Number System Formulas - The Logic Tutorial

Formulas of Number System

Mathematical Formulas

1. 1 + 2 + 3 + 4 + 5 + … + n = n(n + 1)/2
2. (1² + 2² + 3² + ….. + n²) = n ( n + 1 ) (2n + 1)/6
3. (1³ + 2³ + 3³ + ….. + n³) = (n(n + 1)/2)²
4. Entirety of first n odd numbers = n²
5. Entirety of first n even numbers = n(n + 1)

Mathematical Formulas to solve questions

1. (a + b)(a – b) = (a² – b²)
2. (a + b)² = (a² + b² + 2ab)
3. (a – b)² = (a² + b² – 2ab)
4. (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)
5. (a³ + b³) = (a + b)(a² – ab + b²)
6. (a³ – b³) = (a – b)(a² + ab + b²)
7. (a³ + b³ + c³ – 3abc) = (a + b + c)(a² + b² + c² – ab – bc – ac)
8. when a + b + c = 0, then a³ + b³ + c³ = 3abc

Types of Number System

1. Natural Numbers: All positive integers are called natural numbers. N = {1, 2, 3, 4, 5, 6……….∞}
2. Whole Numbers: The set of numbers that includes all natural numbers and the number zero are called whole numbers. W = { 0,1,2,3,4,5,6,7,8,…………..∞}
3. Integers: All numbers that do not have the decimal places in them are called integers. Z = {∞…….-3, -2, -1, 0, 1, 2, 3………∞}
• a. Positive Integers: 1, 2, 3, 4….. is the set of all positive integers.
• b. Negative Integers: −1, −2, −3….. is the set of all negative integers.
• c. Non-Positive and Non-Negative Integers: 0 is neither positive nor negative.
4. Real Numbers: All numbers that can be represented on the number line are called real numbers.
5. Rational Numbers: A rational number is defined as a number of the form a/b where ‘a’ and ‘b’ are integers and b ≠ 0.
6. Irrational Numbers: It is a number that cannot be written as a ratio form (or fraction).
7. Complex Numbers: The complex numbers are the set {a+bi}, where, a and b are real numbers and ‘i’ is the imaginary unit.
8. Imaginary Numbers: A number does not exist on the number line is called imaginary number.
9. Even Numbers: A number divisible by 2 is called an even number.
10. Odd Numbers: A number not divisible by 2 is called an odd number.
11. Prime Numbers: A number greater than 1 is called a prime number, if it has exactly two factors, namely 1 and the number itself.
12. Composite Numbers: Numbers greater than 1 which are not prime, are known as composite numbers.

Formulas for finding the Squares of a number

1. Squares of numbers between 91-100:
   • 97² = 9409
   • 91² = 8281
2. Squares of numbers between 100-109:
   • 102² = 10404
   • 107² = 11449
3. Squares of numbers between 51-60:
   • 53² = 2809
   • 57² = 3249