Number

Real analysis begins with a rigorous construction of numbers. We have several key sets of numbers:

• Natural number
• Integers
• Rational number
• Real number

There are nine key properties of numbers:

• Associative law for addition: $a+(b+c)=(a+b)+c$
• Existence of an additive identity: $a+0=0+a=a$
• Existence of additive inverses: $a+(-a)=(-a)+a=0$
• Commutative law for addition: $a+b=b+a$
• Associative law for multiplication: $a\cdot (b\cdot c)=(a\cdot b)\cdot c$
• Existence of a multiplicative identity: $a\cdot 1=1\cdot a=a$, $1\ne 0$
• Existence of multiplicative inverses: $a\cdot a^{-1}=a^{-1}\cdot a=1$, for $a\ne 0$
• Commutative law for multiplication: $a\cdot b=b\cdot a$
• Distributivity: $a\cdot (b+c)=a\cdot b+a\cdot c$