Meaning of Directed Numbers

Directed numbers are numbers that have both magnitude (size) and direction (positive or negative).

Positive numbers are numbers greater than zero (e.g. +2, +5, +10).

Negative numbers are numbers less than zero (e.g. –2, –5, –10).

Positive numbers are usually used to represent gains, increases, or movements upward, while negative numbers represent losses, decreases, or movements downward.

Multiplication of Directed Numbers

When we multiply directed numbers, we use sign rules to determine whether the answer will be positive or negative.

1. Positive × Positive = Positive
   
   Example: +3 × +2 = +6
   
   Explanation: A positive times a positive gives a positive result.


2. Positive × Negative = Negative
   
   Example: +4 × –2 = –8
   
   Explanation: A positive number multiplied by a negative number gives a negative result.


3. Negative × Positive = Negative
   
   Example: –5 × +3 = –15
   
   Explanation: A negative number multiplied by a positive number gives a negative result.


4. Negative × Negative = Positive
   
   Example: –6 × –2 = +12
   
   Explanation: When two negative numbers are multiplied, the result is positive.

How to Remember the Signs

First Number	Second Number	Result Sign	Example
Positive	Positive	Positive	+2 × +3 = +6
Positive	Negative	Negative	+4 × –2 = –8
Negative	Positive	Negative	–3 × +5 = –15
Negative	Negative	Positive	–6 × –2 = +12

If the signs are the same, the answer is positive. If the signs are different, the answer is negative.

Examples on Multiplication

1. (+6) × (+3) = +18


2. (–5) × (+2) = –10


3. (+4) × (–7) = –28


4. (–8) × (–2) = +16


5. (–3) × (–5) × (+2) = +30
   (Because – × – = +, then + × + = +)

Division of Directed Numbers

Just like multiplication, division of directed numbers also follows sign rules.

Rules for Division of Directed Numbers

1. Positive ÷ Positive = Positive
   Example: +12 ÷ +3 = +4


2. Positive ÷ Negative = Negative
   Example: +12 ÷ –3 = –4


3. Negative ÷ Positive = Negative
   Example: –12 ÷ +3 = –4


4. Negative ÷ Negative = Positive
   Example: –12 ÷ –3 = +4

How to Remember the Signs for Division

First Number	Second Number	Result Sign	Example
Positive	Positive	Positive	+8 ÷ +2 = +4
Positive	Negative	Negative	+8 ÷ –2 = –4
Negative	Positive	Negative	–8 ÷ +2 = –4
Negative	Negative	Positive	–8 ÷ –2 = +4

The rule for multiplication and division of directed numbers is the same.
Same signs → Positive answer
Different signs → Negative answer

Examples on Division

1. (+20) ÷ (+5) = +4


2. (+15) ÷ (–3) = –5


3. (–16) ÷ (+4) = –4


4. (–18) ÷ (–6) = +3


5. (–48) ÷ (+12) = –4


6. (+36) ÷ (–9) = –4

Combining Multiplication and Division

Sometimes, a question may contain both multiplication and division.
You solve from left to right, following the same sign rules.

Examples

1. (–6) × (+3) ÷ (–2) = (–18) ÷ (–2) = +9


2. (+12) ÷ (–3) × (–2) = (–4) × (–2) = +8


3. (–8) × (–4) ÷ (+2) = (+32) ÷ (+2) = +16

Word Problems

1. Example 1:
   A submarine goes 300 metres below sea level each hour for 4 hours.
   
   How far below sea level will it be after 4 hours?
   
   Movement downward = negative direction
   
   Distance per hour = –300 m
   
   Time = 4 hours
   
   Total distance = –300 × 4 = –1,200 metres
   
   Answer: 1,200 metres below sea level.


2. Example 2:
   
   The temperature was –5°C in the morning. It fell 3 times as much in the evening.
   
   Temperature fall = (–5) × 3 = –15°C
   
   Answer: –15°C


3. Example 3:
   
   A man loses ₦500 daily for 5 days. What is his total loss?
   
   Daily loss = –₦500
   
   Number of days = 5
   
   Total loss = –500 × 5 = –₦2,500
   
   Answer: ₦2,500 loss


4. Example 4:
   
   A diver descends 8 metres below sea level per minute for 3 minutes, then rises 4 metres per minute for 2 minutes.
   
   Descent = –8 × 3 = –24 m
   
   Rise = +4 × 2 = +8 m
   
   Total depth = –24 + 8 = –16 m
   
   Answer: 16 metres below sea level.

Common Mistakes Students Make

1. Forgetting the sign rules.


2. Assuming multiplication and addition use the same sign rules (they don’t).


3. Ignoring brackets, which can change the meaning of an expression.


4. Forgetting that division follows the same sign rules as multiplication.


5. Mixing up subtraction and negative signs.

Summary of Rules

Operation	Signs	Result Sign	Example
Multiplication	+ × +	+	+3 × +2 = +6
Multiplication	+ × –	–	+3 × –2 = –6
Multiplication	– × +	–	–3 × +2 = –6
Multiplication	– × –	+	–3 × –2 = +6
Division	+ ÷ +	+	+8 ÷ +2 = +4
Division	+ ÷ –	–	+8 ÷ –2 = –4
Division	– ÷ +	–	–8 ÷ +2 = –4
Division	– ÷ –	+	–8 ÷ –2 = +4

Conclusion

Multiplication and division of directed numbers are very important in mathematics. They help us understand how to work with negative and positive quantities, especially in real-life situations such as loss and gain, rise and fall, increase and decrease, or movement up and down.

Always remember:
Same signs → Positive result
Different signs → Negative result