types (e.g., speed in $\text{km/h}$, fuel efficiency in $\text{km/L}$, unit cost in $\text{\$/item}$).

  • A unit rate is a rate where the second quantity is reduced to 1 unit.

  • Calculate unit rate for $\$10$ for 4 chocolate bars:

\frac{\10}{4\,\text{bars}} = \$2.50\text{ per bar}$

• Calculate unit rate for $5\,\text{km}$ in 25 minutes:

$\frac{5\,\text{km}}{25\,\text{min}} = 0.2\,\text{km/min} = 12\,\text{km/h}$

• Consumer Best Buy Comparison:

  • Option A: $\$8.49$ for $0.45\,\text{kg}$

\text{Unit Price A} = \frac{\8.49}{0.45\,\text{kg}} \approx \$18.87/\text{kg}$

- Option B: $\$34.81$ for $2.14\,\text{kg}$

\text{Unit Price B} = \frac{\34.81}{2.14\,\text{kg}} \approx \$16.27/\text{kg}$

- Option B offers the lower price per unit mass, making it the better value.

• Percentages and Financial Applications:

  • Percent means "per hundred" ($\% = \frac{1}{100}$).

  • Conversion Rules:

  • Percentage to Decimal: Divide by 100 (move decimal point two places left).

  • Fraction/Decimal to Percentage: Multiply by 100.

  • Finding percentage of a number: Multiply the total amount by the decimal representation of the percentage.

  • Hockey Stick Price Calculation:

  • Retail price = $\$430$. Discount = $25\%$.

  • A discount of $25\%$ means paying $75\%$ of the base price ($100\% - 25\% = 75\%$

  • Savings amount:

$0.25 \times \$430 = \$107.50$

- Sale price:

$0.75 \times \$430 = \$322.50$

- Total cost including $13\%$ Harmonized Sales Tax (HST):

\322.50 \times (1 + 0.13) = \322.50 \times 1.13 = \364.425 \approx \$364.43$

Sets, Subsets, and Number Systems

• Historical Numeration Systems:

  • Hindu-Arabic System: Positional system utilizing digits 0-9; includes zero.

  • Roman Numerals: Additive/subtractive notation (I, V, X, L, C, D, M) lacks a representation for zero.

  • Binary System: Base-2 positional notation utilizing digits 0 and 1.

• Mathematical Definitions of Sets and Subsets:

  • A Set is a collection of distinct objects called elements (enclosed in braces $\{\}$). Order of elements is irrelevant.

  • A Subset ($A \subseteq B$) is a set whose elements are all contained within another set $B$.

  • Example: For set $\{1, 2\}$, the complete list of subsets is $\emptyset, \{1\}, \{2\}, \{1, 2\}$ (total $2^2 = 4$ subsets).

  • Example: For set $\{\text{red}, \text{purple}, \text{blue}\}$, total subsets = $2^3 = 8$.

• Number System Hierarchy:

  • Natural Numbers ($\mathbb{N}$): Counting numbers starting at 1: $\{1, 2, 3, 4, \dots\}$.

  • Whole Numbers ($\mathbb{W}$): Natural numbers including zero: $\{0, 1, 2, 3, 4, \dots\}$.

  • Integers ($\mathbb{Z}$): Whole numbers and their negative opposites: $\{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\}$. Developed during the Han Dynasty in China (202 BCE - 220 CE).

  • Rational Numbers ($\mathbb{Q}$): Numbers expressible as a ratio of two integers $\frac{a}{b}$ where $b \neq 0$. Represented as terminating decimals ($0.75$) or repeating decimals ($0.\overline{3}$).

  • Irrational Numbers ($\mathbb{I}$): Non-terminating, non-repeating decimal numbers that cannot be represented as a fraction of two integers. Examples: $\sqrt{2} \approx 1.41421356\dots$, $\pi \approx 3.14159\dots$

  • Real Numbers ($\mathbb{R}$): The union of all Rational and Irrational number sets ($\mathbb{R} = \mathbb{Q} \cup \mathbb{I}$).

Infinity, Density, and Limits

• Part A: Infinity:

  • A set is infinite if its elements cannot be completely counted or listed, having no boundary or end.

  • Natural numbers between 1 and 2: None ($\emptyset$

  • Real numbers between 1 and 2: Infinitely many (e.g., $1.5, 1.25, 1.1, 1.05, 1.001, \dots$

• Part B: Density Property:

  • A set is dense if between any two distinct elements in the set, there exists another element belonging to the same set.

• Density Classification:

  • Dense Sets: Rational Numbers ($\mathbb{Q}$), Irrational Numbers ($\mathbb{I}$), Real Numbers ($\mathbb{R}$).

  • Non-Dense Sets: Natural Numbers ($\mathbb{N}$), Whole Numbers ($\mathbb{W}$), Integers ($\mathbb{Z}$).

  • Part C: Limits:

• The limit of a mathematical pattern or function represents the value that a sequence, table, or graph approaches asymptotically.

• Sequence Limit Example: $\{8.1, 8.01, 8.001, 8.0001, \dots\} \rightarrow 8$

• Decimal Sequence Limit Example: $0.2, 0.22, 0.222, 0.2222, \dots \rightarrow 0.\overline{2} = \frac{2}{9}$

• Graph Limit Example: As $x \to \infty$, the curve shown in the figure approaches a horizontal asymptote at $y = 2$. Thus, the limiting value is $2$.