1. Integers are a bigger collection of numbers which is formed by whole numbers and their negatives.
2. Properties of addition and subtraction :
- Integers are closed for addition and subtraction both. That is, a + b and a - b are again integers, where a and b are any integers.
- Addition is commutative for integers, i.e., a + b = b + a for all integers a and b.
- Addition is associative for integers, i.e., (a + b) + c = a + (b + c) for all integers a, b and c.
- Integer 0 is the identity under addition. That is, a + 0 = 0 + a = a for every integer a.
3. The product of two negative integers is a positive integer. For example, -2 ✕ 7 = -14 and -3 ✕ -8 = 24.
4. Product of even number of negative integers is positive, whereas the product of odd number of negative integers is negative.
5. Integers show some properties under multiplication.
- Integers are closed under multiplication. That is, a ✕ b is an integer for any two integers a and b.
- Multiplication is commutative for integers. That is, a ✕ b = b ✕ a for any integers a and b.
- The integer 1 is the identity under multiplication, i.e., 1 ✕ a = a ✕ 1 = a for any integer a.
- Multiplication is associative for integers, i.e., (a ✕ b) ✕ c = a ✕ (b ✕ c) for any three integers a, b and c.
6. Under addition and multiplication, integers show a property called distributive property. That is, a ✕ (b + c) = a ✕ b + a ✕ c for any three integers a, b and c.
7. The properties of commutativity, associativity under addition and multiplicatio, and the distributive property help us to make our calculations easier.
8. When a positive integer is divided by a negative integer, the quotient obtained is negative and vice - versa.
9. Division of a negative integer by another negative integer gives positive as quotient.
10. For any integer a, we have
- a ÷ 0 is not defined
- a ÷ 1 = a