Numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 together form all the numbers.
Kind of numbers:
- Natural Numbers: The numbers which are used for counting are called Natural numbers. Ex. 1, 2, 3, 4, 5, 6, 7, ……….
- Whole Numbers: Zero included with natural numbers form whole numbers. Ex.- 0, 1, 2, 3, 4, 5, 6, 7, ……….
- Integers: Negative numbers along with Zero and positive natural numbers are called Integers. Ex.- -7, -6, -5, -4, -3, -2, -1, 0 , 1, 2 , 3, 4 , 5, 6, 7, …….
- Rational Numbers: The numbers which can be expressed in the form of p/q where q is not equal to 0., are called Rational numbers. Ex. ½, -3/4, 9,
- Irrational Numbers: The numbers which can not be expressed as Rational numbers are called Irrational numbers. Ex.
,
,
- Real numbers: The group of Rational and Irrational numbers is called Real Numbers. Ex:
,
, 999, -67/34
- Even Numbers: The numbers which are divisible by 2 and contains 0, 2, 4, 6, 8 at unit place, are called even numbers. Ex: 34, 68, 86, 44, 30
- Odd Numbers: The numbers which are not divisible by 2 and contains 1, 3, 5, 7, 9 at unit place, are called odd numbers. EX: 43, 79, 35, 39, 57
- Prime Numbers: The numbers which are only divisible by 1 and themselves are called as Prime Numbers. Ex.: 2 [Only even number which is also a prime number], 3, 5, 7, 11, 13, 17, 19, 23
- Composite Numbers: The numbers which are divisible by 1 and at least 1 more number, are called as Composite Numbers. Ex.: 4, 6,8,9,38,21
- Twin Prime numbers: Pair of prime numbers whose difference is 2, are called as twin prime numbers. Ex. [3,5] [11,13] [17.19]
- Perfect Numbers: The number whose factors’ sum is doble of the number, are called as the perfect numbers. Ex: 6 [Factors: 1, 2, 3, 6 Sum: 12 = 6*2]
Place Value and Face Value: In the number 4567, Place value of 5 is 500 [ Hundredth place] and face value is 5 [the number itself].
Rules/Test for divisibility:
- Divisibility by 2: A number is divisible by 2 if unit digit contains 0,2,4,6,8.
- Divisibility by 3: A number is divisible by 3 if sum of its digits is divisible by 3. Ex: 456789 Sum = 39, which is divisible by 3.
- Divisibility by 4: A number is divisible by 4 if last 2 digits is either 00 or divisible by 4. Ex: 500, 676
- Divisibility by 5: A number is divisible by 5 if unit digit contains 0 or 5.
- Divisibility by 6: A number is divisible by 6, if it is divisible by both 2 & 3.
- Divisibility by 7: A number is divisible by 7 if last 3 digits are divisible by 7.
- Divisibility by 8: A number is divisible by 8 if last 3 digits is either 000 or divisible by 8. Ex: 5000, 6760
- Divisibility by 9: A number is divisible by 9 if sum of its digits is divisible by 9. Ex: 5678901 [ 5+6+7+8+9+0+1 = 36, which is divisible by 9]
- Divisibility by 11: If the sum of odd place numbers and sum of even place numbers are equal or if their difference is multiple of 11, then the number is divisible by 11. Ex: 391457 [ 3 + 1 + 5 = 9; 9 + 4 + 7 = 20; Difference: 20-9 = 11, which is divisible by 11]