Integers: CBSE Class 7 Maths

Welcome to the world of Integers! In Class 6, you learned about whole numbers (0, 1, 2, 3...). But what about numbers less than zero? Imagine the temperature in Shimla dropping below 0°C, or owing your friend ₹50. These situations need negative numbers! Integers are a bigger collection of numbers that include all the positive whole numbers, their negative opposites, and zero. In this chapter, you will master the rules for adding, subtracting, multiplying, and dividing these positive and negative numbers. Understanding integers class 7 NCERT concepts is crucial because they are the foundation for algebra and many other advanced topics in mathematics. By the end of this guide, you'll be able to solve any integer problem with confidence.

Understanding Integers: Beyond Zero

Integers are the complete set of positive numbers, negative numbers, and zero. We can represent them as: ..., -4, -3, -2, -1, 0, 1, 2, 3, 4, ...

A number line is the best way to visualize integers. Zero is in the center. Positive integers (1, 2, 3...) are on the right side of zero, and their values increase as you move further right. Negative integers (-1, -2, -3...) are on the left side of zero, and their values decrease as you move further left. This means -10 is smaller than -2!

Key Concepts:

  • Positive Integers: All integers greater than 0 (1, 2, 3, ...).
  • Negative Integers: All integers less than 0 (..., -3, -2, -1).
  • Zero: An integer that is neither positive nor negative.
  • Absolute Value: The absolute value of an integer is its distance from 0 on the number line, regardless of direction. It is always positive or zero. We denote it with two vertical bars, like |-5| = 5 and |5| = 5.

Step-by-Step: Adding and Subtracting Integers

  1. Rule 1: Adding Integers with the Same Sign — When adding two positive integers or two negative integers, simply add their absolute values and keep the common sign. Example (Both Positive): (+5) + (+3) = 8 Example (Both Negative): (-5) + (-3). Add 5 and 3 to get 8. Since both are negative, the answer is -8.
  2. Rule 2: Adding Integers with Different Signs — When adding a positive and a negative integer, subtract the smaller absolute value from the larger absolute value. The result takes the sign of the integer with the larger absolute value. Example: (-9) + (+4). The absolute values are 9 and 4. Subtract: 9 - 4 = 5. Since 9 is larger than 4 and its sign is negative, the answer is -5. Another Example: (+9) + (-4). Subtract 9 - 4 = 5. Since 9 is larger and positive, the answer is +5.
  3. Rule 3: Subtracting Integers (Keep-Change-Change) — To subtract an integer, add its opposite. This is often called the 'Keep-Change-Change' method. 1. Keep the first number. 2. Change the subtraction sign to an addition sign. 3. Change the sign of the second number to its opposite. * Example: (+7) - (-3). 1. Keep +7. 2. Change - to +. 3. Change -3 to +3. The problem becomes (+7) + (+3), which equals 10.

Worked Examples: Multiplying and Dividing Integers

  • Rule 1: Same Signs give a Positive answer. When you multiply or divide two integers with the same sign (both positive or both negative), the result is always POSITIVE. Multiplication: (-6) × (-4) = +24 Division: (-20) ÷ (-5) = +4
  • Rule 2: Different Signs give a Negative answer. When you multiply or divide two integers with different signs (one positive and one negative), the result is always NEGATIVE. Multiplication: (+7) × (-3) = -21 Division: (+18) ÷ (-9) = -2
  • Combining Rules: Multiple Integers For a problem like (-2) × (3) × (-4), multiply two numbers at a time. First, (-2) × (3) = -6 (Different signs give negative). Then, (-6) × (-4) = +24 (Same signs give positive). The final answer is 24.

Exam Tips for Integers

Examiners love to test the rule for subtracting a negative number! Always remember that a - (-b) is the same as a + b. For example, 10 - (-5) = 10 + 5 = 15. Don't get tricked! Also, pay close attention to the number of negative signs in a multiplication problem. An even number of negative signs results in a positive product (e.g., (-1)×(-1) = 1), while an odd number of negative signs results in a negative product (e.g., (-1)×(-1)×(-1) = -1).

Practice Questions with Solutions

  • Q: Find the value of: (-20) + 13 - (-5) A: Step 1: First, handle the subtraction of a negative number. - (-5) becomes + 5. The expression is now: (-20) + 13 + 5. Step 2: Add the positive numbers together: 13 + 5 = 18. The expression becomes: (-20) + 18. Step 3: Add the numbers with different signs. Subtract their absolute values (20 - 18 = 2). The number with the larger absolute value is -20, so the result is negative. Final answer: -2
  • Q: Calculate: (-8) × (-3) × (-10) A: Step 1: Multiply the first two integers. (-8) × (-3). Since both signs are negative, the result is positive. 8 × 3 = 24. The expression is now: 24 × (-10). Step 2: Multiply the result by the last integer. 24 × (-10). The signs are different (one positive, one negative), so the result is negative. Final answer: -240
  • Q: In a quiz, +5 marks are given for every correct answer and -2 marks for every incorrect answer. Rohan attempts all questions and gets 8 correct answers and 4 incorrect answers. What is his total score? A: Step 1: Calculate the marks for correct answers. 8 correct answers × 5 marks/answer = 40 marks. Step 2: Calculate the marks deducted for incorrect answers. 4 incorrect answers × (-2) marks/answer = -8 marks. Step 3: Calculate the total score by adding the marks from correct and incorrect answers. Total score = 40 + (-8). Step 4: Solve 40 - 8. Final answer: 32 marks
  • Q: Verify the property a × (b + c) = (a × b) + (a × c) for a = -4, b = 2, c = -3. A: Step 1: Solve the Left Hand Side (LHS): a × (b + c). Substitute the values: (-4) × (2 + (-3)). First, solve the bracket: 2 + (-3) = -1. Now multiply: (-4) × (-1) = 4. So, LHS = 4. Step 2: Solve the Right Hand Side (RHS): (a × b) + (a × c). Substitute the values: ((-4) × 2) + ((-4) × (-3)). Solve each bracket: (-4) × 2 = -8 and (-4) × (-3) = 12. Now add the results: (-8) + 12 = 4. So, RHS = 4. Step 3: Compare the LHS and RHS. Since LHS = RHS (4 = 4), the property is verified. Final answer: Verified, as LHS = 4 and RHS = 4.

Frequently Asked Questions

What is the difference between whole numbers and integers?

Whole numbers are the set of numbers `{0, 1, 2, 3, ...}`. Integers include all the whole numbers, plus the negative counterparts of the natural numbers. So, integers are `{..., -3, -2, -1, 0, 1, 2, 3, ...}`.

Is zero a positive or a negative integer?

Zero is neither positive nor negative. It is a neutral integer that separates the positive integers (on its right on a number line) from the negative integers (on its left).

What is the smallest integer?

There is no 'smallest' integer. Because the negative numbers go on forever to the left on the number line (`..., -100, -101, ...`), you can always find a smaller integer. Similarly, there is no 'largest' integer.

What happens when you divide an integer by zero?

Division of any integer by zero is not defined in mathematics. You cannot perform this operation as it doesn't have a meaningful answer.