CBSE Class 6 Maths: Integers - Exercise 6.1 Explained

Welcome, young mathematicians! In Class 6, we step into the exciting world of Integers, a concept that helps us understand numbers beyond just positive values. Think about temperatures – sometimes it's 5°C above zero, and sometimes it's 2°C below zero. How do we represent 'below zero'? That's where integers come in!

Exercise 6.1 from your NCERT textbook focuses on introducing you to what integers are, how to represent various real-life situations using positive and negative signs, and understanding the concept of 'opposites'. By the end of this page, you'll be able to confidently solve all problems related to integers ex 6 1 class 6 ncert, mastering the fundamental ideas of positive, negative, and zero. Let's begin our journey to conquer these essential numbers!

What Exactly Are Integers?

Before diving into Exercise 6.1, let's firmly grasp what integers are. You already know about natural numbers (1, 2, 3, ...) which are used for counting, and whole numbers (0, 1, 2, 3, ...) which include zero. Integers take this a step further by including negative numbers along with zero and positive numbers.

So, the set of integers includes:

  • Positive integers: These are numbers greater than zero (1, 2, 3, ...). They are usually written with a '+' sign (like +5) or no sign at all (like 5, which means +5).
  • Negative integers: These are numbers less than zero (-1, -2, -3, ...). They are always written with a '-' sign (like -5).
  • Zero: Zero is an integer that is neither positive nor negative. It acts as the reference point on the number line.

Integers are crucial for representing quantities that have direction or show an opposite meaning, such as profit/loss, above/below sea level, increase/decrease, or credits/debits in a bank account. Mastering this concept is key to understanding more advanced mathematics.

Representing Real-Life Situations with Integers

  1. Understand the Situation's Direction — First, read the situation carefully and identify if it indicates a movement, change, or state that can be positive or negative. Words like 'above', 'gain', 'increase', 'deposit' usually suggest a positive value. Words like 'below', 'loss', 'decrease', 'withdrawal' usually suggest a negative value. 'Sea level' or 'zero level' is often the reference point for zero.
  2. Identify the Numerical Value — Next, extract the numerical value given in the situation. For example, in '200 meters above sea level', the numerical value is 200.
  3. Assign the Correct Sign — Based on the direction identified in Step 1, assign either a '+' sign (for positive direction) or a '-' sign (for negative direction) to the numerical value. Remember, for positive numbers, the '+' sign is often optional. For negative numbers, the '-' sign is always mandatory. So, '200 meters above sea level' becomes +200 or simply 200. 'Loss of ₹500' becomes -500.
  4. Form the Integer — Combine the sign and the numerical value to form the integer that represents the given situation. This simple process allows us to translate everyday language into mathematical integers.

Understanding and Finding Opposites

  • The opposite of a number is the number with the same numerical value but the opposite sign. It's the same distance from zero on the number line, but in the other direction. For example, if you walk 5 steps forward (+5), the opposite is walking 5 steps backward (-5).
  • Example 1: Find the opposite of +10. Step 1: The given integer is +10. Step 2: Change its sign. Final Answer: The opposite of +10 is -10.
  • Example 2: Find the opposite of -7. Step 1: The given integer is -7. Step 2: Change its sign. Final Answer: The opposite of -7 is +7 (or simply 7).
  • Example 3: Find the opposite of 0. Step 1: The given integer is 0. Step 2: Zero is neither positive nor negative. Its position on the number line is unique. Final Answer: The opposite of 0 is 0 itself.

Exam Tip: Don't Get Confused with Signs!

A common mistake in integers ex 6 1 class 6 ncert is getting confused between 'positive' and 'negative' signs, especially when representing situations. Always associate a 'gain', 'increase', 'above', 'deposit', 'east', 'right' with a positive integer (+), and a 'loss', 'decrease', 'below', 'withdrawal', 'west', 'left' with a negative integer (-). Remember that 'opposite' simply means changing the sign of the number, not changing the number itself (like thinking the opposite of 5 is 1/5, which is its reciprocal, not opposite). Pay close attention to the keywords in the problem statement!

Practice Questions with Solutions

  • Q: Write the opposite of the following: (a) Increase of 30 km (b) 100 m below sea level (c) Spending ₹500 A: Step 1: For (a) 'Increase of 30 km' represents a positive change. Step 2: The opposite of an increase is a decrease. So, the opposite is 'Decrease of 30 km'. Step 3: For (b) '100 m below sea level' represents a negative position. Step 4: The opposite of 'below sea level' is 'above sea level'. So, the opposite is '100 m above sea level'. Step 5: For (c) 'Spending ₹500' represents a loss or decrease in money, which is negative. Step 6: The opposite of spending is gaining or saving. So, the opposite is 'Saving ₹500' or 'Earning ₹500'. Final Answer: (a) Decrease of 30 km (b) 100 m above sea level (c) Saving ₹500
  • Q: Represent the following numbers as integers with appropriate signs: (a) A deposit of ₹2500 in a bank account (b) A withdrawal of ₹800 from a bank account. A: Step 1: For (a) 'Deposit' means adding money, which is a positive action. Step 2: The amount is ₹2500. Step 3: Combine the sign and amount: +2500. Step 4: For (b) 'Withdrawal' means taking money out, which is a negative action. Step 5: The amount is ₹800. Step 6: Combine the sign and amount: -800. Final Answer: (a) +2500 (b) -800
  • Q: Write the opposites of: (a) +67 (b) -15 (c) - (-8) A: Step 1: For (a) The opposite of a positive number is the same number with a negative sign. Step 2: Opposite of +67 is -67. Step 3: For (b) The opposite of a negative number is the same number with a positive sign. Step 4: Opposite of -15 is +15 (or 15). Step 5: For (c) First, simplify -(-8). Two negative signs make a positive, so -(-8) is +8. Step 6: Now, find the opposite of +8, which is -8. Final Answer: (a) -67 (b) +15 (c) -8
  • Q: Identify which of the following situations can be represented by a negative integer: (a) 5 steps to the right (b) 10 degrees Celsius above 0 (c) A debt of ₹1000 (d) Earning ₹750 A: Step 1: Analyze each situation for keywords indicating positive or negative. Step 2: (a) '5 steps to the right' indicates a positive direction. (Not negative). Step 3: (b) '10 degrees Celsius above 0' indicates a positive value. (Not negative). Step 4: (c) 'A debt of ₹1000' means money owed, which is a financial loss or negative balance. (This is negative). Step 5: (d) 'Earning ₹750' means gaining money, which is a positive income. (Not negative). Final Answer: (c) A debt of ₹1000

Frequently Asked Questions

What is the difference between whole numbers and integers?

Whole numbers include zero and all positive counting numbers (0, 1, 2, 3,...). Integers, on the other hand, include whole numbers, and also their negative counterparts (-1, -2, -3,...). So, all whole numbers are integers, but not all integers are whole numbers (e.g., -5 is an integer but not a whole number).

Why is zero considered an integer?

Zero is considered an integer because it is a complete unit (not a fraction or decimal) and it serves as the crucial neutral point on the number line, separating positive integers from negative integers. It's unique because it's neither positive nor negative, but it is a fundamental part of the integer set.

How do I know if I should use a positive or negative sign for a situation?

You determine the sign by identifying the 'direction' or 'effect' of the situation relative to a zero point. Use a positive sign (+) for increases, gains, above, right, east, or deposits. Use a negative sign (-) for decreases, losses, below, left, west, or withdrawals. Think of it like moving up or down, or forward or backward.

What is the opposite of a negative integer, for example, -15?

The opposite of a negative integer is the positive version of that same number. So, the opposite of -15 is +15 (or simply 15). They are both the same distance from zero on the number line, but in opposite directions.