Q1: What type of number is (-8)? (a) Natural number (b) Whole number (c) Negative integer (d) Positive integer
Solution:
Ans: (c) Numbers with a negative sign are less than zero and are called negative integers. The number (-8) has a negative sign before it.
Q2: What is the result of adding two positive integers? (a) Always a negative integer (b) Always zero (c) Always a positive integer (d) Can be positive or negative
Solution:
Ans: (c) When we add two positive integers, we add their values and the result is always a positive integer, as shown with 6 + 4 = 10.
Q3: What is the multiplicative identity for integers? (a) 1 (b) 0 (c) -1 (d) 10
Solution:
Ans: (a) When we multiply any integer by 1, the product is the integer itself. Therefore, 1 is the multiplicative identity for integers.
Q4: Which property states that a + b = b + a for integers? (a) Associative property (b) Distributive property (c) Closure property (d) Commutative property
Solution:
Ans: (d) The commutative property of addition states that if a and b are two integers, then a + b = b + a, meaning we can add integers in any order.
Q5: What is the product of (-6) × (-8)? (a) -48 (b) +48 (c) -14 (d) +14
Solution:
Ans: (b) When we multiply two negative integers with the same sign, we find the product of their values and put a positive sign before the product.
Fill in the Blanks
Q1: Numbers with a negative sign are less than zero and are called _____.
Solution:
Ans: negative numbers
Q2: Zero is the _____ identity for integers.
Solution:
Ans: additive
Q3: When we subtract one integer from another, we add the _____ inverse of the integer being subtracted.
Solution:
Ans: additive
Q4: The collection of natural numbers including zero is called _____.
Solution:
Ans: whole numbers
Q5: The product of a negative integer and zero is always _____.
Solution:
Ans: zero
True or False
Q1: Integers are a set of whole numbers and negative of all natural numbers.
Solution:
Ans: True Integers include whole numbers (0, 1, 2, 3...) and the negative of all natural numbers (-1, -2, -3...) as defined in the content.
Q2: Subtraction of integers follows the commutative property.
Solution:
Ans: False For any two integers a and b, a - b ≠ b - a, which means subtraction does not follow the commutative property.
Q3: When the number of negative integers in a product is even, the product is positive.
Solution:
Ans: True When the number of negative integers in a product is even, then the product is a positive integer as stated in the multiplication rules.
Q4: Division of integers follows the associative property.
Solution:
Ans: False For any three integers a, b and c, (a ÷ b) ÷ c ≠ a ÷ (b ÷ c), which shows division is not associative.
Q5: When we add zero to any integer, the result is the same integer again.
Solution:
Ans: True Zero has a special role in addition. Adding zero to any integer gives the same integer, making zero the additive identity for integers.
Match the Following
Column A
Column B
1. Natural Numbers
A. Product of two negative integers
2. Additive Identity
B. Set of counting numbers
3. Commutative Property
C. Zero (0)
4. Positive Integer
D. a × b = b × a
5. Multiplicative Identity
E. One (1)
Solution:
Ans:
1 - B: Natural numbers are the set of counting numbers used for counting objects, denoted by N in mathematical notation.
2 - C: Zero is the additive identity because when we add zero to any integer, the result is the same integer again.
3 - D: The commutative property of multiplication states that for integers a and b, a × b = b × a, meaning order doesn't matter.
4 - A: When we multiply two negative integers, we get a positive integer according to the multiplication rules for integers with same signs.
5 - E: One is the multiplicative identity because multiplying any integer by 1 gives the integer itself as the product.
Short Answer Questions
Q1: What happens when we add two integers with different signs? Explain with an example.
Solution:
Ans: When we add two integers with different signs, we find the difference of their numerical values. The result takes the sign of the integer with the greater value. For example, when adding 6 + (-9), we find 9 - 6 = 3. Since 9 is greater and negative, the result is -3.
Q2: Explain the closure property under addition for integers.
Solution:
Ans: The closure property under addition states that if a and b are two integers, then a + b is also an integer. This means that when we add any two integers, the result will always be an integer. This property is true for all integers without any exceptions.
Q3: How do we find the product of two integers with unlike signs? Give an example.
Solution:
Ans: To find the product of two integers with unlike signs, we multiply their numerical values and put a negative sign before the product. For example, 26 × (-13) = -338. We multiply 26 and 13 to get 338, then place a negative sign before it.
Q4: What is the distributive property of multiplication over addition? Provide the formula.
Solution:
Ans: The distributive property of multiplication over addition states that for any integers a, b and c, a × (b + c) = a × b + a × c. This means we can distribute the multiplication over addition by multiplying each term separately and then adding the results.
Q5: Explain the rules for dividing integers with different signs and same signs.
Solution:
Ans: When dividing two integers with different signs, we divide them as whole numbers and give a negative sign to the quotient. When dividing two integers with the same signs (both positive or both negative), we divide them as whole numbers and give a positive sign to the quotient.
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