The Set of Real Numbers

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 multiplication

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) 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 addition

Proof:

(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 addition

Proof:

(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 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 associative rule for multiplication. [see Lemma 8, here]

QED

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

Proof:

(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 identity

Proof:

(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 property

Proof:

(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 property

Proof:

(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 property

Proof:

(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