INTEGERS- ESSENTIAL POINTS
- Two fractions are multiplied by multiplying their numerators and denominators separately and writing the product as product of numerators by product of denominators. For example, 2/3 x 5/7 = (2 x 5) / (3 x 7) = 10 / 21
- A fraction acts as an operator ‘of’. Example, ½ of 2 is 1/2 x 2 = 1
- Integers are a bigger collection of numbers which is formed by whole numbers and their negatives.
- All natural numbers are whole numbers and all whole numbers are integers.
- Properties satisfied by addition and subtraction of integers
- Integers are closed for addition and subtraction both. 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.
- Product of a positive and a negative integer is a negative integer, whereas the product of two negative integers is a positive integer. For example, – 2 × 7 = – 14 and – 3 × – 8 = 24.
- Product of even number of negative integers is positive, whereas the product of odd number of negative integers is negative.
- Following are the properties of Integers 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.
- Integers show distributive property under multiplication and addition. That is, a × (b + c) = a × b + a × c for any three integers a, b and c.
- Properties of division of integers:
- When a positive integer is divided by a negative integer, the quotient obtained is a negative integer and vice-versa.
- Division of a negative integer by another negative integer gives a positive integer as quotient.
- For any integer a, we have
- a / 0 is not defined
- a / 1 = a
- Absolute value of a number is the positive value of the number.
