what we have learnt in the chapter integer class 7
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Representation of integers on the number line.
Integers are closed under addition. In general, for any two integers a and b, a + b is an integer.
Integers are closed under subtraction. Thus, if a and b are two integers then a – b is also an integer.
Addition is commutative for integers. In general, for any two integers a and b, we can say a + b = b + a
Subtraction is not commutative for integers.
Addition is associative for integers.
In general for any integers a, b and c, we can say a + (b + c) = (a + b) + c
Zero is an additive identity for integers. In general, for any integer a a + 0 = a = 0 + a
While multiplying a positive integer and a negative integer, we multiply them as whole numbers and put a minus sign (-) before the product. We thus get a negative integer. In general, for any two positive integers a and b we can say a × (-b) = (-a) × b = -(a × b)
Product of two negative integers is a positive integer. We multiply the two negative integers as whole numbers and put positive sign before the product. In general, for any two positive integers a and b, (-a) × (-b) = a × b
Multiplication is commutative for integers. In general, for any two integers a and b, a × b = b × a
The product of a negative integer and zero is zero a × 0 = 0 × a=0
1 is the multiplicative identity for integers.
The distributivity of multiplication over addition is true for integers.
a × (b + c) = a × b + a × c
When we divide a positive integer by a negative integer, we first divide them as whole numbers and then put a minus sign (-) before the quotient.
a ÷ (-b) = (-a) ÷ b where b ≠ 0
Natural numbers, whole numbers and integers: The numbers 1, 2, 3,……… which we use for counting are known as natural numbers. The natural numbers along with zero forms the collection of whole numbers.Properties of Addition and Subtraction of Integers
We know that the addition of two whole numbers is again a whole number. For example, 17 + 24 = 41 which is a whole number. This property is known as the closure property for the addition of whole numbers.
This property is true for integers also, i.e., the sum of two integers is always an integer. We cannot find a pair of integers whose addition is not an integer. Since additions of integers give integers, we can say integers are closed under’addition just like whole numbers. In general, for any two integers a and b, a + b is also an integer.
Closure Under Subtraction
If we subtract two integers, then their difference is also an integer. We cannot find any pair of integers whose difference is not an integer. Since subtraction of integers gives integers, we can say integers are closed under subtraction. In general, for any two integers a and b, a – b is also an integer.
Commutativity of Subtraction: We know that the subtraction is not commutative for whole numbers.
For example, 10 – 20 = -10 and 20 – 10 = 10
So, 10 – 20 ≠ 20 – 10
Similarly, the subtraction is not commutative for integers.
Product of Three or More Negative Integers
We find that if the number of negative integers in a product is even, the product is a positive integer; if the number of negative integers in a product is odd, the product is a negative integer.
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