Number System Formulas

Formulas of Number System:

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:

• Natural Numbers
  • All positive integers are called natural numbers. All counting numbers from 1 to infinity are natural numbers. N = {1, 2, 3, 4, 5, 6……….∞}

• Whole Numbers
  • The set of numbers that includes all natural numbers and the number zero are called whole numbers. They are also called as Non-negative integers. W = { 0,1,2,3,4,5,6,7,8,…………..∞}

• 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.

• Real Numbers
  • All numbers that can be represented on the number line are called real numbers.

• Rational Numbers
  • A rational number is defined as a number of the form a/b where ‘a’ and ‘b’ are integers and b ≠ 0. The rational numbers that are not integers will have decimal values. These values can be of two types
  • a. Terminating decimal fractions: For example: \frac{1}{5} = 0.5,\frac{125}{4} = 31.25

• • b. Non-Terminating decimal fractions: For example:\frac{19}{6} = 3.1666666, \frac{21}{9} = 2.33333

• Irrational Numbers
  •  It is a number that cannot be written as a ratio \frac{x}{y} form (or fraction). An Irrational numbers are non-terminating and non-periodic fractions. For example: \sqrt{2} = 1.414

• Complex Numbers
  • The complex numbers are the set {a+bi}, where, a and b are real numbers and ‘i’  is the imaginary unit.

• Imaginary Numbers
  • A number does not exist on the number line is called imaginary number. For example square root of negative numbers are imaginary numbers. It is denoted by ‘i’ or ‘j.

• 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 is called a prime number, if it has exactly two factors, namely 1 and the number itself.
  • For example: 2, 3, 5, 7, 11, 13, 17, etc.

• Composite numbers
  • Numbers greater than 1 which are not prime, are known as composite numbers. For example: 4, 6, 8, 10, etc.