Properties of Real Numbers

Closure Law

• If $a, b, c \in \mathbb{R}$a,b,c∈R, then:

  • $a + b \in \mathbb{R}$a+b∈R

  • $a \cdot b \in \mathbb{R}$a⋅b∈R

• The set of real numbers is closed under addition and multiplication.

Commutative Laws

• Addition: $a + b = b + a$a+b=b+a

• Multiplication: $a \cdot b = b \cdot a$a⋅b=b⋅a

• The order of addition or multiplication does not change the result.

Associative Laws

• Addition: $a + (b + c) = (a + b) + c$a+(b+c)=(a+b)+c

• Multiplication: $a \cdot (b \cdot c) = (a \cdot b) \cdot c$a⋅(b⋅c)=(a⋅b)⋅c

• Grouping of numbers does not affect the sum or product.

Distributive Law

• $a(b + c) = ab + ac$a(b+c)=ab+ac

• Multiplication distributes over addition.

Identity Elements

• Additive Identity: $a + 0 = 0 + a = a$a+0=0+a=a (0 is the identity for addition)

• Multiplicative Identity: $a \cdot 1 = 1 \cdot a = a$a⋅1=1⋅a=a (1 is the identity for multiplication)

Inverse Elements

• Additive Inverse: For any $a$a, there exists $-a$−a such that:
  $a + (-a) = 0$a+(−a)=0

• Multiplicative Inverse: For any $a \neq 0$a=0, there exists $a^{-1}$a−1 (or $\frac{1}{a}$a1​) such that:
  $a \cdot a^{-1} = 1$a⋅a−1=1