Integers

A. Defining the Set of Integers ($\mathbb{Z}$):

The set of integers consists of positive natural numbers, zero, and negative natural numbers:

• Positive Integers: $\mathbb{Z}^+ = \{1, 2, 3, 4, \dots\}$ (lie to the right of 0 on the number line).
• Negative Integers: $\mathbb{Z}^- = \{-1, -2, -3, -4, \dots\}$ (lie to the left of 0 on the number line).
• Zero ($0$): An integer that is neither positive nor negative. It acts as the neutral origin.

B. Absolute Value (Modulus) of an Integer:

The Absolute Value of an integer $x$, denoted by $|x|$, represents the pure geometrical distance of $x$ from the origin $0$ on the number line, regardless of direction:

Examples: $|+7| = 7$; $|-15| = -(-15) = 15$; $|0| = 0$. The absolute value of any integer is always non-negative ($|x| \ge 0$).

A. Addition of Integers:

• Like Signs: Add their absolute values and prefix the common sign:
  • $(+14) + (+8) = +22$
  • $(-14) + (-8) = -(14 + 8) = -22$
• Unlike Signs: Subtract the smaller absolute value from the larger absolute value, and prefix the sign of the integer with the greater absolute value:
  • $(-25) + (+9) = -(25 - 9) = -16$ (since $|-25| > |+9|$)
  • $(+40) + (-15) = +(40 - 15) = +25$

B. Subtraction of Integers:

To subtract an integer $b$ from $a$, add the Additive Inverse of $b$ to $a$:

Example: Subtract $-18$ from $-12$: $(-12) - (-18) = -12 + 18 = +6$.

C. Multiplication & Division Rules:

For any three integers $a, b, c \in \mathbb{Z}$, the following properties hold:

1. Closure Property:
   • Integers are closed under addition, subtraction, and multiplication ($a + b \in \mathbb{Z}$, $a - b \in \mathbb{Z}$, $a \times b \in \mathbb{Z}$).
   • Integers are NOT closed under division (e.g., $5 \div 2 = 2.5 \notin \mathbb{Z}$).
2. Commutative Property:
   • Addition: $a + b = b + a$
   • Multiplication: $a \times b = b \times a$
   • Subtraction and division are NOT commutative: $a - b \neq b - a$, $a \div b \neq b \div a$.
3. Associative Property:
   • Addition: $(a + b) + c = a + (b + c)$
   • Multiplication: $(a \times b) \times c = a \times (b \times c)$
4. Distributive Property of Multiplication over Addition & Subtraction: $$a \times (b + c) = (a \times b) + (a \times c)$$ $$a \times (b - c) = (a \times b) - (a \times c)$$
5. Identity & Inverse Elements:
   • Additive Identity: $0$ ($a + 0 = 0 + a = a$).
   • Additive Inverse: $-a$ ($a + (-a) = 0$).
   • Multiplicative Identity: $1$ ($a \times 1 = 1 \times a = a$).
   • Multiplicative Property of Zero: $a \times 0 = 0 \times a = 0$.

To evaluate complex arithmetic expressions involving multiple operations and brackets, follow the strict BODMAS hierarchy:

• B — Brackets: Evaluate brackets from the innermost to outermost:
  1. Line bar / Vinculum: $\overline{a - b}$
  2. Parentheses (Round brackets): $( \dots )$
  3. Braces (Curly brackets): $\{ \dots \}$
  4. Square brackets (Box brackets): $[ \dots ]$
• O — Orders / Of: Powers, exponents, or multiplication implied by "of" ($25\% \text{ of } 80$).
• D / M — Division & Multiplication: Computed from left to right.
• A / S — Addition & Subtraction: Computed from left to right.