CBSE Class 7 Maths: Integers Exercise 1.1 - NCERT Solutions & Concepts

Welcome, young mathematicians! In Class 6, you were introduced to integers – the family of numbers that includes all positive whole numbers, all negative whole numbers, and zero. Now, in Class 7, we're going to dive deeper into how these numbers behave, especially when we add and subtract them. Exercise 1.1 from your NCERT textbook is your first step in mastering operations with integers. This exercise helps you solidify your understanding of positive and negative numbers and how they interact. By the end of this lesson, you'll be able to confidently solve problems involving addition and subtraction of integers, use the number line effectively, and apply these concepts to real-life situations. Get ready to build a strong foundation for more complex topics ahead!

Revisiting Integers: The Number Line and Operations

Integers are simply whole numbers, but they also include their negative counterparts. So, ..., -3, -2, -1, 0, 1, 2, 3, ... are all integers. They are super useful for describing things like temperature (above or below zero), bank balances (deposits and withdrawals), and elevation (above or below sea level). The most helpful tool for understanding integers and their operations is the number line. Imagine a straight line with zero in the middle. Positive integers (1, 2, 3, ...) are to the right of zero, and negative integers (-1, -2, -3, ...) are to the left. The further a number is to the right, the greater its value. For example, 5 is greater than -2, and -1 is greater than -5. When we add a positive integer, we move to the right on the number line. When we add a negative integer (or subtract a positive integer), we move to the left. Understanding this visual movement is key to solving problems in Exercise 1.1. Remember, 'minus a minus makes a plus' is a crucial rule for subtraction!

Step-by-Step: Adding and Subtracting Integers

  1. Identify the Numbers and Operation — First, clearly identify the integers involved and the operation (addition or subtraction) you need to perform. Pay close attention to the signs (+ or -) of each number.
  2. Use the Number Line (Optional, but Helpful) — Mentally, or actually, draw a number line. Start at the position of the first integer. If you are adding a positive number, move to the right. If you are adding a negative number (or subtracting a positive number), move to the left. If you are subtracting a negative number, convert it to addition of a positive number (e.g., a - (-b) becomes a + b) and then move right.
  3. Apply Rules of Signs — For addition: If signs are the same (both + or both -), add the absolute values and keep the common sign. If signs are different (one + and one -), subtract the smaller absolute value from the larger absolute value, and keep the sign of the number with the larger absolute value. For subtraction: Change the subtraction sign to addition and change the sign of the number being subtracted. Then follow the addition rules (e.g., 5 - (-3) becomes 5 + 3).
  4. Calculate the Final Result — Perform the calculation carefully based on the rules applied. Double-check your signs!

Worked Examples from Integers Ex 1.1 Type Problems

  • Example 1: Find the value of (-7) + 12. Step 1: We have two integers with different signs: -7 (negative) and 12 (positive). Step 2: According to the rule for different signs, we subtract the smaller absolute value from the larger absolute value: |12| - |-7| = 12 - 7 = 5. Step 3: The number with the larger absolute value is 12 (which is positive). So, the result will be positive. Final Answer: 5
  • Example 2: Solve: (-15) - (-8). Step 1: This is a subtraction problem involving a negative number. Recall that subtracting a negative number is the same as adding its positive counterpart. Step 2: So, (-15) - (-8) becomes (-15) + 8. Step 3: Now we have two integers with different signs: -15 (negative) and 8 (positive). Step 4: Subtract the smaller absolute value from the larger absolute value: |-15| - |8| = 15 - 8 = 7. Step 5: The number with the larger absolute value is -15 (which is negative). So, the result will be negative. Final Answer: -7
  • Example 3: A diver is 10 metres below sea level. He ascends 4 metres. What is his new position? Step 1: Represent the initial position. 10 metres below sea level can be represented as -10 metres. Step 2: Represent the change in position. Ascending 4 metres means moving upwards, which is a positive change, so +4 metres. Step 3: Formulate the operation: Initial position + Change in position = New position. So, (-10) + 4. Step 4: We have two integers with different signs. Subtract the smaller absolute value from the larger: 10 - 4 = 6. Step 5: The number with the larger absolute value is -10 (which is negative). So, the result will be negative. Final Answer: The diver's new position is -6 metres, or 6 metres below sea level.

Exam Tip: Avoiding Common Mistakes with Integers

One of the most frequent errors students make while solving problems with integers is confusing the rules for addition/subtraction with multiplication/division. Remember, for addition and subtraction:

  1. Signs are the same: If both numbers are positive or both are negative, add their absolute values and keep the common sign. For example, -3 - 5 = -8 (add 3 and 5 to get 8, keep the negative sign). 3 + 5 = 8.
  2. Signs are different: Subtract the smaller absolute value from the larger absolute value. The result takes the sign of the number with the larger absolute value. For example, -7 + 10 = 3 (subtract 7 from 10 to get 3, 10 is larger and positive, so result is positive). 7 - 10 = -3 (subtract 7 from 10 to get 3, 10 is larger and negative, so result is negative).
  3. Subtracting a negative: Always remember that a - (-b) is equivalent to a + b. This is a crucial rule that often trips students up. Practice these sign rules thoroughly!

Practice Questions with Solutions

  • Q: Evaluate: (-25) + (-10). A: Step 1: Identify the operation as addition. Both integers are negative. Step 2: When signs are the same (both negative), add their absolute values: | -25 | + | -10 | = 25 + 10 = 35. Step 3: Keep the common sign, which is negative. Final answer: -35
  • Q: Solve: 18 - (-6). A: Step 1: Identify the operation as subtraction. Notice we are subtracting a negative number. Step 2: Change subtraction of a negative to addition of a positive: 18 - (-6) becomes 18 + 6. Step 3: Perform the addition. Final answer: 24
  • Q: Find the value of 5 + (-13). A: Step 1: Identify the operation as addition. The integers have different signs: 5 (positive) and -13 (negative). Step 2: When signs are different, subtract the smaller absolute value from the larger: | -13 | - | 5 | = 13 - 5 = 8. Step 3: The number with the larger absolute value is -13 (which is negative). So, the result will be negative. Final answer: -8
  • Q: The temperature in Shimla was -4°C on Monday. It dropped by 2°C on Tuesday. What was the temperature on Tuesday? A: Step 1: Identify the initial temperature: -4°C. Step 2: Identify the change in temperature: 'dropped by 2°C' means a decrease, so this is -2°C. Step 3: Formulate the equation: Initial temperature + Change in temperature = Temperature on Tuesday. So, (-4) + (-2). Step 4: Both integers are negative. Add their absolute values: |-4| + |-2| = 4 + 2 = 6. Step 5: Keep the common negative sign. Final answer: The temperature on Tuesday was -6°C.

Frequently Asked Questions

What are integers, and why are they important?

Integers are whole numbers, including positive numbers (1, 2, 3...), negative numbers (-1, -2, -3...), and zero. They are important because they help us describe quantities that can be above or below a reference point, like temperature, altitude, or financial transactions.

How do I add integers with different signs?

To add integers with different signs, you subtract the smaller absolute value from the larger absolute value. The result then takes the sign of the integer that had the larger absolute value. For example, 10 + (-3) = 7, and -10 + 3 = -7.

What happens when I subtract a negative integer?

Subtracting a negative integer is equivalent to adding its positive counterpart. For instance, if you have 5 - (-3), it becomes 5 + 3, which equals 8. This rule is crucial for correct integer operations.

Can I always use a number line for integer operations?

Yes, a number line is a fantastic visual aid for understanding integer operations, especially when you're starting out. While you might learn mental shortcuts later, using the number line can help you visualize movements and confirm your answers, reducing sign errors.