Quantitative Aptitude - Number - Formula & Concept Tutorial
Formula
- (a + b)^{2 }+ (a – b)^{2} = 2(a^{2} + b^{2)}
- (a + b)^{2 }- (a – b)^{2} = 4ab
- (a² – b²) = (a – b)(a + b)
- (a + b)² = (a² + b² + 2ab)
- (a – b)² = (a² + b² – 2ab)
- (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)
- (a + b)^{3} = (a^{3} + b^{3} + 3ab(a+b))
- (a³ + b³) = (a + b)(a² – ab + b²)
- (a³ – b³) = (a – b)(a² + ab + b²)
- (a³ + b³ + c³ – 3abc) = (a + b + c)(a² + b² + c² – ab – bc – ac)
- When a + b + c = 0, then a³ + b³ + c³ = 3abc.
Types Of number
- Natural Numbers
- All positive integers are called natural numbers except 0.
- They are denoted by N. N = {1, 2, 3, 4, 5, 6……….∞}
- Whole Numbers
- All positive integers are called Whole numbers including 0.
- They are also called as Non-negative integers and denoted by W. W = { 0,1,2,3,4,5,6,7,8,…………..∞}
- Integers
- All numbers including negative and positive are called integers. Z = {∞…….-3, -2, -1, 0, 1, 2, 3……∞}
- Number including decimal is not integers.
- Rational Numbers
- A number of the form a/b where ‘a’ and ‘b’ are integers and b ≠ 0 is known as rational number.
- For Example 3/5, 7/9, 8/9, 13/15 etc.
- Irrational Numbers
- It is a number that cannot be written as a form a/b where ‘a’ and ‘b’ are integers and b ≠ 0 is known as irrational number
- For Example 2, 3, 7, etc..
- Real Numbers
- A Real numbers are the numbers which include both rational and irrational numbers.
- Imaginary Numbers
- A number satisfying the equation i^{2} = −1. Because no real number satisfies this equation, i is called an imaginary number.
- Complex Numbers
- The numbers in the form of {a+bi}, where, a and b are real numbers and ‘i’ is the imaginary number, this format is called Complex number.
- Even Numbers
- A number divisible by 2 is called an even number.
- For example: 2, 6, 8, 14, 18, 246, etc.
- Odd Numbers
- A number not divisible by 2 is called an odd number.
- For example: 3, 7, 9, 15, 17, 373, etc.
- Prime numbers
- A number greater than 1 and has exactly two factors, namely 1 and the number itself. Then the number is called prime number.
- For example: 2, 3, 5, 7, 11, 13, 17, etc.
- Composite numbers
- A numbers greater than 1 which are not prime, are known as composite numbers. For example: 4, 6, 8, 10, etc.