Number Systems in Math

Natural Numbers: Counting numbers, 1, 2, 3, ...
Whole Numbers: All natural numbers including 0, 0, 1, 2, 3, ...
Integers: All whole numbers and their negative counterparts, ..., -3, -2, -1, 0, 1, 2, 3, ...
Rational Numbers: Numbers that can be expressed as a ratio of two integers, e.g., 3/4, 22/7
Irrational Numbers: Numbers that cannot be expressed as a ratio of two integers, e.g., π, e
Real Numbers: The set of all rational and irrational numbers

Rational Numbers:
- Can be expressed as a finite decimal or a ratio of integers
- Examples: 3/4, 22/7, 0.5, 0.25
Irrational Numbers:
- Cannot be expressed as a finite decimal or a ratio of integers
- Examples: π, e, √2, √3

Properties of Real Numbers:
- Commutative Property: a + b = b + a, a × b = b × a
- Associative Property: (a + b) + c = a + (b + c), (a × b) × c = a × (b × c)
- Distributive Property: a × (b + c) = a × b + a × c
Real Number Line: A visual representation of real numbers on a number line, where each point corresponds to a unique real number

Addition:
- Closure Property: a + b is always a real number
- Commutative Property: a + b = b + a
- Associative Property: (a + b) + c = a + (b + c)
Subtraction:
- Not commutative, a - b ≠ b - a
- Not associative, (a - b) - c ≠ a - (b - c)
Multiplication:
- Closure Property: a × b is always a real number
- Commutative Property: a × b = b × a
- Associative Property: (a × b) × c = a × (b × c)
Division:
- Not defined for division by zero
- Not commutative, a ÷ b ≠ b ÷ a
- Not associative, (a ÷ b) ÷ c ≠ a ÷ (b ÷ c)

Natural numbers are counting numbers starting from 1
Whole numbers include all natural numbers and 0
Integers include all whole numbers and their negative counterparts
Rational numbers can be expressed as a ratio of two integers, such as 3/4
Irrational numbers cannot be expressed as a ratio of two integers, such as π and e
Real numbers are the set of all rational and irrational numbers

Rational numbers can be expressed as a finite decimal or a ratio of integers, with examples including 3/4, 22/7, 0.5, and 0.25
Irrational numbers cannot be expressed as a finite decimal or a ratio of integers, with examples including π, e, √2, and √3

The commutative property of real numbers states that a + b = b + a and a × b = b × a
The associative property of real numbers states that (a + b) + c = a + (b + c) and (a × b) × c = a × (b × c)
The distributive property of real numbers states that a × (b + c) = a × b + a × c
The real number line is a visual representation of real numbers, where each point corresponds to a unique real number

Addition of real numbers has the closure property, meaning a + b is always a real number
Addition of real numbers has the commutative property, meaning a + b = b + a
Addition of real numbers has the associative property, meaning (a + b) + c = a + (b + c)
Subtraction of real numbers is not commutative, meaning a - b ≠ b - a
Subtraction of real numbers is not associative, meaning (a - b) - c ≠ a - (b - c)
Multiplication of real numbers has the closure property, meaning a × b is always a real number
Multiplication of real numbers has the commutative property, meaning a × b = b × a
Multiplication of real numbers has the associative property, meaning (a × b) × c = a × (b × c)
Division of real numbers is not defined for division by zero
Division of real numbers is not commutative, meaning a ÷ b ≠ b ÷ a
Division of real numbers is not associative, meaning (a ÷ b) ÷ c ≠ a ÷ (b ÷ c)

Natural Numbers: start from 1 and go on indefinitely (1, 2, 3,...).
Whole Numbers: include 0 and all natural numbers (0, 1, 2, 3,...).
Integers: include all whole numbers and their negative counterparts (...,-3, -2, -1, 0, 1, 2, 3,...).
Rational Numbers: can be expressed as the ratio of two integers (e.g., 3/4, 22/7).
Irrational Numbers: cannot be expressed as the ratio of two integers (e.g., π, e).
Real Numbers: include all rational and irrational numbers.

Commutative Property: the order of numbers does not change the result of an operation (e.g., a + b = b + a).
Associative Property: the order in which numbers are grouped does not change the result of an operation (e.g., (a + b) + c = a + (b + c)).
Distributive Property: the multiplication of a number with the sum of two numbers is the same as the sum of the multiplication of the number with each of the two numbers (e.g., a(b + c) = ab + ac).

Addition: the sum of two or more real numbers is always a real number.
Subtraction: the difference of two real numbers is always a real number.
Multiplication: the product of two or more real numbers is always a real number.
Division: the division of two real numbers is always a real number, except when the divisor is zero.

Closure Property: the result of an operation on two real numbers is always a real number.
Existence of Additive Identity: there exists a real number, 0, such that a + 0 = a for every real number a.
Existence of Multiplicative Identity: there exists a real number, 1, such that a × 1 = a for every real number a.