U1 Directed Numbers: Essential Formulas

Directed numbers are numbers with a sign (positive or negative) that indicates direction or value relative to zero. This topic covers the fundamental rules for arithmetic operations with positive and negative numbers.

1 Basic Operations and Rules

Addition and Subtraction

To add two numbers with the same sign, add their absolute values and keep the common sign. To add two numbers with different signs, subtract the smaller absolute value from the larger one and take the sign of the number with the larger absolute value. Subtraction is equivalent to adding the opposite: $a - b = a + (-b)$.

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

Multiplication and Division

The product or quotient of two numbers with the same sign is positive. The product or quotient of two numbers with different signs is negative.

$$ (+a) \times (+b) = +ab $$
$$ (-a) \times (-b) = +ab $$
$$ (+a) \times (-b) = -ab $$
$$ \frac{+a}{+b} = +\frac{a}{b}, \quad \frac{-a}{-b} = +\frac{a}{b}, \quad \frac{+a}{-b} = -\frac{a}{b} $$

2 Order of Operations and Applications

BODMAS/PEMDAS with Directed Numbers

The standard order of operations (Brackets, Orders, Division/Multiplication, Addition/Subtraction) applies strictly when evaluating expressions involving directed numbers. Pay close attention to signs.

$$ -3 + 4 \times (-2) = -3 + (-8) = -11 $$
$$ (-5)^2 - \frac{10}{-2} = 25 - (-5) = 30 $$

Real-world Context: Temperature Change

Directed numbers are often used to model changes. For example, if the temperature is $5^{\circ}C$ and it drops by $8^{\circ}C$, the new temperature is $5 + (-8) = -3^{\circ}C$.

5 -3 Temperature Change Drop of 8°C

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