Introduction to Integers
Integers form a fundamental concept in mathematics, particularly important in Class 7 as students build on their previous knowledge while preparing for more advanced mathematical concepts. R.D. Sharma's Class 7 Mathematics textbook provides comprehensive coverage of integers, with well-structured exercises that gradually increase in complexity. This guide provides detailed solutions to help students understand the concepts thoroughly and master the operations with integers.
Understanding Integers
Before diving into the exercises, it's essential to understand what integers are. Integers are a set of whole numbers, including zero, positive numbers, and their negative counterparts. The set of integers is denoted by Z and can be written as {..., -3, -2, -1, 0, 1, 2, 3, ...}. This infinite set extends indefinitely in both the positive and negative directions on the number line.
Exercise 1.1: Introduction to Integers
This introductory exercise helps students familiarize themselves with integers and their representation on the number line.
Question 1: Identify the integers among the following numbers: 5, -3, 0, 7.5, -8.2, 12, -15
Solution: The integers in this list are: 5, -3, 0, 12, -15. The numbers 7.5 and -8.2 are not integers because they have decimal or fractional parts.
Question 2: Represent -5, 0, and 3 on the number line.
Solution: On the number line:
- -5 would be marked 5 units to the left of zero
- 0 is at the center of the number line
- 3 would be marked 3 units to the right of zero
Exercise 1.2: Addition and Subtraction of Integers
This exercise focuses on the fundamental operations of addition and subtraction with integers.
Question 1: Add -7 and 5
Solution: -7 + 5 = -2
Explanation: When adding a negative number (-7) and a positive number (5), we find the difference between their absolute values (| -7 | - | 5 | = 2) and assign the sign of the number with the greater absolute value (negative in this case).
Question 2: Subtract -4 from 10
Solution: 10 - (-4) = 10 + 4 = 14
Explanation: Subtracting a negative number is the same as adding its absolute value. When we subtract -4 from 10, it's equivalent to adding 4 to 10.
Question 3: Find the sum: (-8) + (-3) + (-10)
Solution: (-8) + (-3) + (-10) = -(8 + 3 + 10) = -21
Explanation: When adding multiple negative numbers, we add their absolute values and keep the negative sign.
Tips for Addition and Subtraction:
- When adding integers with the same sign, add their absolute values and keep the common sign.
- When adding integers with different signs, find the difference between their absolute values and use the sign of the integer with the greater absolute value.
- Subtracting a number is the same as adding its opposite: a - b = a + (-b).
Exercise 1.3: Properties of Addition
This exercise explores the various properties that govern addition of integers.
Question 1: Verify the commutative property of addition for -8 and 5.
Solution: The commutative property states that a + b = b + a for all integers a and b.
For a = -8 and b = 5:
Left side: a + b = -8 + 5 = -3
Right side: b + a = 5 + (-8) = 5 - 8 = -3
Since -8 + 5 = 5 + (-8), the commutative property holds true.
Question 2: Find the additive inverse of -9.
Solution: The additive inverse of a number is the value that, when added to the original number, gives zero.
For -9, we need a number x such that -9 + x = 0.
Therefore, x = 9, because -9 + 9 = 0.
The additive inverse of -9 is 9.
Exercise 1.4: Multiplication of Integers
This section covers the rules and properties of multiplying integers.
Question 1: Multiply -6 and -4.
Solution: (-6) (-4) = 24
Explanation: When multiplying two negative integers, the product is always positive. We multiply the absolute values and get a positive result.
Question 2: Find the product: 7 (-3) (-2).
Solution: 7 (-3) (-2) = (7 (-3)) (-2) = (-21) (-2) = 42
Explanation: First, we multiply 7 and -3, resulting in -21. Then, we multiply -21 by -2. Since we're multiplying two negative numbers, the result is positive, giving us 42.
Multiplication Rules for Integers:
- Positive Positive = Positive
- Positive Negative = Negative
- Negative Positive = Negative
- Negative Negative = Positive
Exercise 1.5: Properties of Multiplication
This exercise focuses on understanding the properties that govern multiplication of integers.
Question 1: Verify the distributive property of multiplication over addition for -5, 6, and -2.
Solution: The distributive property states that a (b + c) = a b + a c.
For a = -5, b = 6, and c = -2:
Left side: a (b + c) = -5 (6 + (-2)) = -5 4 = -20
Right side: a b + a c = (-5) 6 + (-5) (-2) = -30 + 10 = -20
Since -5 (6 + (-2)) = (-5) 6 + (-5) (-2), the distributive property holds true.
Exercise 1.6: Division of Integers
This exercise covers the division of integers, which follows specific rules based on the signs of the dividend and divisor.
Question 1: Divide -48 by 6.
Solution: -48 6 = -8
Explanation: When dividing a negative integer by a positive integer, the quotient is negative. We divide the absolute values and assign a negative sign to the result.
Question 2: Find the quotient: -36 (-4).
Solution: -36 (-4) = 9
Explanation: When dividing two negative integers, the quotient is positive. We divide the absolute values and get a positive result.
Exercise 1.7: Word Problems
This section applies the concepts of integer operations to real-world situations.
Question 1: A diver dives 15 meters below sea level and then rises up 8 meters. What is his current position?
Solution: Let's represent the sea level as 0 on the number line.
Initial position: -15 meters (15 meters below sea level)
Change in position: +8 meters (rising up)
Current position: -15 + 8 = -7 meters
The diver is now 7 meters below sea level.
Question 2: At 8 AM, the temperature was -4C. By 2 PM, it had risen by 12C. What was the temperature at 2 PM?
Solution: Initial temperature: -4C
Change in temperature: +12C
Temperature at 2 PM: -4 + 12 = 8C
The temperature at 2 PM was 8C.
Tips for Mastering Integer Operations
Common Mistakes to Avoid
- Forgetting to change the sign of the second number when subtracting
- Confusing the rules for addition and multiplication when dealing with signs
- Attempting to divide by zero, which is undefined
- Not using parentheses correctly in problems with multiple operations
- Misinterpreting the position of integers on the number line
Conclusion
The chapter on Integers in R.D. Sharma's Class 7 Mathematics textbook provides a solid foundation for understanding and working with integers. The carefully designed exercises guide students through the fundamental concepts, operations, and properties of integers, progressively building their skills and confidence. By following the solutions provided and practicing regularly, students will develop a strong command of integers, which is essential for success in higher mathematics.
