Number System in Mathematics

In mathematics, the Number System is the study of different types of numbers, the operations performed on them, and their interrelationships. Let’s understand it step-by-step:

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🔹 Types of Numbers

1. Natural Numbers (N)
   1. Used for counting.
   1. Only positive numbers.
   1. Example: 1, 2, 3, 4, 5, …
   1. ⚡ Note: 0 is not a natural number.
2. Whole Numbers (W)
   1. Natural numbers including 0.
   1. Example: 0, 1, 2, 3, 4, 5, …
3. Integers (Z or T)
   1. Whole numbers and their negatives.
   1. Example: …, -3, -2, -1, 0, 1, 2, 3, …
   1. Classifications:
      1. Positive Integers: 1, 2, 3, …
      1. Negative Integers: -1, -2, -3, …
      1. Zero is neither positive nor negative.
4. Even Numbers
   1. Divisible by 2 without remainder.
   1. Example: 2, 4, 6, 8, …
5. Odd Numbers
   1. Not completely divisible by 2.
   1. Example: 1, 3, 5, 7, …
6. Prime Numbers
   1. Exactly two distinct divisors: 1 and the number itself.
   1. Example: 2, 3, 5, 7, 11, …
   1. ⚡ Note: 2 is the only even prime number.
7. Composite Numbers
   1. More than two divisors.
   1. Example: 4, 6, 8, 9, 12, …
8. Co-Prime Numbers
   1. Two numbers whose only common divisor is 1.
   1. Example: 15 and 16.
9. Rational Numbers (Q)
   1. Expressed in the form pq\frac{p}{q}, where pp and qq are integers and q≠0q \neq 0.
   1. Example: 35\frac{3}{5}, −72\frac{-7}{2}, 4 (since 4 = 41\frac{4}{1}).
10. Irrational Numbers
    1. Cannot be expressed as pq\frac{p}{q}.
    1. Example: 2\sqrt{2}, 5\sqrt{5}, π\pi.
11. Real Numbers (R)
    1. Combination of Rational and Irrational numbers.
    1. Example: 3¾, π\pi, 5\sqrt{5}.

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🔹 Fundamental Operations

1. Addition (+)
   1. Combining two or more numbers into one.

Formulas:

• Sum of first nn natural numbers:

Sum=n(n+1)2\text{Sum} = \frac{n(n + 1)}{2}

• Sum of squares of first nn natural numbers:

Sum of squares=n(n+1)(2n+1)6\text{Sum of squares} = \frac{n(n + 1)(2n + 1)}{6}

• Sum of cubes of first nn natural numbers:

Sum of cubes=(n(n+1)2)2\text{Sum of cubes} = \left( \frac{n(n + 1)}{2} \right)^2

• Subtraction (−)
  • Taking one number away from another.
• Multiplication (×)
  • Repeated addition of a number.
• Division (÷)
  • Splitting a number into equal parts.

Formula:

Dividend=(Divisor×Quotient)+Remainder\text{Dividend} = (\text{Divisor} × \text{Quotient}) + \text{Remainder}

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🔹 VBODMAS Rule

Order of operations to be followed in solving expressions:

• V = Vinculum (Bars: | |)
• B = Brackets (Parentheses: ( ), [ ], { })
• O = Orders (Powers and Roots)
• D = Division (÷)
• M = Multiplication (×)
• A = Addition (+)
• S = Subtraction (−)

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🔹 Fractions

1. Simple Fraction
   1. A number expressed as pq\frac{p}{q}, where q≠0q \neq 0.
2. Reduced Fraction
   1. A fraction with no common factor between numerator and denominator except 1.
   1. Example: 34\frac{3}{4}, 59\frac{5}{9}.
3. Proper and Improper Fractions
   1. Proper Fraction: Numerator < Denominator (e.g., 35\frac{3}{5}).
   1. Improper Fraction: Numerator > Denominator (e.g., 94\frac{9}{4}).
4. Mixed Fraction
   1. A whole number combined with a proper fraction.
   1. Example: 4¾.
5. Reciprocal Fraction
   1. Interchanging the numerator and denominator of a fraction.
   1. Example: Reciprocal of 511\frac{5}{11} is 115\frac{11}{5}.