Addition and subtraction of integers

Understanding Integers on the Number Line

Alright class, let's dive into the fascinating world of Integers! You've seen these numbers before, but today, we're going to become masters of moving them around. Think of it like a game. Ready to play?

The Great Number Adventure: Up, Down, and All Around!

Imagine you're playing a video game. You start at ground level (that's our hero, Zero). You find a treasure and get +10 points. You move up. Then, you fall into a trap and get -15 points. You move down, even below where you started! That's exactly what integers are for: to talk about values that can be positive, negative, or zero.

They are everywhere! From the temperature in Shimla dropping to -4°C in winter, to the money in your bank account (a deposit is +₹500, a withdrawal is -₹200), to the floors in a tall building with basement parking (-1, -2 floors). Understanding how to add and subtract them is like having a map for this up-and-down world.

{{VISUAL: diagram: A clear, horizontal number line from -10 to 10. Zero is at the center. Positive integers (1, 2, 3...) are marked to the right in blue. Negative integers (-1, -2, -3...) are marked to the left in red. Arrows indicate 'Positive Direction →' and 'Negative Direction ←'.}}

What Exactly Are Integers?

Let's get our key definition straight. Integers are the complete set of whole numbers and their opposites (the negative numbers).

Integers include:

• Positive Integers: {1, 2, 3, 4, ...} These are the natural numbers we use for counting.
• Negative Integers: {-1, -2, -3, -4, ...} These are the opposites of the positive integers.
• Zero: {0} It's special! It is neither positive nor negative.

So, the whole family of integers, which we often represent with the letter Z, looks like this: {..., -4, -3, -2, -1, 0, 1, 2, 3, 4, ...}.

{{KEY: type=definition | title=The Integer Family (Z) | text=Integers are a collection of all whole numbers and their negative counterparts. This includes all positive counting numbers, all negative counting numbers, and the number zero. Fractions and decimals are NOT integers.}}

Navigating the Number Line

The number line is our most powerful tool for understanding integers. Think of it as a long road with a starting point, Zero. Walking to the right is a positive journey, and walking to the left is a negative one. Every addition and subtraction is simply a move on this road!

The Rules of Movement

Here's the logic for how we move. It's super simple and once you get it, you'll never be confused again.

The last one, subtracting a negative, is the trickiest. Think of it like this: "removing a debt of 5 rupees" is the same as "gaining 5 rupees". So, -(-5) becomes +5. You move to the right! This is a very important rule: two negatives make a positive.

{{CALLOUT: type=memory | text=Remember: Addition means follow the sign of the number you are adding. Subtraction means do the opposite of the sign of the number you are subtracting.}}

Let's Practice: Solved Examples

Time to put our knowledge to the test. We'll start easy and work our way up.

Example 1: Simple Positive Addition (Easy)

Given: The expression 3 + 5.

To Find: The sum using the number line.

Solution:

1. Locate the starting point. We begin at the first number, which is 3.

2. Determine the movement. We are adding a positive integer (+5). The rule says: "Add a Positive Integer → Move to the Right".

3. Perform the move. From 3, we jump 5 steps to the right. 3 → 4 → 5 → 6 → 7 → 8

{{VISUAL: diagram: A number line from 0 to 10. An arrow starts at 3 and makes 5 jumps to the right, landing on 8.}}

4. State the final position. We land on 8.

3 + 5 = 8

Final Answer: 8

Example 2: Adding a Negative Integer (Medium)

Given: The expression 6 + (-4).

To Find: The sum using the number line.

Solution:

1. Locate the starting point. We start at 6.

2. Determine the movement. We are adding a negative integer (-4). The rule says: "Add a Negative Integer → Move to the Left".

3. Perform the move. From 6, we jump 4 steps to the left. 6 → 5 → 4 → 3 → 2

4. State the final position. We land on 2.

6 + (-4) = 2

Final Answer: 2

Example 3: Multi-Step Movement (Hard)

Now for a two-step problem! Let's solve (-3) + 7 - 2 on the number line. This looks tricky, but it's just two separate moves, one after the other. Bachcho, let's take this to the whiteboard to see it clearly.

{{SOLVE: {"problem":"Solve on the number line: (-3) + 7 - 2","type":"numerical","subject":"math","intro":"Chalo, isse whiteboard pe step-by-step solve karte hain.","outro":"Aur ye raha humara final answer! Ab class room mein wapas chalte hain.","steps":[{"explanation":"First, we find our starting position on the number line, which is the first integer, -3.","write":"Start at -3","tough":false},{"explanation":"Next, we need to add 7. Adding a positive number means we move 7 steps to the right.","write":"Move 7 steps Right: (-3) → (+4)","tough":false},{"explanation":"From our new position, 4, we need to subtract 2. Subtracting a positive number means we move 2 steps to the left.","write":"From +4, move 2 steps Left: (+4) → (+2)","tough":false},{"explanation":"The final position on the number line is our answer.","write":"Final Position = 2","tough":false}]}}}

As you saw on the board, by breaking the problem down into simple moves, even a multi-step calculation becomes easy to visualize and solve. The number line keeps us from getting lost!

Example 4: The Submarine Challenge (Tricky Word Problem)

Given: A submarine is at a depth of 400 meters below sea level. It rises 150 meters. What is its new position?

To Find: The final position of the submarine relative to sea level.

Solution:

1. Translate words into integers.

   • "Sea level" is our 0.
   • "400 meters below sea level" is -400. This is our starting point.
   • "Rises 150 meters" means adding a positive value, so +150.

2. Formulate the expression. The problem is asking us to calculate:

   -400 + 150
   

3. Visualize the movement. We start at -400. We are adding a positive number (150), so we move to the right (upwards, in this case) on the number line.

4. Calculate the final position. Moving 150 steps to the right from -400 will land us at:

   -400 + 150 = -250
   

Final Answer: The new position of the submarine is 250 meters below sea level (-250 m).

{{VISUAL: diagram: A vertical number line representing sea level. '0' is marked as 'Sea Level'. The line goes down to -500. An arrow starts at -400 and moves up by 150 units, ending at -250.}}

Tips & Tricks for Speed

While the number line is great for understanding, you'll need to be faster for exams. Here are some mental shortcuts.

Common Mistakes to Avoid

Many students get tripped up on the same few things. Let's look at them so you don't make the same mistakes!

Brain-Teaser Questions

Ready to flex those brain muscles? Try these HOTS (Higher Order Thinking Skills) questions.

1. A number is 15 steps to the left of 4 on the number line. Where is it?

   💡 Answer: "15 steps to the left" means subtracting 15. So, 4 - 15 = -11. The number is -11.

2. What must be subtracted from -3 to get -9?

   💡 Answer: Let the number be x. The equation is -3 - x = -9. This means x = -3 - (-9) = -3 + 9 = 6. You must subtract 6.

3. You are at 0 on a number line. You take 5 steps forward (positive), then 12 steps back (negative), then 3 steps back, and finally 6 steps forward. Where do you end up?

   💡 Answer: The expression is 0 + 5 - 12 - 3 + 6. 5 - 12 = -7 -7 - 3 = -10 -10 + 6 = -4. You end up at -4.

Mini Cheatsheet for Revision

Here is a quick summary of everything we learned today. You can screenshot this for last-minute revision before your exam!