[해석학] Field, Axiom, 실수집합R

 

 

Field

 

• A field is a set F with two operations, called addition and multiplication, which satisfy the following so-called "field axioms"

tAxioms for addition by Principles of mathematical analysis, by W. Rudin, 3rd edition

 

 

Axioms for multiplication by Principles of mathematical analysis, by W. Rudin, 3rd edition

 

 

Distributive law by Principles of mathematical analysis, by W. Rudin, 3rd edition

 

example) 1) Is N a field? No

                 2) Is Z a field? No

                 3) Is Q a field? Yes

                 4) Is R a field? Yes

                 5) Is C a field? Yes

 

 

 

 

 

 

• An ordered field is a field F which is also an ordered set, such that

ordered field property by Principles of mathematical analysis, by W. Rudin, 3rd edition

example) 1) Is Q an ordered field? Yes

                 4) Is R an ordered field? Yes

                 5) Is C an ordered field? No (i*i = -1)

 

 

 

 

 

 

Real number, R

Let F be an ordered field. S⊂F is a subfield of F if S is an ordered field with the same operation "+", "x" and order "<" as F

 

• Theorem: There exists an ordered field R such that 
• 1) R has the least-upper-bound property (largest-lower-bound)
• 2) Q is a subfield of R

 

• Further property of R
• 1) For any x in R, x>0 and any n in N, there exists a unique y in R such that y^n = x.
• 2) For all a,b in R, a,b>0 then (ab)^1/n = a^1/n * a^1/n
• 3) Decimal representations of real numbers: ex) 1=0.9999....

 

 

 

 

 

 

 

Extended real number system

¯R = R U {-∞, +∞}: Some operations and volations with new elements:

 

1. -∞ < x < +∞, for all x in R
2. x+∞ = +∞, x-∞ = -∞, for all x in R
3. x>0 => x(+∞) = +∞, x(-∞) = -∞, and x<0 => x(+∞) = -∞, x(-∞) = +∞
4. x in R, x is not 0, => x/+∞ = x/-∞ = 0

¯R is not a field because +∞-∞, ∞/∞ are not defined!