Numbers Introduction
Natural Number : Counting numbers 1,2,3,4,... are known as Natural Number.
The set of natural numbers can be represented by N = {1,2,3,4,...}
Whole Number : Natural Numbers including 0 is known as Whole number.
- The set of whole numbers can be represented by W = {0,1,2,3,4,...}
- Every natural number is a whole number.
- All whole number is a natural number excluding 0.
Integers : All Counting numbers and their negative including 0 are known as Integers.
The set of integers can be represented by I = {...,-3,-2,-1,0,1,2,3,4,...}
Positive Integers : The set {1,2,3,4,....} is a set of all positive integers.
Positive integers and natural numbers are synonyms.
Negative Integers : The set {-1,-2,-3,-4,....} is a set of all negative integers.
Non-positive and Non-negative Integers : 0 is neither positive and nor negative integers.
Even Number : An even number is a number that is exactly divisible by 2.
Examples : 2,4,6,8,144
Odd Number : An odd number is a number that is not exactly divisible by 2.
Examples : 1,3,5,7,9
Note : 0 is an even number.
Prime Number : A number is greater than 1 and which is divisible by 1 and itself is called Prime number.
Examples : 2,3,5,7,11,13
Note : 2 is only the prime number which is an even and other than 2 are odd prime numbers.
Composite Number : Natural numbers greater than 1 which are not prime are called composite numbers
Examples : 4,6,9,15
Co-Prime Number : Two numbers are Co-prime numbers, if their HCF is 1 is called Co-prime numbers.
Examples : (2,3),(4,5),(8,11)
Rational Number : The numbers of the for a/b, where a and b are integers and b ≠ 0
Examples : 2/5,4/3
Irrational Number : An Irrational Number is a real number that cannot be written as a simple fraction. In others words, Irrational numbers which when expressed in decimal form neither terminating nor repeating decimals.
Examples : √2 ,√5,Π
Algebraic Formulas
- $(a+b)^2=a^2+2ab+b^2$
- $(a-b)^2=a^2-2ab+b^2$
- $(a-b)(a+b)=a^2-b^2$
- $(a-b)^2+(a+b)^2=2(a^2+b^2)$
- $(a-b)^2-(a+b)^2=4ab$
- $(a+b)^3$ $=a^3+b^3+3a^2b+3ab^2$ $= a^3+b^3+3ab(a+b)$
- $(a-b)^3$ $= a^3-b^3-3a^2b+3ab^2$ $= a^3+b^3-3ab(a-b)$
- $a^3+b^3$ $= (a+b)(a^2-ab+b^2)$
- $a^3-b^3$ $= (a-b)(a^2+ab+b^2)$
- $a^3+b^3+c^3-3abc$ $=(a+b+c)$ $(a^2+b^2+c^2-ab-bc-ca)$