Natural Numbers (N):
* (Positive Integers) 1,2,3,4,.............∞.
Whole Numbers (W):
* (set of all natural numbers and 0) 0,1,2,3,4,5,...........∞.
Integers (Z):
* -∞...........-3,-2,-1,0,1,2,3,4,.............∞.
Even Numbers:
* 2,4,6,8,............∞.
* Numbers are exactly divisible by 2. These numbers are end with even number or zero.
Odd Numbers:
* 1,3,5,7,.............∞.
* Numbers are not exactly divisible by 2. These numbers are end with odd number.
* Sum of any two odd number is always an even number.
* Product of any two odd numbers always an odd number.
Rational Numbers:
* Numbers which can be expressed in the form of p / q where p and q both are integers, and q is a non-zero integer.
* A number expressed in the form of p ÷ q is also called a fraction, where p is numerator and q is denominator.
* A fraction when converted into decimal, gives finite or recurring digits after decimal sign.
Ex: ¼,¾,½,
Mixed numbers:
* Numbers which consists of a whole number and a fraction.
Ex: 3¼, 5¼, 7¾, etc.
Irrational Numbers:
* Numbers which cannot be expressed in the form of p ÷ q where p and q both are integers i.e. the numbers which are not Rational.
* These numbers when converted into decimal, give infinite and non-recurring digits after decimal sign.
Ex: π, √2, √3, √25.
Real Numbers:
* Real Numbers are combination of rational and irrational numbers.
Complex or imaginary Numbers:
* These numbers are written as 'a+bi' where i²=-1.
Composite Numbers:
* The number which have more than two factors.
Ex: 4,6,8,9.
Prime Numbers:
* A number which is not divisible by any number other than 1 and itself. such numbers have only two factors, the number itself and 1.
Ex: 2,3,5,7,11,13,17,19.
* 1 is neither a prime number nor a composite number.
* Product of any two prime numbers is always a composite number, since the product is divisible by both the prime numbers.
Twin-Prime Numbers:
* Pairs of such prime numbers whose difference is 2.
Ex: 3 and 5, 11 and 13.
Co-Prime Numbers:
* Co-Prime Numbers are those numbers which are prime to each other i.e they don't have any common factor other than 1.
* Since these numbers do not have any common factor, their HCF is 1 and their LCM is equal to product of the numbers.
* Co-Prime numbers can be prime or composite numbers. Any two prime numbers are always co-prime numbers.
Ex1: 3 and 5 both numbers are prime numbers.
Ex2: 8 and 15 both numbers are composite numbers but they are prime to each other i.e. they don't have any common factor.
Face Value and Base Value:
* Face value is absolute value of the digit in a number.
* Place value or Local Value is value of a digit in relation to its position in the numbers.
Ex: Face value and place value of 9 in 12921 is 9 and 900.
BODMAS Rule:
* Any simplification of an expression may involve various operations, these operations must be performed in the following order of preference.
* B:Bracket:
If more then one type of brackets are found, they are operated in the following order.
1. First: (¯) line or bar bracket or vinculum bracket.
2. Second: ( ) small bracket.
3. Third: {} Mid Bracket.
4. Fourth: [ ] Large Bracket.
* O(of): means multiplication but is operated before division.
* D: Division.
* M: Multiplication.
* A: Addition.
* S: Subtraction.
Note:
1. if sum of two numbers a and b is x and difference between the numbers is y. then a= x+y/2 and b = x-y/2
2. if on adding 9 to a two digit number (say xy), its digits are reversed. Then units digit of the number exceeds its ten's digit by 1. similarly, if on adding 9a to a two digit number its digits are reversed, then units digit of the number exceeds its tens digit by a i.e. x+a = y if digits are reversed on subtracting 9a from the number then x-a = y Difference between two digits of the number added/subtracted ÷ 9.
3. If on adding 99a to a three digit number (say xyz), its digits are reversed, then x+a = z
If digits are reversed on subtracting 99a from the number, then x-a = z.
4. if a composite number C, can be expressed as a^m x b^n x c^p....., where a,b,c,....are prime factors and m,n,p,....are natural numbers,then total number of factors of C = (m+1)x(n+1)x(p+1)....
Total number of factors includes unity and the number itself.
* (Positive Integers) 1,2,3,4,.............∞.
Whole Numbers (W):
* (set of all natural numbers and 0) 0,1,2,3,4,5,...........∞.
Integers (Z):
* -∞...........-3,-2,-1,0,1,2,3,4,.............∞.
Even Numbers:
* 2,4,6,8,............∞.
* Numbers are exactly divisible by 2. These numbers are end with even number or zero.
Odd Numbers:
* 1,3,5,7,.............∞.
* Numbers are not exactly divisible by 2. These numbers are end with odd number.
* Sum of any two odd number is always an even number.
* Product of any two odd numbers always an odd number.
Rational Numbers:
* Numbers which can be expressed in the form of p / q where p and q both are integers, and q is a non-zero integer.
* A number expressed in the form of p ÷ q is also called a fraction, where p is numerator and q is denominator.
* A fraction when converted into decimal, gives finite or recurring digits after decimal sign.
Ex: ¼,¾,½,
Mixed numbers:
* Numbers which consists of a whole number and a fraction.
Ex: 3¼, 5¼, 7¾, etc.
Irrational Numbers:
* Numbers which cannot be expressed in the form of p ÷ q where p and q both are integers i.e. the numbers which are not Rational.
* These numbers when converted into decimal, give infinite and non-recurring digits after decimal sign.
Ex: π, √2, √3, √25.
Real Numbers:
* Real Numbers are combination of rational and irrational numbers.
Complex or imaginary Numbers:
* These numbers are written as 'a+bi' where i²=-1.
Composite Numbers:
* The number which have more than two factors.
Ex: 4,6,8,9.
Prime Numbers:
* A number which is not divisible by any number other than 1 and itself. such numbers have only two factors, the number itself and 1.
Ex: 2,3,5,7,11,13,17,19.
* 1 is neither a prime number nor a composite number.
* Product of any two prime numbers is always a composite number, since the product is divisible by both the prime numbers.
Twin-Prime Numbers:
* Pairs of such prime numbers whose difference is 2.
Ex: 3 and 5, 11 and 13.
Co-Prime Numbers:
* Co-Prime Numbers are those numbers which are prime to each other i.e they don't have any common factor other than 1.
* Since these numbers do not have any common factor, their HCF is 1 and their LCM is equal to product of the numbers.
* Co-Prime numbers can be prime or composite numbers. Any two prime numbers are always co-prime numbers.
Ex1: 3 and 5 both numbers are prime numbers.
Ex2: 8 and 15 both numbers are composite numbers but they are prime to each other i.e. they don't have any common factor.
Face Value and Base Value:
* Face value is absolute value of the digit in a number.
* Place value or Local Value is value of a digit in relation to its position in the numbers.
Ex: Face value and place value of 9 in 12921 is 9 and 900.
BODMAS Rule:
* Any simplification of an expression may involve various operations, these operations must be performed in the following order of preference.
* B:Bracket:
If more then one type of brackets are found, they are operated in the following order.
1. First: (¯) line or bar bracket or vinculum bracket.
2. Second: ( ) small bracket.
3. Third: {} Mid Bracket.
4. Fourth: [ ] Large Bracket.
* O(of): means multiplication but is operated before division.
* D: Division.
* M: Multiplication.
* A: Addition.
* S: Subtraction.
Note:
1. if sum of two numbers a and b is x and difference between the numbers is y. then a= x+y/2 and b = x-y/2
2. if on adding 9 to a two digit number (say xy), its digits are reversed. Then units digit of the number exceeds its ten's digit by 1. similarly, if on adding 9a to a two digit number its digits are reversed, then units digit of the number exceeds its tens digit by a i.e. x+a = y if digits are reversed on subtracting 9a from the number then x-a = y Difference between two digits of the number added/subtracted ÷ 9.
3. If on adding 99a to a three digit number (say xyz), its digits are reversed, then x+a = z
If digits are reversed on subtracting 99a from the number, then x-a = z.
4. if a composite number C, can be expressed as a^m x b^n x c^p....., where a,b,c,....are prime factors and m,n,p,....are natural numbers,then total number of factors of C = (m+1)x(n+1)x(p+1)....
Total number of factors includes unity and the number itself.
