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7th class > Integers > What Have We Discussed?

What Have We Discussed?

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We now study the properties satisfied by addition and subtraction:

(a) Integers are for addition and subtraction both. That is, a + b and a – b are again integers, where a and b are any integers.

(b) Addition is for integers, i.e., a + b = b + a for all integers a and b.

(c) Addition is for integers, i.e., (a + b) + c = a + (b + c) for all integers a, b and c.

(d) Integer 0 is the under addition. That is, a + 0 = 0 + a = a for every integer a.

We studied, how integers could be multiplied, and found that product of a positive and a negative integer is a integer, whereas the product of two negative integers is a integer. For example, – 2 × 7 = – 14 and – 3 × – 8 = 24.

Product of even number of negative integers is , whereas the product of odd number of integers is negative.

Integers show some properties under multiplication. (a) Integers are closed under multiplication. That is, a × b is an integer for any two integers a and b. (b) Multiplication is commutative for integers. That is, a × b = × for any integers a and b. (c) The integer 1 is the identity under multiplication, i.e., 1 × a = × = a for any integer a. (d) Multiplication is associative for integers, i.e., (a × b) × c = × ( × c) for any three integers a, b and c

Under addition and multiplication, integers show a property called property. That is, a × (b + c) = a × b + a × c for any three integers a, b and c.

The properties of commutativity, associativity under addition and multiplication, and the distributive property help us to make our calculations easier.

We also learnt how to divide integers. We found that, (a) When a positive integer is divided by a negative integer, the quotient obtained is and vice-versa. (b) Division of a negative integer by another negative integer gives as quotient.

For any integer a, we have (a) a ÷ 0 = (b) a ÷ 1 =