Operations with Real Numbers

Operations With Real Numbers

Addition and Subtraction

• Commutative Property: a + b = b + a and a - b = -(b - a)
• Associative Property: (a + b) + c = a + (b + c) and (a - b) - c = a - (b + c)
• Distributive Property: a + (b - c) = a + b - c
• The sum of two real numbers is a real number
• The difference of two real numbers is a real number

Multiplication

• Commutative Property: a × b = b × a
• Associative Property: (a × b) × c = a × (b × c)
• Distributive Property: a × (b + c) = a × b + a × c
• The product of two real numbers is a real number
• Multiplication by 0: a × 0 = 0 × a = 0

Division

• Not commutative: a ÷ b ≠ b ÷ a (except when a = b)
• Not associative: (a ÷ b) ÷ c ≠ a ÷ (b ÷ c)
• Division by 0 is undefined: a ÷ 0 is not a real number
• The quotient of two real numbers is a real number (except when dividing by 0)

Properties of Operations

• Closure: The result of any operation on real numbers is always a real number
• The order of operations (PEMDAS) applies: parentheses, exponents, multiplication and division, and addition and subtraction

Inverse Operations

• Addition and subtraction are inverse operations: a + b = c implies a = c - b and b = c - a
• Multiplication and division are inverse operations: a × b = c implies a = c ÷ b and b = c ÷ a

Operations With Real Numbers

Properties of Addition and Subtraction

• Commutative Property of Addition: order of numbers does not change the result
• Associative Property of Addition: grouping numbers does not change the result
• Distributive Property of Addition: combining like terms is allowed
• Result of adding or subtracting two real numbers is always a real number

Properties of Multiplication

• Commutative Property of Multiplication: order of numbers does not change the result
• Associative Property of Multiplication: grouping numbers does not change the result
• Distributive Property of Multiplication: multiplying a number by a sum is the same as multiplying by each addend
• Product of two real numbers is a real number
• Multiplying a number by 0 results in 0

Properties of Division

• Division is not commutative (except when numbers are equal)
• Division is not associative
• Dividing by 0 is undefined
• Result of dividing two real numbers is a real number (except when dividing by 0)

General Properties

• Closure: result of any operation on real numbers is always a real number
• Order of operations (PEMDAS) applies: parentheses, exponents, multiplication and division, and addition and subtraction

Inverse Operations

• Addition and subtraction are inverse operations: adding a number can be undone by subtracting it
• Multiplication and division are inverse operations: multiplying by a number can be undone by dividing by it

Properties of Real Numbers

Closure Property

• The set of real numbers is closed under addition, subtraction, multiplication, and division, meaning that the result of combining two real numbers using these operations is always a real number.

Commutative Property

• Real numbers follow the commutative property, meaning that the order of the numbers does not change the result of addition and multiplication:
  • Addition: a + b = b + a
  • Multiplication: a × b = b × a

Associative Property

• Real numbers follow the associative property, meaning that the order in which operations are performed does not change the result:
  • Addition: (a + b) + c = a + (b + c)
  • Multiplication: (a × b) × c = a × (b × c)

Distributive Property

• Real numbers follow the distributive property, meaning that multiplication can be distributed over addition:
  • a × (b + c) = a × b + a × c

Identities and Inverses

• Additive Identity: 0 is the additive identity, meaning a + 0 = a
• Multiplicative Identity: 1 is the multiplicative identity, meaning a × 1 = a
• Additive Inverses: every real number has an additive inverse, meaning a + (-a) = 0
• Multiplicative Inverses: every non-zero real number has a multiplicative inverse, meaning a × (1/a) = 1