Calculating with Positive and Negative Numbers

• When adding and subtracting, think about directional movement on the number line. Adding moves you right, while subtracting moves you left.
• If two signs appear directly next to each other in a calculation, you must simplify them using the "Two Sign" rule.
• Same signs directly next to each other ($+(+)$ or $-(-)$) become a plus ($+$).
• Different signs directly next to each other ($+(-)$ or $-(+)$) become a minus ($-$).
• You can check your result using an inverse operation (an operation that "undoes" another, like using addition to check subtraction).

Worked Example: Addition (Different Signs)

Calculate $-2 + (-3)$

• Step 1: Simplify adjacent signs. The $+$ and $-$ are different, so they become a minus. The calculation becomes $-2 - 3$.

• Step 2: Use the number line. Start at $-2$ and move $3$ spaces to the left.

• Answer: $-5$

Worked Example: Subtraction (Double Negative)

Calculate $7 - (-9)$

• Step 1: Simplify adjacent signs. The two minus signs are the same, so they become a plus. The calculation becomes $7 + 9$.

• Step 2: Use the number line. Start at $7$ and move $9$ spaces to the right.

• Answer: $16$

Multiplying and Dividing Directed Numbers

Think of playing a video: if you rewind (a negative action) a video of someone walking backwards (a negative direction), they appear to be moving forwards. This explains why a negative times a negative makes a positive!

• The rules for multiplication and division rely on the signs interacting to form a product or quotient.
• Same signs: Multiplying or dividing two positive or two negative numbers always gives a positive result.
• Different signs: Multiplying or dividing a positive and a negative number always gives a negative result.
• For continuous strings of multiplication, an even number of negative signs gives a positive answer, while an odd number gives a negative answer.

Worked Example: Multiplication

Calculate $5 \times (-4)$

• Step 1: Multiply the magnitudes. $5 \times 4 = 20$

• Step 2: Apply the sign rule. Positive $\times$ Negative = Negative.

• Answer: $-20$

• (Number Line Method: To show $3 \times -2$, interpret this as "three lots of $-2$" using repeated addition. Start at $0$ and jump $2$ units left three times: $0 \rightarrow -2 \rightarrow -4 \rightarrow -6$.)

Worked Example: Division

Calculate $-42 \div (-7)$

• Step 1: Divide the magnitudes. $42 \div 7 = 6$

• Step 2: Apply the sign rule. Negative $\div$ Negative = Positive.

• Answer: $6$

Combined Operations with Directed Numbers

Just like assembling a flat-pack wardrobe, executing mathematical steps in the wrong order will leave you with a complete mess.

• When a calculation involves mixed operations, you must follow the standard order of operations (BIDMAS).
• Apply the specific sign rules for directed numbers at each individual stage of the calculation.

Worked Example: Combined Operations (BIDMAS)

Work out $-12 \div (-3) + (-2) \times 5$

• Step 1: Division. $-12 \div -3 = 4$ (applying the same signs rule).

• Step 2: Multiplication. $-2 \times 5 = -10$ (applying the different signs rule).

• Step 3: Substitute back. Replace the calculated parts into the expression to get $4 + (-10)$.

• Step 4: Addition (Simplify signs). The $+(-)$ becomes a $-$, so the final calculation is $4 - 10$.

• Answer: $-6$