Integers are a fundamental part of mathematics, representing positive and negative whole numbers, including zero. Integers practice questions will help students grasp the concepts of addition, subtraction, multiplication, and division with these numbers.
This article will discuss various properties of integers along with practice questions on integers which are designed to strengthen their understanding of these operations.
What are Integers?
Integers are a set of numbers that include all whole numbers (both positive and negative) and zero. They are denoted by the symbol Z. The set of integers can be written as:
Z = { . . . ,−3, −2, −1, 0, 1, 2, 3, . . . }
For example, -2, 0, 1, -1923, 1234, etc are all examples of integers.
Properties of Integers
Closure Property: For any two integers a and b, the result of a * b is also an integer, where * represents arithmetic operations ( +, –, × ).
For example:
- –9 + 7 = 2 is an integer
- – 39 – 7 = – 46 is an integer
- – 3 × 7 = – 21 is an integer
- 12 ÷ 9 = 1.33 is not an integer
Therefore, integers are not closed under division
Associativity Property:
For any integers a, b, and c.
a*(b*c) = (a*b)*c
where * represents arithmetic operations (+ and ×).
For example:
- 4 + (–6 + 9) = 7 = {4 + (–6)} + 9; addition of integers is associative
- –5 × (4 × 3) = –60 = (–5 × 4) × 3; multiplication of integers is associative
- 6 – (8 – 4) = 2 ≠ (6 – 8) – 4
- Hence, subtraction of integers is not associative.
Distributive Property:
For any integers a, b, and c
a × ( b + c) = (a × b) + (a × c)
this is known as the distributive property of multiplication over addition.
Commutativity Property:
For any integers a and b,
a*b = b*a
where * represents the arithmetic operations ( + and ×).
For example:
- 4 + (–5) = –1 = (–5) + 4; addition of integers is commutative.
- 4 × (–9) = –36 = –9 × 4; multiplication of integers is commutative.
Additive Identity:
0 is the additive identity for integers, as for any integer a,
a + 0 = a = 0 + a.
Additive Inverse:
For any integer a, there exists an integer –a such that
a + (–a) = 0 = (–a) + a
Hence –a is the additive inverse of a.
Multiplicative Identity:
1 is the multiplicative identity for integers, as for any integer a,
a × 1 = a = 1 × a.
Integers Practice Questions - Solved
1: Evaluate the following:
(i) 24 – ( –92)
(ii) 210 + ( –15)
(i) 24 – ( –92) = 24 + 92 = 116
(ii) 210 + ( –15) = 210 – 15 = 195
2: Evaluate the following
(i) –18.92 – 32
(ii) – 21 + 36 – 36
(i) –18.92 – 32 = – (18.92 + 32) = – 50.92
(ii) – 21 + 36 – 36 = –21 + 0 = –21
3: Verify a + (b + c) = (a + b) + c for the following:
a = 3, b = 1, c = –10
a = 3, b = 1, c = –10
L.H.S = a + (b + c) = 3 + (1 + 10)
= 3 + 11 = 14
R.H.S = (a + b) + c = (3 + 1) + 10
= 4 + 10 = 14
∴ L.H.S = R.H.S
4: Evaluate using properties:
(i) 92 – 61 + 30 – (–35)
(ii) 198 + 213 – {32 – (372)}
(i) 92 – 61 + 30 – (–35)
= 92 – 61 + 30 + 35
= 92 – 61 + (30 + 35) {associativity}
= 92 – 61 + 65
= (92 + 65) – 61 {commutativity and associativity}
= 157 – 61 = 96
(ii) 198 + 213 – {32 – (372)}
= 198 + 213 – {32 – 372}
= 198 + 213 – (–340)
= 198 + 213 + 340
= (198 + 213) + 340 {associativity}
= 411 + 340 = 751.
5: Evaluate:
(i) (–5) × (–16) × (–36)
(ii) (–1) × (–1) × (–1) × … 101 times
(i) (–5) × (–16) × (–36)
= – (5 × 16 × 36)
= – 2880.
(ii) (–1) × (–1) × (–1) × … 101 times
= (– 1) 101
= –1 {∵ 101 is an odd number}
6: The temperature in a room during the summer is 36 o C. If an air conditioner chills the room by 4 o C/min, what will the temperature be after the air conditioner has been on for 5 minutes?
Temperature before the air conditioner is turned on = 36 o C
Rate of cooling = –4 o C/min
Temperature after the air conditioner has been on for 5 min = 36 – (4 × 5)
= 36 – 20 = 16 o C.
Integers Practice Questions - Unsolved
This worksheet contains Practice Questions on Integers that will help you to test your understanding on the concept.
Conclusion
Understanding the properties of integers and practicing integer problems is essential for mastering basic arithmetic operations. The Integers Practice Questions provided in this article aim to strengthen students' understanding and help them build a solid foundation in mathematics. By working through these examples and practice problems, students will gain confidence in their ability to handle integer operations effectively.
