Properties of Real Numbers

Closure Law

• If 

  $a,b,c\in \mathbb{R}$

  then:

  • $a+b\in \mathbb{R}$
  • $a\cdot b\in \mathbb{R}$

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

Commutative Laws

• Addition: 

  $a+b=b+a$

• Multiplication: 

  $a\cdot b=b\cdot a$

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

Associative Laws

• Addition: 

  $a+(b+c)=(a+b)+c$

• Multiplication: 

  $a\cdot (b\cdot c)=(a\cdot b)\cdot c$

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

Distributive Law

• $a(b+c)=ab+ac$

• Multiplication distributes over addition.

Identity Elements

• Additive Identity: 

  $a+0=0+a=a$

  (0 is the identity for addition)

• Multiplicative Identity: 

  $a\cdot 1=1\cdot a=a$

  (1 is the identity for multiplication)

Inverse Elements

• Additive Inverse: For any

  a, there exists

  −a such that:

  $a+(-a)=0$

• Multiplicative Inverse: For any 

  $a\mathrm{\ne }0$

   there exists 

  $a^{-1}$

  or 

  $\frac{1}{a}$

  such that:

  $a\cdot a^{-1}=1$