Integers and Directed Numbers
# Integers and Directed Numbers ## Learning Objectives By the end of this lesson, you will be able to: - Understand what integers and directed numbers are, and identify them in real-world contexts - Order and compare positive and negative integers on a number line - Add, subtract, multiply, and divide integers using appropriate rules - Solve problems involving directed numbers in practical situations - Apply the concept of absolute value and understand the magnitude of integers ## Introduction Imagine you're checking the weather forecast for a winter holiday in Moscow. The temperature shows -15°C. Or perhaps you're looking at your bank account after buying a new game, and it shows a balance of -£20 (you're overdrawn!). These situations involve numbers that go below zero – we call these **negative numbers**. While counting numbers (1, 2, 3...) and whole numbers (0, 1, 2, 3...) helped us with many everyday tasks, they couldn't describe everything. What happens when temperatures drop below freezing? What about depths below sea level, or floors in a building below ground? This is where **integers** and **directed numbers** become essential. They allow us to describe quantities that have direction – above or below, profit or loss, forward or backward. The set of integers includes all whole numbers and their negative counterparts: {..., -3, -2, -1, 0, 1, 2, 3, ...}. Understanding how to work with these numbers is crucial not just for mathematics, but for science, geography, finance, and many other subjects you'll encounter in your studies. ## Key Concepts ### What are Integers and Directed Numbers? **Integers** are the set of whole numbers that include: - Positive integers: 1, 2, 3, 4, 5, ... - Zero: 0 - Negative integers: -1, -2, -3, -4, -5, ... **Directed numbers** is another term for integers that emphasizes their direction (positive or negative). We use a **plus (+)** or **minus (−)** sign to show this direction. ### The Number Line A number line helps us visualize integers. Zero sits in the middle, positive numbers extend to the right, and negative numbers extend to the left: ``` ... ←─────┼─────┼─────┼─────┼─────┼─────┼─────┼─────┼─────→ ... -4 -3 -2 -1 0 1 2 3 4 ``` **Key points:** - Numbers increase in value as you move right - Numbers decrease in value as you move left - For example: -1 > -5 (negative one is greater than negative five) - The further left you go, the smaller the value becomes ### Operations with Integers #### Addition Rules 1. **Same signs**: Add the absolute values and keep the sign - (+5) + (+3) = +8 - (−5) + (−3) = −8 2. **Different signs**: Subtract the smaller absolute value from the larger, and take the sign of the number with larger absolute value - (+5) + (−3) = +2 - (−5) + (+3) = −2 #### Subtraction Rules Subtracting a number is the same as adding its opposite: - (+5) − (+3) = (+5) + (−3) = +2 - (+5) − (−3) = (+5) + (+3) = +8 - (−5) − (−3) = (−5) + (+3) = −2 - (−5) − (+3) = (−5) + (−3) = −8 **Remember**: Two negatives make a positive when subtracting! #### Multiplication and Division Rules The rules are simpler for multiplication and division: - **Same signs** = **Positive result** - (+4) × (+3) = +12 - (−4) × (−3) = +12 - (+12) ÷ (+3) = +4 - (−12) ÷ (−3) = +4 - **Different signs** = **Negative result** - (+4) × (−3) = −12 - (−4) × (+3) = −12 - (+12) ÷ (−3) = −4 - (−12) ÷ (+3) = −4 ### Absolute Value The **absolute value** of a number is its distance from zero, regardless of direction. We write it as |n|. Examples: - |+5| = 5 - |−5| = 5 - |0| = 0 ## Worked Examples ### Example 1: Temperature Change **Problem**: The temperature in Edinburgh at 6 AM was −3°C. By noon, it had risen by 8°C. What was the temperature at noon? **Solution**: - Starting temperature: −3°C - Change: +8°C (risen means adding) - Calculation: (−3) + (+8) - Think: different signs, so subtract: 8 − 3 = 5 - Take the sign of the larger absolute value (8 is positive) - **Answer: +5°C or 5°C** ### Example 2: Multiple Operations **Problem**: Calculate: (−6) + (+4) − (−3) × (+2) **Solution**: Step 1: Follow order of operations (BIDMAS/PEMDAS) – multiplication first - (−3) × (+2) = −6 Step 2: Rewrite the expression - (−6) + (+4) − (−6) Step 3: Work left to right with addition and subtraction - (−6) + (+4) = −2 - (−2) − (−6) = (−2) + (+6) = +4 **Answer: +4** ### Example 3: Real-World Application **Problem**: A submarine is at a depth of −250 metres (250 metres below sea level). It ascends 180 metres, then descends 95 metres. What is its final depth? **Solution**: Step 1: Starting position: −250 m Step 2: Ascends 180 m (add because going up) - (−250) + (+180) = −70 m Step 3: Descends 95 m (subtract because going down) - (−70) + (−95) = −165 m **Answer: −165 metres (165 metres below sea level)** ## Practice Questions 1. Calculate the following: - a) (−8) + (+15) - b) (+6) − (−9) - c) (−4) × (−7) 2. The temperature at midnight was −7°C. By 8 AM it had risen by 5°C, then by noon it rose another 6°C. What was the temperature at noon? 3. Arrange these integers in ascending order (smallest to largest): −12, +5, −3, 0, −8, +2, −1 4. A diver starts at sea level (0 m), dives down 45 metres, comes up 20 metres, then dives down another 15 metres. What is the diver's final position? 5. Calculate: (−5) + (+3) × (−2) − (−4) --- ### Answers to Practice Questions 1. a) +7 | b) +15 | c) +28 2. −7 + 5 + 6 = +4°C 3. −12, −8, −3, −1, 0, +2, +5 4. 0 + (−45) + (+20) + (−15) = −40 metres (40 metres below sea level) 5. (−5) + (−6) − (−4) = (−5) + (−6) + (+4) = −7 ## Summary - **Integers** include all positive whole numbers, negative whole numbers, and zero - On a number line, numbers increase in value moving right and decrease moving left - **Adding same signs**: add absolute values, keep the sign - **Adding different signs**: subtract absolute values, take the sign of the larger - **Subtracting**: add the opposite (change subtraction to addition and flip the sign) - **Multiplying/dividing**: same signs give positive, different signs give negative - **Absolute value** |n| represents distance from zero - Always follow the order of operations (BIDMAS/PEMDAS) in complex calculations ## Exam Tips - **Draw a number line** when comparing or ordering integers – visualizing helps prevent errors, especially with negative numbers. Remember: −1 > −10 because −1 is to the right on the number line. - **Watch for double negatives** in subtraction problems. The phrase "subtract a negative" means you add the positive equivalent. Write it out: 5 − (−3) = 5 + 3 = 8. This is one of the most common exam mistakes! - **Check your signs in multiplication and division**. Create a quick mental reminder: "same signs = positive, different signs = negative." If you're rushing through multi-step problems, circle or underline signs before calculating to avoid careless errors.
Why This Matters
This lesson explores integers and directed numbers, including positive and negative whole numbers. Students learn to represent, compare, and order integers on number lines, and perform operations including addition, subtraction, multiplication, and division with directed numbers.
Key Words to Know
Introduction
Integers and directed numbers form a fundamental part of mathematics that extends our understanding of numbers beyond simple counting. While natural numbers (1, 2, 3...) are sufficient for basic arithmetic, real-world situations often require us to express values that go in opposite directions, such as temperatures above and below freezing, profit and loss, or elevations above and below sea level.
Integers include all whole numbers and their negative counterparts: ...-3, -2, -1, 0, 1, 2, 3... Directed numbers include integers as well as positive and negative fractions and decimals, representing any value that has both magnitude (size) and direction (positive or negative). Understanding these numbers is essential for solving problems in science, geography, finance, and many other fields.
This topic builds on your knowledge of whole numbers and introduces the concept of negative values. You will learn to compare, order, and perform operations with these numbers, using number lines as a visual aid. Mastering integers and directed numbers opens the door to algebra, coordinate geometry, and more advanced mathematical concepts that you'll encounter throughout your studies.
Core Concepts
Understanding integers and directed numbers requires familiarity with several core concepts:
- Positive and negative numbers: Positive numbers are greater than zero and typically written without a sign (e.g., 5 or +5). Negative numbers are less than zero and always include a minus sign (e.g., -5).
- Zero: Zero is neither positive nor negative; it serves as the central reference point on the number line.
- The number line: A horizontal line where numbers are positioned according to their value, with negative numbers to the left of zero and positive numbers to the right.
- Absolute value: The distance of a number from zero, regardless of direction. The absolute value of both 7 and -7 is 7, written as |7| = 7 and |-7| = 7.
- Opposites: Two numbers that are the same distance from zero but in opposite directions (e.g., 4 and -4).
- Ordering: As you move right on the number line, numbers increase in value; -10 < -5 < 0 < 3 < 8.
These foundational concepts help us interpret and work with directed numbers in various contexts.
Key Skills
Developing proficiency with integers and directed numbers involves mastering these essential skills:
- Comparing and ordering: Determine which of two directed numbers is greater or arrange multiple numbers in ascending or descending order using the number line as reference.
- Addition and subtraction: Add and subtract integers using rules such as "adding a negative is the same as subtracting a positive" and "subtracting a negative is the same as adding a positive."
- Multiplication and division: Apply sign rules: multiplying or dividing two numbers with the same sign gives a positive result, while different signs give a negative result.
- Using number lines: Visualize operations by moving right for addition or positive direction, and left for subtraction or negative direction.
- Solving real-world problems: Apply directed numbers to contexts like temperature changes, bank transactions, elevation changes, and time zones.
- Working with brackets: Simplify expressions involving multiple operations and negative signs, such as -(-5) = 5 or -(+3) = -3.
Regular practice of these skills builds confidence and fluency in working with directed numbers across various mathematical situations.
Worked Examples
Example 1: Calculate: -8 + 5
Start at -8 on the number line. Move 5 spaces to the right (addi...
Common Mistakes
Students often encounter these pitfalls when working with integers and directed numbers:
- Confus...
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Exam Tips
- 1.Remember: two negatives make a positive when multiplying or dividing, but subtracting a negative means adding a positive
- 2.Always use a number line to visualize problems when unsure about the direction of operations
- 3.Check your answer makes sense in context - temperatures below zero are negative, debts are negative values