22 Questions
The product of two __________ is again an integer.
integers
In general, a × b is an __________, for all integers a and b.
integer
(–30) × 12 = __________.
-360
Multiplication is __________ for integers.
commutative
(–15) × (–10) = __________.
150
(–17) × 0 = __________.
0
We know that division is the inverse operation of ______.
multiplication
(–1) × (–1) = ______.
1
2 × (– 6) = ______.
(–12)
When dividing a negative integer by a positive integer, we divide them as whole numbers and then put a ______ sign before the quotient.
minus
(–8) × (–9) = ______.
72
(–3) × (–7) = ______.
21
(–8) × 4 = ______.
(–32)
5 × (– 9) = ______.
(–45)
When a positive integer is divided by a ______ integer, the quotient obtained is negative and vice-versa.
negative
The product of an ______ number of negative integers is positive.
even
Integers show a property called ______ property under addition and multiplication.
distributive
The ______ 1 is the identity under multiplication, i.e., 1 × a = a × 1 = a for any integer a.
integer
The properties of commutativity, associativity under addition and multiplication, and the ______ property help us to make our calculations easier.
distributive
Integers are ______ for addition and subtraction, i.e., a + b and a – b are again integers, where a and b are any integers.
closed
Multiplication of integers is ______ for integers, i.e., a × b = b × a for any integers a and b.
commutative
Division of a ______ integer by another negative integer gives positive as quotient.
negative
Study Notes
Properties of Integers
- Integers are closed under addition and subtraction, meaning that the result of adding or subtracting two integers is always an integer.
- Addition is commutative, meaning that the order of the integers being added does not change the result (a + b = b + a).
- Addition is associative, meaning that the order in which integers are added does not change the result ((a + b) + c = a + (b + c)).
- Integer 0 is the identity under addition, meaning that adding 0 to any integer does not change the value (a + 0 = 0 + a = a).
Multiplication of Integers
- Integers are closed under multiplication, meaning that the product of two integers is always an integer.
- Multiplication is commutative, meaning that the order of the integers being multiplied does not change the result (a × b = b × a).
- Multiplication is associative, meaning that the order in which integers are multiplied does not change the result ((a × b) × c = a × (b × c)).
- The integer 1 is the identity under multiplication, meaning that multiplying any integer by 1 does not change the value (1 × a = a × 1 = a).
Distributive Property
- Integers exhibit the distributive property, meaning that a × (b + c) = a × b + a × c.
Division of Integers
- Division is the inverse operation of multiplication.
- When a positive integer is divided by a negative integer, the quotient is negative, and vice versa.
- When a negative integer is divided by another negative integer, the quotient is positive.
Examples and Observations
- The product of two integers is always an integer.
- Multiplication is commutative for integers, meaning that the order of the integers being multiplied does not change the result.
- Division statements can be written corresponding to multiplication statements for integers.
Quiz about the properties of integers including closure, commutative and associative properties, and the identity element under addition.
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