Number Theory Fundamentals

Fundamental Theorem of Arithmetic

• Every positive integer can be expressed as a product of prime numbers in a unique way, except for the order in which the prime numbers are listed.
• This theorem provides a way to write a positive integer as a product of prime numbers, known as the prime factorization of the integer.
• The prime factorization of a positive integer is unique, up to the order of the prime numbers.

Irrational Numbers

• An irrational number is a real number that cannot be expressed as a finite decimal or fraction.
• Irrational numbers have an infinite number of digits that never repeat in a predictable pattern.
• Examples of irrational numbers include:
  • Pi (π)
  • Euler's number (e)
  • The square root of 2 (√2)
  • The square root of 3 (√3)

Properties of Real Numbers

• Commutative Property of Addition: The order in which real numbers are added does not change the result.
  • a + b = b + a
• Commutative Property of Multiplication: The order in which real numbers are multiplied does not change the result.
  • a × b = b × a
• Associative Property of Addition: The order in which real numbers are added does not change the result, even when more than two numbers are involved.
  • (a + b) + c = a + (b + c)
• Associative Property of Multiplication: The order in which real numbers are multiplied does not change the result, even when more than two numbers are involved.
  • (a × b) × c = a × (b × c)
• Distributive Property: The product of a real number and the sum of two real numbers is equal to the sum of the products of the real number and each of the two real numbers.
  • a × (b + c) = a × b + a × c

Fundamental Theorem of Arithmetic

• Every positive integer has a unique prime factorization, excluding the order of prime numbers.
• The prime factorization of a positive integer is unique, up to the order of the prime numbers.

Irrational Numbers

• An irrational number is a real number that cannot be expressed as a finite decimal or fraction.
• Irrational numbers have an infinite number of digits that never repeat in a predictable pattern.
• Examples of irrational numbers include Pi (π), Euler's number (e), the square root of 2 (√2), and the square root of 3 (√3).

Properties of Real Numbers

• The commutative property of addition states that the order of real numbers being added does not change the result: a + b = b + a.
• The commutative property of multiplication states that the order of real numbers being multiplied does not change the result: a × b = b × a.
• The associative property of addition states that the order of real numbers being added does not change the result, even when more than two numbers are involved: (a + b) + c = a + (b + c).
• The associative property of multiplication states that the order of real numbers being multiplied does not change the result, even when more than two numbers are involved: (a × b) × c = a × (b × c).
• The distributive property states that the product of a real number and the sum of two real numbers is equal to the sum of the products of the real number and each of the two real numbers: a × (b + c) = a × b + a × c.

Fundamental Theorem of Arithmetic

• Every positive integer can be expressed as a product of prime numbers in a unique way, except for the order in which they are listed.
• Example: 12 = 2 × 2 × 3 (unique prime factorization)

Irrational Numbers

• A real number that cannot be expressed as a finite decimal or fraction (ratio of integers).
• Examples: π (pi), e (Euler's number), √2 (square root of 2)
• Irrational numbers are non-terminating and non-repeating decimals.
• Irrational numbers cannot be expressed as a simple fraction (numerator and denominator are integers).

Properties of Real Numbers

Closure Properties

• The sum of two real numbers is always a real number.
• The product of two real numbers is always a real number.

Commutative Properties

• a + b = b + a (addition)
• a × b = b × a (multiplication)

Associative Properties

• (a + b) + c = a + (b + c) (addition)
• (a × b) × c = a × (b × c) (multiplication)

Distributive Property

• a × (b + c) = a × b + a × c

Existence of Identities

• Additive identity: 0 (a + 0 = a)
• Multiplicative identity: 1 (a × 1 = a)

Existence of Additive Inverses

• For each real number a, there exists a number -a such that a + (-a) = 0