Welcome to the World of Directed Numbers!

Hey there! Get ready to explore a super useful part of mathematics called directed numbers. You've been using numbers your whole life, but now we're going to give them a direction: positive (forward) and negative (backward).

Why is this important? Because directed numbers are everywhere in the real world! They help us talk about:
• Temperatures above and below zero.
• Money you have (credits) and money you owe (debts).
• Going up or down in a lift, or even heights above and below sea level.

By the end of these notes, you'll be a pro at understanding, comparing, and calculating with these numbers. Don't worry if it seems new at first – we'll go step-by-step!

Section 1: What Are Directed Numbers?

Meet the Numbers: Positive, Negative, and Zero

A directed number is simply a number that has a direction, shown by a sign.

• Positive Numbers (+): These are the numbers you're already friends with! They are any number greater than zero. They show a gain, an increase, or a forward direction. We can write a plus sign (+) in front of them, but we usually don't.
For example, 5 is the same as +5.

• Negative Numbers (-): These are numbers less than zero. They are the "opposites" of positive numbers. They show a loss, a decrease, or a backward direction. They always have a minus sign (-) in front of them.
For example, -5 (we say "negative five").

• Zero (0): Zero is special! It's neither positive nor negative. It's our starting point or neutral ground.

Real-World Analogy: Temperature

Imagine a thermometer.
• A hot day of +25°C means it's 25 degrees ABOVE zero.
• A freezing day of -10°C means it's 10 degrees BELOW zero.
• 0°C is the freezing point of water.

Section 2: The Number Line - Our Mathematical Map

Visualising Numbers on a Number Line

The best way to understand directed numbers is to see them on a number line. It's like a map for all our numbers.

• Zero (0) is in the center.
• Positive numbers go to the right of zero and get bigger.
• Negative numbers go to the left of zero and get smaller.

Imagine this line:
... -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5 ...

Comparing Directed Numbers

Which number is bigger? The number line makes this easy!

The Golden Rule: Any number on the right is always greater than any number on its left.

Examples:

• Which is greater, 2 or -1?
Find them on the number line. Since 2 is to the right of -1, 2 is greater than -1. We write this as $$2 > -1$$.

• Which is greater, -5 or -2?
Find them on the number line. -2 is to the right of -5, so -2 is greater than -5. We write this as $$-2 > -5$$. This can feel strange, but think about temperature: -2°C is warmer (a greater temperature) than -5°C!

Common Mistake Alert!

Don't think that because 5 is bigger than 2, -5 must be bigger than -2. It's the opposite! With negative numbers, the one that *looks* smaller is actually the greater number.

Key Takeaway

The number line is your best friend for directed numbers. Further right = Greater value.

Section 3: Adding and Subtracting Directed Numbers

Addition: Moving Along the Number Line

Think of adding as taking steps on the number line.

• Adding a positive number means you walk to the right.
• Adding a negative number means you walk to the left.

Let's walk through some examples:

1. Positive + Positive: $$3 + 2$$
Start at 3. Walk 2 steps to the right. You land on 5.
$$3 + 2 = 5$$

2. Positive + Negative: $$5 + (-3)$$
Start at 5. Walk 3 steps to the left. You land on 2.
$$5 + (-3) = 2$$

3. Negative + Positive: $$-4 + 6$$
Start at -4. Walk 6 steps to the right. You land on 2.
$$-4 + 6 = 2$$

4. Negative + Negative: $$-2 + (-5)$$
Start at -2. Walk 5 steps to the left. You land on -7.
$$-2 + (-5) = -7$$

Subtraction: The Magic Trick

Subtraction can be tricky, so here's a simple trick: Subtracting a number is the same as adding its opposite.

The Rule: "Keep, Change, Change" 1. Keep the first number the same.
2. Change the subtraction sign to an addition sign.
3. Change the sign of the second number to its opposite.

Let's try it:

1. Simple Subtraction: $$9 - 4$$
Keep 9, Change - to +, Change 4 to -4. It becomes $$9 + (-4)$$.
Start at 9, move 4 steps left. The answer is 5.

2. Subtracting a bigger number: $$6 - 10$$
Keep 6, Change - to +, Change 10 to -10. It becomes $$6 + (-10)$$.
Start at 6, move 10 steps left. The answer is -4.

3. Subtracting a Negative: $$8 - (-3)$$
This is the most important one! Keep 8, Change - to +, Change -3 to +3. It becomes $$8 + 3$$.
The answer is 11.

Memory Aid: When you see two minuses next to each other, like in $$ -(-3) $$, they join together to make a big plus! So, $$8 - (-3)$$ just becomes $$8 + 3$$.

Key Takeaway

For addition, use the number line. For subtraction, change the problem into an addition problem first ("Keep, Change, Change").

Section 4: Multiplying and Dividing Directed Numbers

Multiplication and Division: The Sign Rules

Good news! For multiplication and division, you don't need the number line. You just need to learn two simple rules about the signs.

Rule 1: If the signs are the SAME, the answer is POSITIVE.
Positive × Positive = Positive ($$3 \times 4 = 12$$)
Negative × Negative = Positive ($$(-3) \times (-4) = 12$$)
Negative ÷ Negative = Positive ($$(-10) \div (-2) = 5$$)

Rule 2: If the signs are DIFFERENT, the answer is NEGATIVE.
Positive × Negative = Negative ($$3 \times (-4) = -12$$)
Negative × Positive = Negative ($$(-3) \times 4 = -12$$)
Positive ÷ Negative = Negative ($$10 \div (-2) = -5$$)

Quick Review Box

Multiplication & Division Signs:
• Same Signs = Positive (+)
• Different Signs = Negative (-)

Step-by-step process:

1. Ignore the signs and just multiply or divide the numbers.
2. Look at the signs in the original question to decide if your answer should be positive or negative.

Example: Calculate $$(-20) \div 4$$

Step 1: Just divide the numbers: $$20 \div 4 = 5$$.
Step 2: Look at the signs. We have a negative (-20) and a positive (4). The signs are different, so the answer must be negative.
Final Answer: $$-5$$.

Did you know?

This works because multiplication is just repeated addition. For example, $$3 \times (-4)$$ means adding -4 three times: $$(-4) + (-4) + (-4) = -12$$. The sign rules are a shortcut!

Section 5: Mixed Operations & Solving Problems

Order of Operations (BODMAS)

When you have a long problem with different operations, you must follow the correct order. You might know this as BODMAS or PEMDAS. The rules are the same for directed numbers!

1. Brackets
2. Orders (powers - we don't need to worry about this yet)
3. Division and Multiplication (from left to right)
4. Addition and Subtraction (from left to right)

Step-by-step worked example: $$(-2) \times (8 - 3) + (-12) \div 2$$

Don't panic! Just follow the steps.
Step 1: Brackets
First, solve what's inside the brackets: $$(8 - 3) = 5$$.
Our problem is now: $$(-2) \times 5 + (-12) \div 2$$

Step 2: Division and Multiplication (Left to Right)
Next, we do the multiplication: $$(-2) \times 5 = -10$$ (different signs give a negative).
Then we do the division: $$(-12) \div 2 = -6$$ (different signs give a negative).
Our problem is now: $$-10 + (-6)$$

Step 3: Addition and Subtraction
Finally, we do the addition. Start at -10 and move 6 steps to the left.
$$-10 + (-6) = -16$$

The final answer is -16. You've got this! Just take it one step at a time.

Solving Real-World Problems

Let's use our new skills to solve some real-life puzzles.

Problem 1: Temperature Change
The temperature in Moscow was -8°C at midnight. By noon, it had risen by 12°C. What was the temperature at noon?
Setup: We start at -8 and "risen by 12" means we add 12. So, $$-8 + 12$$.
Solution: Start at -8 on the number line and move 12 places to the right. You will land on 4.
The temperature at noon was 4°C.

Problem 2: Bank Account
You have $20 in your bank account. You use your card to buy a game that costs $50. What is your new bank balance?
Setup: We start with 20 and take away 50. So, $$20 - 50$$.
Solution: Using "Keep, Change, Change", this becomes $$20 + (-50)$$. Start at 20 and move 50 steps left. You pass 0 and end up at -30.
Your new bank balance is -$30. This means you owe the bank $30.

Problem 3: Diving Depth
A diver is at a depth of 15 metres below sea level. This can be represented as -15m. If she dives down to a depth 3 times as far, what is her new position?
Setup: We need to find 3 times -15. So, $$3 \times (-15)$$.
Solution: First, multiply the numbers: $$3 \times 15 = 45$$. The signs are different (one positive, one negative), so the answer is negative.
Her new position is -45m, or 45 metres below sea level.

Chapter Recap: Your Directed Numbers Toolkit

Fantastic work! You've learned the essentials of directed numbers. Here are the key points to remember:

• Directed Numbers have a sign (+ or -) that shows their direction from zero.

• The Number Line is your visual guide. Numbers on the right are always greater.

• For Addition, move right for positive and left for negative on the number line.

• For Subtraction, change the problem to addition by adding the opposite ("Keep, Change, Change"). Remember that two minuses together make a plus!

• For Multiplication and Division, remember the sign rules: Same signs give a positive answer, and different signs give a negative answer.

• In complex problems, always follow the Order of Operations (BODMAS).

Keep practising, and you'll find that directed numbers are a powerful tool for solving all sorts of problems!