RBSE Solutions Class 7 Maths Chapter 1 Integers More Ques

Integers In Text Exercise

Question. Check if integers are closed for addition. Also, verify if this property holds true only for positive integers or if it applies to negative integers as well.
Answer: We will check this property by adding different types of integers (positive and negative) and observing the results.

From the table above, it is clear that when you add any two integers, whether they are positive or negative, the result is always an integer. This confirms that integers are closed under addition. The closure property ensures that the sum of any two integers will always be another integer.
In simple words: When you add two whole numbers (positive or negative), the answer is always another whole number. It works for all kinds of integers.

🎯 Exam Tip: Remember the closure property means that performing an operation on numbers from a set always produces a result that is also in that set. For integers, addition always stays within the set of integers.

Question. What happens when we subtract one integer from another? Does their difference is also an integer? Complete the following table by observation:

Answer: When we subtract one integer from another, the difference is always an integer. This means that the set of integers is closed under subtraction. It is impossible to find two integers whose difference is not an integer. This property ensures that any subtraction operation between integers will always yield an integer as the result.
In simple words: Subtracting any two whole numbers (integers) always gives you another whole number. You won't get a fraction or a decimal.

🎯 Exam Tip: The closure property holds for both addition and subtraction of integers. This is a fundamental concept for understanding integer operations.

Question. Check whether the following statements are same? Does integers also follow commutative property of addition? Find some other additions also.
(i) \( (-8) + (-4) \) and \( (-4) + (-8) \)
(ii) \( (-2) + 5 \) and \( 5 + (-2) \)
(iii) \( 12 + 0 \) and \( 0+12 \)

Answer: Yes, the statements are the same. From the table, it is clear that integers follow the commutative property of addition. This means that the order in which you add two integers does not change the sum. For any two integers \(a\) and \(b\), \(a + b = b + a\). This property simplifies calculations as you can arrange numbers in any order before adding them.
In simple words: When you add two numbers, it doesn't matter which one you put first; the answer will be the same. Like \(2 + 3\) is the same as \(3 + 2\).

🎯 Exam Tip: The commutative property is crucial for understanding how numbers behave under certain operations. Always check if a property holds true for different number types (like positive, negative, or zero) to confirm it.

Question. Observe the following and fill in the blanks, and confirm that 0 is an additive identity for integers using examples:
(i) \( (-4) + 0 = -4 \)
(ii) \( 7 + 0 = 7 \)
(iii) \( 0+ (-14) = .... \)
(iv) \( -8 + .... = - 8 \)
(v) \( .... + 0 = 15 \)
(vi) \( -23 + .... = - 23 \)

Answer: Let's fill in the blanks and then confirm the property:

(iii) \( 0 + (-14) = -14 \)
(iv) \( -8 + 0 = -8 \)
(v) \( 15 + 0 = 15 \)
(vi) \( -23 + 0 = -23 \)

From these examples, it is clear that when you add 0 to any integer, the integer remains unchanged. Therefore, 0 is the additive identity for integers. This means that 0 has a special role in addition, acting like a neutral element that doesn't change the value of other numbers. Here are more examples confirming that 0 is the additive identity:

(i) \( 5 + 0 = 5 \)
(ii) \( (-10) + 0 = -10 \)
(iii) \( (-9) + 0 = -9 \)
(iv) \( 23 + 0 = 23 \)
(v) \( 0 + 20 = 20 \)
(vi) \( 0 + (-2) = -2 \)
In simple words: When you add zero to any whole number, the number itself doesn't change. That's why zero is called the "additive identity."

🎯 Exam Tip: The additive identity is always 0. It's a key concept to remember for properties of integers and other number systems.

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Question. What happens when we multiply a negative integer with a positive integer?
Answer: When a negative integer is multiplied by a positive integer, the product is always a negative integer. This can be verified using the following examples. This rule is consistent and helps in determining the sign of the product in multiplication involving integers.
(i) \( (-1) \times 4 = -4 \)
(ii) \( (-2) \times 3 = -6 \)
(iii) \( (-5) \times 2 = -10 \)
In simple words: If you multiply a negative number by a positive number, the answer will always be negative.

🎯 Exam Tip: Remember the rule: "Unlike signs (one positive, one negative) give a negative product." This is fundamental for integer multiplication.

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Question. Observe the following patterns and complete the blanks by filling in the values for multiplication of negative integers with other negative integers:
\( -3 \times 4 = -12 \)
\( -3 \times 3 = -9 = -12 - (-3) \)
\( -3 \times 2 = -6 = -9 - (-3) \)
\( -3 \times 1 = -3 = -6 - (-3) \)
\( -3 \times 0 = 0 = -3 - (-3) \)
\( -3 \times (-1) = 3 = 0 - (-3) \)
\( -3 \times (-2) = 6 = 3 - (-3) \)
(i) \( -3 \times (-3) = \) ........
(ii) \( -3 \times (-4) = \) ........
Fill in the blanks in a similar manner:
\( -5 \times 3 = -15 \)
\( -5 \times 2 = -10 = -15 - (-5) \)
\( -5 \times 1 = -5 = -10 - (-5) \)
(i) \( -5 \times (-1) = \) ........
(ii) \( -5 \times (-2) = \) ........
(iii) \( -5 \times (-3) = \) ........

Answer: By observing the pattern, we can see that multiplying a negative integer by another negative integer results in a positive integer. Each step in the pattern adds the first negative integer to the previous result, moving towards more positive numbers. This demonstrates the rule that "like signs give a positive product."

Here are the completed calculations:
(i) \( -3 \times (-3) = 9 = 6 - (-3) \)
(ii) \( -3 \times (-4) = 12 = 9 - (-3) \)

And for the second set of blanks:
(i) \( -5 \times (-1) = 5 = 0 - (-5) \)
(ii) \( -5 \times (-2) = 10 = 5 - (-5) \)
(iii) \( -5 \times (-3) = 15 = 10 - (-5) \)
In simple words: When you multiply two negative numbers, the answer always becomes positive. It's like two negatives making a positive.

🎯 Exam Tip: Remember the rules of signs for multiplication: "Like signs (both positive or both negative) give a positive product, while unlike signs (one positive, one negative) give a negative product."

Question. Observe the multiplication statements and their corresponding division statements, then complete the division rules for integers:

Complete the following division rules for integers:
(i) Negative integer \( \div \) Positive integer \( = \) Negative integer
(ii) Positive integer \( \div \) Negative integer \( = \) Negative integer
(iii) Positive integer \( \div \) Positive integer \( = \) Positive integer
(iv) Negative integer \( \div \) Negative integer \( = \) Positive integer

Answer: The patterns in the table clearly demonstrate the rules for dividing integers. Division is the inverse of multiplication, so the sign rules are similar: like signs result in a positive quotient, and unlike signs result in a negative quotient. This makes it easier to predict the sign of the answer when dividing numbers.

Here are the completed division rules, confirmed by the table:
(i) Negative integer \( \div \) Positive integer \( = \) Negative integer \( (\checkmark) \)
(ii) Positive integer \( \div \) Negative integer \( = \) Negative integer \( (\checkmark) \)
(iii) Positive integer \( \div \) Positive integer \( = \) Positive integer \( (\checkmark) \)
(iv) Negative integer \( \div \) Negative integer \( = \) Positive integer \( (\checkmark) \)
In simple words: When dividing numbers, if both numbers have the same sign (both positive or both negative), the answer is positive. If they have different signs, the answer is negative.

🎯 Exam Tip: The sign rules for division are the same as for multiplication. Always apply these rules carefully to ensure the correct sign for your quotient.

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Question. Complete the following table:

Answer: The table above shows various products of integers. From the completed table, it is clear that the product of any two integers, whether positive or negative, is always an integer. This confirms that the set of integers is closed under multiplication. The closure property ensures that multiplying integers will always result in another integer.
In simple words: When you multiply two whole numbers together, the answer is always another whole number.

🎯 Exam Tip: The closure property holds for multiplication of integers. This means the result of multiplying two integers will always be an integer.

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Question. Observe and complete the following table:

Answer: From the completed table, the outcome clearly shows that the product of integers does not depend on their order. This property is known as the commutative property of multiplication. For any two integers \(a\) and \(b\), \(a \times b = b \times a\). This is a helpful rule because it means you can multiply numbers in any sequence and still get the same answer.
In simple words: When you multiply two numbers, changing their order does not change the answer. Like \(2 \times 3\) is the same as \(3 \times 2\).

🎯 Exam Tip: The commutative property applies to both addition and multiplication of integers. Understanding this property can simplify calculations and problem-solving.

Question. Check the effect of multiplying integers by 1:
(i) \( (-3) \times 1 = -3 \)
(ii) \( 1 \times 5 = 5 \)
(iii) \( (-4) \times 1 = \) ........
(iv) \( 1 \times 8 = \) ........
(v) \( 1 \times (-5) = \) ........
(vi) \( 3 \times 1 = \) ........

Answer: Let's complete the multiplications:

(iii) \( (-4) \times 1 = -4 \)
(iv) \( 1 \times 8 = 8 \)
(v) \( 1 \times (-5) = -5 \)
(vi) \( 3 \times 1 = 3 \)

From these examples, it is evident that when any integer is multiplied by 1, the integer remains unchanged. Therefore, 1 is the multiplicative identity for integers. This property is very useful as it shows that 1 acts as a neutral element in multiplication.
In simple words: When you multiply any whole number by 1, the number stays the same. So, 1 is called the "multiplicative identity."

🎯 Exam Tip: Remember that 1 is the multiplicative identity for integers, just as 0 is the additive identity. These identities are fundamental to algebra and number theory.

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Question. Observe and complete the following table to determine if integers are closed under division:

Answer: From the completed table, it is evident that dividing one integer by another does not always result in an integer. For example, \( \frac{1}{4} \) and \( \frac{3}{-8} \) are not integers. Therefore, integers are not closed under division. This means that if you divide two integers, the answer might be a fraction or decimal, which is not an integer. This shows that the closure property does not apply to division for integers.
In simple words: When you divide two whole numbers, the answer might not always be a whole number. So, whole numbers are not "closed" for division.

🎯 Exam Tip: Remember that division is not closed for integers. This is an important distinction compared to addition, subtraction, and multiplication, which are closed.

Do and Learn

Question 1. In which direction one should move on the number line to add \( -5 \)?
Answer: To add \( -5 \) on a number line, one should move to the left. Adding a negative number is equivalent to subtracting a positive number, which always means moving in the negative direction on the number line. This helps visualize how negative numbers affect position.
In simple words: To add \( -5 \), you move to the left on the number line.

🎯 Exam Tip: Moving right on a number line means adding positive numbers, and moving left means adding negative numbers (or subtracting positive numbers).

Question 3. In which direction we will move and on which number will we reach by adding 5 to 3?
Answer: When adding 5 to 3 on a number line, we will move to the right. Starting from 3, moving 5 units to the right will lead us to the number 8. So, \( 3 + 5 = 8 \). This is a basic illustration of addition with positive integers.
In simple words: To add 5 to 3, you move right on the number line and land on 8.

🎯 Exam Tip: Always remember that adding a positive number on the number line means moving to the right, increasing the value.

Question 4. In which direction we will move and on which number will we reach by subtracting \( +5 \) from \( -3 \)?
Answer: To subtract \( +5 \) from \( -3 \), we will move to the left on the number line. Starting at \( -3 \), and moving 5 units to the left (because subtracting a positive is like adding a negative), we will reach \( -3 - (+5) = -3 - 5 = -8 \). This shows how subtraction, especially with negative starting points, moves you further into the negative region.
In simple words: To subtract 5 from \( -3 \), you move left on the number line and end up at \( -8 \).

🎯 Exam Tip: Subtracting a positive number always moves you to the left on the number line, decreasing the value.

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Question. Solve the following:
(i) \( 4 \times (8) = .... = .... \)
(ii) \( 3 \times (-3) = .... = ...... \)
(iii) \( 5 \times (-9) = .... = ..... \)

Answer: Let's solve each multiplication problem:
(i) \( 4 \times (8) = 32 \). This is a simple multiplication of two positive integers.
(ii) \( 3 \times (-3) = -9 \). When a positive integer is multiplied by a negative integer, the result is negative.
(iii) \( 5 \times (-9) = -45 \). Similar to the previous one, a positive multiplied by a negative gives a negative.
In simple words: Multiply the numbers. If one is negative and one is positive, the answer is negative. If both are positive, the answer is positive.

🎯 Exam Tip: Always pay close attention to the signs of the numbers when performing multiplication. A common mistake is to get the sign wrong.

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Question. Find the following products:
(i) \( 15 \times (-5) \)
(ii) \( 27 \times (-10) \)
(iii) \( -12 \times 12 \)
(iv) \( -7 \times 4 \)

Answer: Let's calculate each product:
(i) \( 15 \times (-5) = -75 \). (Positive \( \times \) Negative \( = \) Negative)
(ii) \( 27 \times (-10) = -270 \). (Positive \( \times \) Negative \( = \) Negative)
(iii) \( -12 \times 12 = -144 \). (Negative \( \times \) Positive \( = \) Negative)
(iv) \( -7 \times 4 = -28 \). (Negative \( \times \) Positive \( = \) Negative). These calculations help reinforce the rules of integer multiplication.
In simple words: When multiplying numbers with different signs, the answer is always negative.

🎯 Exam Tip: Consistency in applying the sign rules (positive \( \times \) negative \( = \) negative) is key to solving integer multiplication problems correctly.

Question. Calculate the following products:
(i) \( (-12) \times (-15) \)
(ii) \( (-25) \times (-4) \)
(iii) \( (-17) \times (-11) \)

Answer: Let's calculate each product:
(i) \( (-12) \times (-15) = 180 \). When two negative integers are multiplied, the product is positive.
(ii) \( (-25) \times (-4) = 100 \). Similarly, negative times negative equals positive.
(iii) \( (-17) \times (-11) = 187 \). This also follows the rule of two negative numbers making a positive product. It's a fundamental rule in integer arithmetic.
In simple words: When you multiply two negative numbers, the answer always becomes positive.

🎯 Exam Tip: Remember that a negative number multiplied by a negative number always results in a positive product. This is a common rule used in algebra.

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Question. Find the following product:
(i) \( (-1) \times (-1) \times (-1) = \) ........
(ii) \( (-1) \times (-1) \times (-1) \times (-1) = \) ........

Answer: Let's find the products by applying the multiplication rules step-by-step:
(i) \( (-1) \times (-1) \times (-1) = [(-1) \times (-1)] \times (-1) = (+1) \times (-1) = -1 \). When there is an odd number of negative signs, the product is negative.
(ii) \( (-1) \times (-1) \times (-1) \times (-1) = [(-1) \times (-1)] \times [(-1) \times (-1)] = (+1) \times (+1) = 1 \). When there is an even number of negative signs, the product is positive. This is a crucial pattern to observe for multiplying multiple negative numbers.
In simple words: If you multiply \( -1 \) many times, count how many \( -1 \)s there are. If it's an odd number, the answer is \( -1 \). If it's an even number, the answer is \( 1 \).

🎯 Exam Tip: The sign of a product with multiple negative factors depends on the count of negative factors: an odd count yields a negative product, an even count yields a positive product.