Post by Ryan on Jun 4, 2011 22:48:55 GMT -5
First:
Axioms of real numbers (R)
R is closed under addition - if a and b are real numbers and a+b=c then c is a real number
R has an additive identity - there exists a real number I such that if a is a real number, a+I=a
R has additive inverses - for each real number a, other than I, there exists -a such that a+(-a) = I
R is associative under addition - if a,b, and c are in R then a+(b+c) = (a+b)+c.
R is closed under multiplication - if a and b are real numbers and a*b=c then c is a real number
R has a multiplicative identity - there exists a real number i such that if a is a real number, a*i=a
R has multiplicative inverses - for each real number, other than I, there exists 1/a such that a*(1/a)=i (note this I/i switch - I has no multiplicative identity).
R is associative under multiplication - if a,b, and c are in R then a*(b*c) = (a*b)*c
R is ordered - there exists total order so that for real numbers x,y, and z, if x>=y then x+z>=y+z and if x>=0, y>=0, x*y>=0
R is complete - for any bounded subset of R, there exists a 1) supremum or least upper bound if bounded above, and/or a 2) infimum or greatest lower bound if bounded below.
Anyways, we don't really need most of these, but after this we must build the real numbers!
Let us start with 0 and 1. We will let 0 be the additive identity and 1 be the multiplicative identity. Since the 1 is in there we get -1 so that 1+(-1)=0. From this we get all the integers (positive and negative whole numbers) by adding and subtracting 1. So 4 is a real number, as is 2.
Not only this, but we also get that 1+1=2 and that 3+1 = 4 and that 2+1 = 3. (built from 0 and 1 - 1+1 is called 2 so 1+1=2 1+1+1 is called 3 so 1+1+1 = 3 but 1+1=2 so (1+1)+1 = 3 thus 2+1 = 3. 1+1+1+1 is called 4 so 1+1+1+1= 4 but 1+1+1=3 so 1+1+1+1=3+1).
From this it is easy to see that 1+1+1+1 also called 4 is really (1+1)+(1+1). This grouping is allowed due the the associativity of real numbers under addition. 1+1 is called 2 so (1+1)+(1+1)=2+2=4.
Therefore
2+2=4
Proved from the axioms of real numbers.
This post is by request. I need some practice proving things anyway.
Axioms of real numbers (R)
R is closed under addition - if a and b are real numbers and a+b=c then c is a real number
R has an additive identity - there exists a real number I such that if a is a real number, a+I=a
R has additive inverses - for each real number a, other than I, there exists -a such that a+(-a) = I
R is associative under addition - if a,b, and c are in R then a+(b+c) = (a+b)+c.
R is closed under multiplication - if a and b are real numbers and a*b=c then c is a real number
R has a multiplicative identity - there exists a real number i such that if a is a real number, a*i=a
R has multiplicative inverses - for each real number, other than I, there exists 1/a such that a*(1/a)=i (note this I/i switch - I has no multiplicative identity).
R is associative under multiplication - if a,b, and c are in R then a*(b*c) = (a*b)*c
R is ordered - there exists total order so that for real numbers x,y, and z, if x>=y then x+z>=y+z and if x>=0, y>=0, x*y>=0
R is complete - for any bounded subset of R, there exists a 1) supremum or least upper bound if bounded above, and/or a 2) infimum or greatest lower bound if bounded below.
Anyways, we don't really need most of these, but after this we must build the real numbers!
Let us start with 0 and 1. We will let 0 be the additive identity and 1 be the multiplicative identity. Since the 1 is in there we get -1 so that 1+(-1)=0. From this we get all the integers (positive and negative whole numbers) by adding and subtracting 1. So 4 is a real number, as is 2.
Not only this, but we also get that 1+1=2 and that 3+1 = 4 and that 2+1 = 3. (built from 0 and 1 - 1+1 is called 2 so 1+1=2 1+1+1 is called 3 so 1+1+1 = 3 but 1+1=2 so (1+1)+1 = 3 thus 2+1 = 3. 1+1+1+1 is called 4 so 1+1+1+1= 4 but 1+1+1=3 so 1+1+1+1=3+1).
From this it is easy to see that 1+1+1+1 also called 4 is really (1+1)+(1+1). This grouping is allowed due the the associativity of real numbers under addition. 1+1 is called 2 so (1+1)+(1+1)=2+2=4.
Therefore
2+2=4
Proved from the axioms of real numbers.
This post is by request. I need some practice proving things anyway.
