A number system is a system of writing numbers. It defines how numbers are expressed and understood. The most commonly used number system is the decimal system, which is based on ten digits (0-9). However, there are different types of number systems, such as:

Types of Number Systems:

1. Natural Numbers (N):
   • Definition: The set of counting numbers that starts from 1, 2, 3, and so on.
   • Symbol: N
   • Examples: {1,2,3,4,… }
   • Key Point: Natural numbers do not include 0 or negative numbers.
2. Whole Numbers (W):
   • Definition: The set of natural numbers along with 0.
   • Symbol: W
   • Examples: {0,1,2,3,… }
   • Key Point: Whole numbers are non-negative.
3. Integers (Z):
   • Definition: The set of all positive and negative numbers, including zero.
   • Symbol: Z (from the German word “Zahlen,” meaning numbers)
   • Examples: {−3,−2,−1,0,1,2,3,… } 
   • Key Point: Integers can be positive, negative, or zero, but they do not include fractions or decimals.
4. Rational Numbers (Q):
   • Definition: Numbers that can be expressed as a fraction of two integers, where the denominator is not zero.
   • Symbol: Q
   • Examples:
   • Key Point: Every integer is a rational number since it can be expressed as  ​. Rational numbers can be terminating or repeating decimals.
5. Irrational Numbers (I):
   • Definition: Numbers that cannot be expressed as a simple fraction or ratio of two integers. Their decimal expansions are non-terminating and non-repeating.
   • Symbol: I
   • Examples: π(3.14159…),  (1.4142…) 
   • Key Point: Irrational numbers cannot be written in fraction form.
6. Real Numbers (R):
   • Definition: The set of all rational and irrational numbers combined. Essentially, any number that can be placed on the number line.
   • Symbol: RRR
   • Examples: 2, −5, 0.75, π,
   • Key Point: Real numbers include natural numbers, whole numbers, integers, rational numbers, and irrational numbers.

Properties of Numbers:

1. Even and Odd Numbers:
   • Even: Numbers divisible by 2 (e.g., 2, 4, 6).
   • Odd: Numbers not divisible by 2 (e.g., 1, 3, 5).
2. Prime and Composite Numbers:
   • Prime Numbers: Numbers greater than 1 that have no divisors other than 1 and themselves (e.g., 2, 3, 5, 7).
   • Composite Numbers: Numbers that have more than two divisors (e.g., 4, 6, 8).
3. Significant Numbers:
   • Zero (0): Neither positive nor negative, it plays a critical role in place-value systems.
   • One (1): The multiplicative identity, meaning any number multiplied by 1 remains unchanged.
4. Co-prime numbers (also called relatively prime or mutually prime numbers) are two or more numbers that have no common factors other than 1. In other words, their greatest common divisor (GCD) is 1.

Or

Two numbers are said to be co-prime if the only positive integer that divides both of them is 1.

Example: Numbers 8 and 15 are co-prime because their factors are:

• Factors of 8: 1,2,4,81, 2, 4, 81,2,4,8
• Factors of 15: 1,3,5,151, 3, 5, 151,3,5,15
• The only common factor is 1, so 8 and 15 are co-prime.

Important Points:

1. Any two prime numbers are always co-prime. For example, 3 and 7 are prime and have no common factors other than 1.
2. 1 is co-prime with every number. This is because the only divisor of 1 is 1, which makes it co-prime with any other number.
3. Two consecutive numbers are always co-prime. For example, 14 and 15, or 101 and 102, are co-prime because consecutive numbers do not share any factors other than 1.