In a

previous blog, I showed how the Dedekind cut could be used to define the real numbers.

In today's blog, I will show that the real numbers form a field.

Lemma 1: The real numbers are closed on addition.Proof:

(1) We can define the real numbers based on a Dedekind cut. [see Definition 2,

here]

(2) Let

x,y be the real numbers.

(3) From the definition of the Dedekind cut,

x is the set of rational numbers that are less than

x and

y is the set of rational numbers that are less than

y.

(4)

x+y is defined as the set of rational numbers in

x added to the set of rational numbers in

y so that

x+y is the set of all of possible sums.

(5) Since the rational numbers are closed on addition [see Lemma 2,

here], it follows that

x+y is also closed on addition.

QED

Lemma 2: The real numbers are closed on multiplicationProof:

(1) We can define the real numbers based on a Dedekind cut. [see Definition 2,

here]

(2) Let

x,y be the real numbers.

(3) From the definition of the Dedekind cut,

x is the set of rational numbers that are less than

x and

y is the set of rational numbers that are less than

y.

(4)

xy is defined as the set of rational numbers in

x multiplied to the set of rational numbers in

y so that

xy is the set of all of possible products.

(5) Since the rational numbers are closed on multiplication [see Lemma 3,

here], it follows that

xy is also closed on multiplication.

QED

Lemma 3: The set of real numbers support the commutative rule for additionProof:

(1) By the definition of addition for real numbers, addition follows the properties of the set of rationals. [see Definition 5,

here]

(2) So, the conclusion follows from the fact that the rational numbers support the commutative rule for addition. [see Lemma 4,

here]

QED

Lemma 4: The set of real numbers support the associative rule for additionProof:

(1) By the definition of addition for real numbers, addition follows the properties of the set of rationals. [see Definition 5,

here]

(2) So, the conclusion follows from the fact that the rational numbers support the associative rule for addition. [see Lemma 5,

here]

QED

Lemma 5: The set of real numbers support the commutative rule for multiplication.Proof:

(1) By the definition of multiplication for real numbers, multiplication follows the properties of the set of rationals. [see Definition 7,

here]

(2) So, the conclusion follows from the fact that the rational numbers support the commutative rule for multiplication. [see Lemma 11,

here]

QED

Lemma 6: The set of real numbers support the associative rule for multiplicationProof:

(1) By the definition of multiplication for real numbers, multiplication follows the properties of the set of rationals. [see Definition 7,

here]

(2) So, the conclusion follows from the fact that the rational numbers support the associative rule for multiplication. [see Lemma 8,

here]

QED

Lemma 7: The set of real numbers support the distributive ruleProof:

(1) The properties of multiplication of reals is based on the properties of rational numbers [see Definition 7,

here] and the properties of addition of reals is based on the properties of rational numbers [see Definition 5,

here].

(2) So, the conclusion follows from the fact that the rational numbers support the associative rule for multiplication. [see Lemma 10,

here]

QED

Lemma 8: The set of real numbers have an additive identityProof:

(1) The additive identity is the set of all rational numbers less than

0.

(2) Let

x be any real number.

(3) The

x+0 be the set of all rational numbers less than

x added to all rational numbers less than

0.

(4) Let

a be any rational number less than

x.

(5) Let

b be any rational number less than

0 so that be must be negative.

(6)

a + b is thus less than

a which is less than

x.

(7) Since

b can be as close to

0 as we want,

a+b can be as close to

x as we want.

QED

Lemma 9: The real numbers support an additive inverse propertyProof:

(1) Let

x be any real number

(2) Let

-x be the set of rational numbers that are less than

-x.

(3) Using the definition for addition (see Definition 5,

here):

x+-x is the set of all rational numbers less than

0.

QED

Lemma 10: The set of real numbers supports a multiplicative identity propertyProof:

(1) Let

x be any real number

(2) The multiplicative inverse is

1 which is the set of rational numbers less than

1.

(3) It is clear that

x*1 = {the set of rational numbers less than

x }

= x.

QED

Lemma 11: The set of real numbers supports a multiplicative inverse propertyProof:

(1) Let

x = be any nonzero real number

(2) The multiplicative inverse is

1/x(3) This is clear since the set defined by

x*1/x is the set of all rational numbers less than

1.

QED

Theorem 12: The real numbers form a field(1) The real numbers are closed on addition [see Lemma 1 above] and multiplication [see Lemma 2 above].

(2) The real numbers support the commutative property of addition [see Lemma 3 above], the associative property of addition [see Lemma 4 above], the commutative property of multiplication [see Lemma 5 above], an associative property of multiplication [see Lemma 6 above], and a distributive property [see Lemma 7 above].

(3) The set of real numbers has an additive identity property [see Lemma 8 above], an additive inverse property [see Lemma 9 above], a multiplicative identity property [see Lemma 10 above], and a multiplicative inverse property [see Lemma 11 above].

(4) From all these properties, the real numbers form a field. [see Definition 3,

here]

QED