Real Numbers- Axioms Click Here for a quick overview of Real numbers before proceeding with this note!!! :)

﻿Field Axioms

 i. Closure Property : The sum of real number is always real value.i.e., If {a, b} ϵ R then a+b ϵ R ii. Commutative Property : The sum of two real numbers is the same whatever order they are added.i.e., If {a, b} ϵ R then a + b = b + a iii. Associative Property : The sum of any two of the three real numbers with the third will be the same.﻿i.e., if ﻿{a, b, c} ϵ R then, a+(b+c)=(a+b)+ c iv. Additive Identity : If a ϵ R, then a+ 0 = 0 + a =aHere, 0 is the additive identity. v. Additive Inverse : For every a ϵ R, there exists -a ϵ R such that:a+(-a) = (-a) +a = 0Here, -a is the additive inverse of a

﻿2. Multiplicative Axiom

 i. Closure Property : The product of real number is always real value.i.e., If {a, b} ϵ R then a.b ϵ R ii. Commutative Property : The product of two real numbers is the same whatever order they are multiplied.i.e., If {a, b} ϵ R then a.b = b.a iii. Associative Property : The product of any two of the three real numbers with the third will be the same.﻿i.e., if ﻿{a, b, c} ϵ R then, a.(bc)=(ab). c iv. Multiplicative Identity : If a ϵ R, then a.1 = 1.a =aHere, 1 is the multiplicative identity. v. Multiplicative Inverse : For every aϵ R(a≠0), there exists a-1 ϵ Rsuch that:a. a-1 = a-1. a = 1Here, a-1 is the multiplicative inverse of a vi. Distributive Property : If {a, b, c} ϵ R, thena(b+c)=ab+bc

﻿Order Axioms
﻿i. If {a, b} ϵ R such that a>0 and b>0 then ab>0.
﻿ii. If {a, b} ϵ R then one and only one of the following relation holds
﻿a<b,     a=b,     a>b
iii. If {a, b, c} ϵ R such that a>b and b>c then a>c.(same is the case for '<')
iv.
If {a, b, c} ϵ R such that a>b then a+c > b+c.(same is the case for '<')
v.
{a, b, c} ϵ R
→If a>b            ﻿then             ﻿ac>bc            ﻿when c>0
→If a>b            ﻿then             ﻿ac<bc            ﻿when c<0
→If a>b            ﻿then             ﻿a/c>b/c        ﻿when c>0
→If a>b            ﻿then             ﻿a/c<b/c        ﻿when c<0
(same is the case for '<')
vi.
If {a, b} ϵ R such that a<b, then there exists a real number such that: a<c<b