📌 Integers & Rational Numbers

1. The Number Line Axis

• The Origin ($0$): Zero is the central reference point on the axis.
• Positive Numbers ($> 0$): Located to the right of zero ($1, 2, 3, \dots$). As you move right, numbers get larger.
• Negative Numbers ($< 0$): Located to the left of zero ($-1, -2, -3, \dots$). As you move left, numbers get smaller.

• Additive Inverse Rule: For any number $p$, its opposite is $-p$, and their sum is always zero:
  
  $$p + (-p) = 0$$
• Real-World Example: A gain of $50 combined with a loss of $50 equals a net change of $0.
  
• Number Line Representation: $p + q$ represents starting at position $p$ and moving a distance $|q|$ units:
  - To the right if $q > 0$ (positive direction).
  - To the left if $q < 0$ (negative direction).

• Notation: Absolute value is written with vertical bars around the number: $|x|$.
• Opposites Property: Opposite numbers have the exact same distance from zero, so their absolute values are equal:
  
  $$|-a| = |a|$$
  
  Examples: $|-5| = 5$ and $|+5| = 5$.
  
• Evaluating Expressions with Absolute Value:
  Treat absolute value bars $| \dots |$ like grouping symbols (parentheses). First evaluate the expression inside the bars, then take the non-negative absolute value:
  Example: Evaluate $|-18 + 7| - |-5|$:
  1. Evaluate inside first: $-18 + 7 = -11$.
  2. Take absolute value: $|-11| = 11$ and $|-5| = 5$.
  3. Subtract: $11 - 5 = 6$.

• Subtraction Rule:
  
  $$p - q = p + (-q)$$
• Distance on a Number Line:
  The distance between any two rational numbers $p$ and $q$ on a number line is the absolute value of their difference:
  
  $$\text{Distance} = |p - q|$$
  
  Example: The distance between $-5$ and $3$ is $|-5 - 3| = |-8| = 8$ units.

• Commutative Property of Addition: $a + b = b + a$. Changing the order of addends does not change the sum.
• Associative Property of Addition: $(a + b) + c = a + (b + c)$. Changing the grouping of addends does not change the sum.

1. Rules of Signs

• Rules of Signs for Multiplication:
  ◦ $\text{positive} \times \text{positive} = \text{positive}$ ($(+) \times (+) = +$)
  ◦ $\text{positive} \times \text{negative} = \text{negative}$ ($(+) \times (-) = -$)
  ◦ $\text{negative} \times \text{negative} = \text{positive}$ ($(-) \times (-) = +$)
  
• Derivation via Distributive Property:
  Why does $(-1)(-1) = 1$? Since $1 + (-1) = 0$, multiplying by $(-1)$ gives:
  $(-1)(1 + (-1)) = (-1)(1) + (-1)(-1)$
  $0 = -1 + (-1)(-1)$
  $\implies (-1)(-1) = 1$
  
• Rules of Signs for Division:◦ $\dfrac{\text{positive}}{\text{positive}} = \text{positive}$ ($(+) \div (+) = +$)◦ $\dfrac{\text{negative}}{\text{negative}} = \text{positive}$ ($(-) \div (-) = +$)◦ $\dfrac{\text{positive}}{\text{negative}} = \text{negative}$ ($(+) \div (-) = -$)◦ $\dfrac{\text{negative}}{\text{positive}} = \text{negative}$ ($(-) \div (+) = -$)

• Division by Zero: Division by zero is undefined ($\frac{p}{0}$ is undefined).
  
• Rational Numbers: Every quotient of integers $\frac{p}{q}$ (where $q \neq 0$) is a rational number.
  
• Equivalent Forms of Negative Fractions: The negative sign can be placed on the numerator, denominator, or in front of the fraction:
  
  $$-\frac{p}{q} = \frac{-p}{q} = \frac{p}{-q}$$
  
  Example: $\frac{-12}{4} = \frac{12}{-4} = -\frac{12}{4} = -3$.

3. Converting Rational Numbers to Decimals

• Terminating Decimal: The long division reaches a remainder of $0$.
  Example: $\frac{3}{8} = 3 \div 8 = 0.375$.
  
• Repeating Decimal & How to Find It:
  When dividing $p \div q$, track the remainders after each step. Once a remainder repeats, the subsequent digits in the quotient repeat in the exact same order forever!
  
  Step-by-Step Example: Finding $\frac{5}{11}$ as a Repeating Decimal:
  1. Divide $5.0$ by $11$: $50 \div 11 = \mathbf{4}$ with a remainder of $6$ ($50 - 44 = 6$).
  2. Bring down $0 \rightarrow 60$: $60 \div 11 = \mathbf{5}$ with a remainder of $\mathbf{5}$ ($60 - 55 = 5$).
  3. Identify Repeating Remainder: The remainder $5$ is identical to our starting number $5$! Continuing long division will yield remainders of $6, 5, 6, 5\dots$ forever.
  4. Write with Overline Notation: The quotient digits $45$ repeat indefinitely:
  
  $$\frac{5}{11} = 0.4545\dots = 0.\overline{45}$$
  
  
  💡 Pro-Tip / Shortcut (Prime Factors of Denominator):
  For any simplified fraction $\frac{p}{q}$:
• If the prime factors of denominator $q$ contain only $2$'s and/or $5$'s (e.g. $8 = 2^3$, $20 = 2^2 \times 5$), it will always be a terminating decimal.
• If denominator $q$ has any other prime factors (like $3, 7, 11$), it will always be a repeating decimal (e.g. $\frac{5}{11} = 0.\overline{45}$).

• Bar / Overline notation: Type \overline over repeating digits (e.g. 0.\overline{45} or 0.1\overline{6}).
• Expanded Decimal: Type repeating digits (e.g. 0.4545 or 0.4545...).
• Fraction form: Enter the exact fraction (e.g. 5/11 or \frac{5}{11}).

4. Solving Multi-Step Real-World Problems