2.3 Directed Numbers (Integers), Order of Operations, and Inverse Operations

Directed Numbers in Real-World Contexts

Directed numbers (including positive and negative integers as well as signed decimals) encode both magnitude (distance from zero) and direction (above or below a reference baseline of $0$). In school and community data, zero represents a chosen reference point rather than the absence of quantity:

• Temperature Scales: Zero degrees Celsius ($0^\circ\text{C}$) is the freezing point of water. A frosty Canberra winter overnight minimum of $-4.5^\circ\text{C}$ lies $4.5^\circ\text{C}$ below freezing, whereas a daytime maximum of $13.2^\circ\text{C}$ lies $13.2^\circ\text{C}$ above freezing.
• School Finance and Account Balances: A positive balance (+$1,250) represents a budget surplus or credit, whereas a negative balance (-$420) indicates a deficit, over-expenditure, or amount owing.
• Cohort Benchmark Comparisons: Student assessment scores are often reported relative to a state or national benchmark (e.g., $+14\text{ points}$ above the mean vs. $-9\text{ points}$ below the mean).
• Elevation and Depth: Mean sea level is defined as $0\text{ m}$. Mount Kosciuszko stands at $+2\,228\text{ m}$, while Lake Eyre (Kati Thanda) lies at $-15\text{ m}$ ($15\text{ m}$ below sea level).

On a horizontal number line, numbers increase from left to right. Consequently, $-2.4^\circ\text{C}$ is warmer (greater) than $-6.1^\circ\text{C}$ because $-2.4$ sits to the right of $-6.1$ and is closer to zero, even though $6.1$ has a larger absolute magnitude.

Arithmetic with Signed Numbers

Understanding how positive and negative signs interact prevents sign errors in both mental and calculator-assisted problems:

• Adding a negative number moves left (downwards) on the number line and is equivalent to subtracting a positive number: $a + (-b) = a - b$. A canteen account with a surplus of $350 that incurs a loss of $120 has a new balance of $350 + (-120) = 350 - 120 = 230$ ($230).
• Subtracting a negative number reverses direction and is equivalent to adding a positive number: $a - (-b) = a + b$.
• Multiplying or dividing signed numbers: When the two signs are the same (both positive or both negative), the result is positive ($-6 \times -4 = +24$ and $-36 \div -9 = +4$). When the two signs are different, the result is negative ($-6 \times 4 = -24$ and $36 \div -9 = -4$).

Why Subtracting a Negative Equals Adding a Positive

To find the change or range between two directed values, always subtract the starting (or lower) value from the final (or higher) value:

$$\text{Change (or Range)} = \text{Final (or Higher) Value} - \text{Initial (or Lower) Value}$$

Suppose Canberra's temperature rises from an overnight minimum of $-4.5^\circ\text{C}$ to a daytime maximum of $13.2^\circ\text{C}$. Warming from $-4.5^\circ\text{C}$ up to $0^\circ\text{C}$ requires an increase of $+4.5^\circ\text{C}$, and warming from $0^\circ\text{C}$ to $13.2^\circ\text{C}$ requires an additional $+13.2^\circ\text{C}$:

$$13.2 - (-4.5) = 13.2 + 4.5 = 17.7^\circ\text{C}$$

A frequent error is subtracting the positive magnitudes ($13.2 - 4.5 = 8.7^\circ\text{C}$), which ignores the $4.5^\circ\text{C}$ climb from below zero up to zero. Likewise, the vertical distance between a coastal lookout at $+85\text{ m}$ and a reef dive site at $-18\text{ m}$ is $85 - (-18) = 85 + 18 = 103\text{ m}$.

Order of Operations (BODMAS / BIMDAS)

When an expression combines multiple operations, evaluate them according to the hierarchy known in Australian schools as BODMAS (or BIMDAS):

Implicit Brackets in Fractions and Basic Calculator Entry

A horizontal fraction bar acts as an implicit pair of brackets enclosing the entire numerator and the entire denominator:

$$\frac{36 + 12}{10 - 4} = (36 + 12) \div (10 - 4) = 48 \div 6 = 8$$

Typing 36 + 12 ÷ 10 - 4 = continuously into a calculator without brackets produces the wrong result ($33.2$ on an order-aware calculator, or $0.8$ on a sequential calculator). ACER's Section 1 on-screen calculator follows the order of operations, so it would show $33.2$. Either type the brackets—(36 + 12) ÷ (10 - 4) =—or evaluate the numerator ($36 + 12 = 48$) and the denominator ($10 - 4 = 6$) separately, write both down, and then divide ($48 \div 6 = 8$).

Working Backwards with Inverse Operations

When a word problem applies a chain of operations to an unknown starting value and gives the final result, recover the starting number by working backwards from the final figure using inverse operations in reverse chronological order:

• The inverse of adding $k$ is subtracting $k$ (and vice versa).
• The inverse of multiplying by $k$ is dividing by $k$ (and vice versa).
• The inverse of squaring is taking the square root.