Axioms Real Numbers

8/13/2019 Axioms Real Numbers

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MATH H1B – AXIOMS FOR REAL NUMBERS

Later we will see how to construct real numbers by starting with positive integers as undefined objects,then using those to build positive rational numbers. From positive rational numbers, we can constructpositive irrational numbers, and then we complete the picture by adding in zero and negatives.

For now, though, we will think of the set of real numbers as “undefined objects” which satisfy certain

basic properties that we call “axioms”. We will use R to denote the set of real numbers, and we assume thatwe have two operations called addition and multiplication . Given two real numbers x and y , we assume thattheir sum x + y and their product xy are both uniquely determined by x and y. (I.e. exactly one number isthe sum x + y and exactly one number is the product xy.)

Basic properties of equality

• REFLEXIVE PROPERTY: For any real number x, x = x.

• SYMMETRIC PROPERTY: For any real numbers x and y, if x = y, then y = x.

• TRANSITIVE PROPERTY: For any real numbers x, y, and z, if x = y and y = z , then x = z .

The field axioms

We treat the following axioms as our basic facts about real numbers.

• AXIOM 1 (Commutative laws): x + y = y + x and xy = yx for all real numbers x and y .

• AXIOM 2 (Associative laws): x + (y + z) = (x + y) + z for all real numbers x, y, and z.

• AXIOM 3 (Distributive law): x(y + z) = xy + xz for all real numbers x, y, and z.

• AXIOM 4 (Existence of identity elements): There exist two distinct real numbers, which we call 0 and1, such that for every real number x we have x + 0 = x and x · 1 = x.

• AXIOM 5 (Existence of negatives): For every real number x there is a real number y such that x+y = 0.

• AXIOM 6 (Existence of reciprocals): For every real number x = 0 there is a real number y such thatxy = 1.

The order axioms

We next assume there is a subset R+ ⊂ R called the set of positive numbers satisfying the following properties.

• AXIOM 7: If x and y are in R+, then so are x + y and xy.

• AXIOM 8: For every real x = 0, either x ∈ R+ or −x ∈ R+, but not both.

• AXIOM 9: The additive identity 0 is not in R+.

Definitions of some symbols

Now we can define the symbols <, >, ≤, and ≥:

• x < y means that y − x ∈R+;

• y > x means that x < y;

• x ≤ y means that either x < y or x = y ;

• y ≥ x means that x ≤ y .



The least upper bound axiom

So far, all of the properties we listed have been essentially algebraic . In fact, you might have noticed thatthe set Q of rational numbers satisfies all of these properties. To do calculus, however, we really want tothink about all the real numbers. So, we will add in one more property, an analytic one.

We need to make a couple definitions first. Suppose S is a set of numbers. A number B is an upper

bound for S if ∀x ∈ S , we have x ≤ B . The number B is called the least upper bound or supremum of S if B is an upper bound for S and B is less than or equal to every other upper bound for S .

• AXIOM 10: Every nonempty set S of real numbers which is bounded above has a real number as itssupremum. That is, there is a real number B such that B = sup(S ).

Note that the set Q of rational numbers does not satisfy this axiom. For example, if we take an infinitesequence of rational numbers which are smaller than

√ 2, but getting closer and closer to

√ 2, that set is

bounded above, and it does not have rational number as its supremum. (We will prove that√

2 is notrational.)

Some theorems about real numbers

Using the axioms and basic logic as our building blocks, we can now prove some facts about real numbers.Facts that we can prove using our axioms and previously proved facts will be called theorems . This firstsection consists of theorems we can prove using just the field axioms.

• THEOREM 1 (Cancellation law for addition): If a + b = a + c, then b = c.

• THEOREM 2 (Possibility of subtraction or uniqueness of negatives): Given real numbers a and b, thereis exactly one number x such that a + x = b. We usually write b − a instead of x, and in particular, wewrite −a for 0 − a and call that the negative of a. (Note: we only refer to the blah-blah-blah if thereis exactly one blah-blah-blah. I.e., it is the unique such object.)

• THEOREM 3: b − a = b + (−a)

• THEOREM 4:

−(

−a) = a

• THEOREM 5: a(b − c) = ab − ac

• THEOREM 6: 0 · a = a · 0 = 0

• THEOREM 7 (Cancellation law for multiplication): If ab = ac and a = 0, then b = c.

• THEOREM 8 (Possibility of division or uniqueness of reciprocals): Given real numbers a and b suchthat a = 0, there is exactly one number x such that ax = b. We usually write b

a instead of x. In

particular, 1a

or a−1 is the reciprocal of a.

• THEOREM 9: If a = 0, then b

a = b · a−1.

• THEOREM 10: If a = 0, then (a−1)−1 = a.

• THEOREM 11: If ab = 0, then a = 0 or b = 0.

• THEOREM 12: (−a)b = −(ab) and (−a)(−b) = ab.

• THEOREM 13: a

b + c

d = (ad+bc)

bd if b = 0 and d = 0.

• THEOREM 14: a

b

c

d = ac

bd if b = 0 and d = 0.

• THEOREM 15:a

bc

d

= ad

bc if b = 0, c = 0, and d = 0.

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The next batch of theorems also uses the order axioms.

• THEOREM 16 (Trichotomy law): For arbitrary real numbers a and b, exactly one of the following istrue: a < b, b < a, or a = b.

• THEOREM 17 (Transitivity): If a < b and b < c, then a < c.

• THEOREM 18: If a < b, then a + c < b + c.

• THEOREM 19: If a < b and c > 0, then ac < bc.

• THEOREM 20: If a = 0, then a2 > 0.

• THEOREM 21: 1 > 0.

• THEOREM 22: If a < b and c < 0, then ac > bc.

• THEOREM 23: If a < b, then −a > −b. In particular, if a < 0, then−a > 0.

• THEOREM 24: If ab > 0, then a and b are both positive or a and b are both negative.

• THEOREM 25: If a < c and b < d, then a + b < c + d.

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Axioms Real Numbers

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