OUR FRIENDS THE REAL NUMBERS
Everyone is familiar with the real numbers. They
we
√ are the numbers
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think of living on the number line, including 0, 2, 17, −5, 10(10 ) , π, and
many others. We are also all familiar with arithmetic with the real numbers,
facts like 2(1 + 3) = 2 · 1 + 2 · 3, and if a < b, then −b < −a. It turns out that
our familiar real numbers can be uniquely characterized by twelve axioms.
We won’t prove that the real numbers are the unique structure that satisfies
these axioms, nor will be prove that the real numbers actually exist, but
we will use these axioms to derive several familiar formulas, and eventually
some less familiar formulas.
1. The Axioms
In this section, our universe of discourse will be the set of real numbers,
R (except in axiom twelve, where the universe of discourse is bigger). Our
set R has two operations on it, called + and ·. So, if a, b ∈ R then a + b
and a · b are real numbers. We’ll often write xy instead of x · y. The real
numbers also have a comparison relation called <. These fit together in the
following manner.
(1) Associativity: ∀x∀y∀z[(x + y) + z = x + (y + z) ∧ (xy)z = x(yz)].
(2) Commutativity: ∀x∀y(x + y = y + x ∧ xy = yx).
(3) Distributivity: ∀x∀y∀z[x(y + z) = xy + xz].
(4) Additive identity: ∃y∀x(y + x = x). We will prove that this y is
unique and we will call it 0.
(5) Additive inverses: ∀x∃y(x + y = 0). We will prove that this y is
unique and we will call it −x, or the additive inverse of x.
(6) Multiplicative identity ∃y∀x(y 6= 0 ∧ yx = x). We will prove that
this y is unique and call it 1.
(7) Multiplicative inverses: ∀x[x 6= 0 → (∃y(xy = 1))]. We will prove
that this y is unique and call it 1/x, or the multiplicative inverse of
x.
(8) Translation invariance of order: ∀x∀y∀z(x < y → x + z < y + z).
(9) Transitivity: ∀x∀y[(x < y ∧ y < z) → x < z].
(10) Trichotomy: ∀x∀y[(x < y ∨ y < x ∨ x = y) ∧ ¬[(x < y ∧ y <
x) ∨ (x < y ∧ x = y) ∨ (y < x ∧ x = y)]]. In other words, for all
x, y ∈ R exactly one of the following is true: x < y, y < x, or x = y.
(11) Scale invariance: ∀x∀y∀z[(x < y ∧ 0 < z) → xz < yz].
(12) Say that a set A ⊂ R is bounded above if there is some y ∈ R so
that for all x ∈ A, x < y. A least upper bound for a set A that
is bounded above is a real number y 0 so that for all x ∈ A, x < y 0 ,
Date: June 5th, 2015.
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OUR FRIENDS THE REAL NUMBERS
and if z is any real number greater than every element of A, either
y 0 < z or y 0 = z. Our final axiom states that every nonempty set of
real numbers that is bounded above has a least upper bound.
Mathematicians study sets of axioms because they have found that axioms either characterize something they like to study, or that many different
things they like to study all follow the same basic rules.
A structure that satisfies the first six axioms above is called a commutative
ring. For example, the integers satisfy the first 6 axioms (plus the last 5),
but they don’t satisfy axiom 7. If a structure also satisfies seven, then it is
a field. Fields and commutative rings are ubiquitous in mathematics, and
you will eventually meet a lot of examples of field and commutative rings
that look nothing like R.
A structure that satisfies the first eleven axioms is called an ordered field.
For example, the rational numbers Q form an ordered field.
2. More Properties of the Real Numbers
The following rules follow from the 12 axioms for the real numbers. It is
easiest to go in order as you prove these. We’ll do some of these in class, I
encourage you to do the rest on your own.
(1) There is a unique additive identity.
(2) Each real number has a unique additive inverse.
(3) There is a unique multiplicative identity.
(4) Every nonzero real number has a unique multiplicative inverse.
(5) For all a ∈ R, 0 · a = 0.
(6) If ab = 0, then either a = 0 or b = 0.
(7) We have that −0 = 0.
(8) For all a ∈ R, −(−a) = a.
(9) For all a, b ∈ R, (−a)b = −(ab) = a(−b).
(10) For all a, b ∈ R, (−a)(−b) = ab.
(11) For all a ∈ R, −a = −1(a).
(12) For all a ∈ R, a + a = 2a (2 is defined to be 1 + 1).
(13) For all a, b, c ∈ R, (a + b)c = ac + bc.
(14) If a 6= 0, 1/a 6= 0.
(15) We have that 1−1 = 1.
(16) For all nonzero a, (a−1 )−1 = a.
(17) For all nonzero a, (−a)−1 = −(a)−1 .
(18) For all a, b ∈ R\{0}, (ab)−1 = a−1 b−1 .
(19) For all a, b ∈ R\{0}, (a·1/b)−1 = b·1/a. So, we’ll write a·1/b = a/b.
(20) For all a, b, c, d ∈ R, (a/b)(c/d) = (ac)/(bd).
(21) For all a, b, c ∈ R\{0}, (ab)/(ac) = b/c.
(22) For all a, b, c ∈ R, a(b/c) = (ab)/c.
(23) For all a, b, c ∈ R, (ab)/b = a.
(24) For all a, b ∈ R with b 6= 0, (−a)/b = −(a/b) = a/(−b), and
(−a)/(−b) = a/b.
OUR FRIENDS THE REAL NUMBERS
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(25) For all a, b, c, d ∈ R with b, d 6= 0, a/b + c/d = (ad + bc)/bd, and
a/b − c/d = (ad − bc)/(bd).
(26) For all a ∈ R, either a2 > 0 or a = 0 (a2 means a · a).
(27) For all a ∈ R, a2 = 0 if and only if a = 0.
(28) For all a ∈ R, (−a)2 = a2 .
(29) For all a ∈ R, (a−1 )2 = 1/(a2 ).
(30) If a2 = b2 , either a = b or a = −b.
(31) If a, b are both greater than 0, then a < b implies that a2 < b2 .
(32) If a, b are both negative, then a < b implies that b2 < a2 .
(33) If a ≤ b and b ≤ a then a = b (a ≤ b means a < b ∨ a = b).
(34) If a < b, then −b < −a.
(35) We have that 0 < 1.
(36) If a > 0 then a−1 > 0.
(37) If a < 0, then a−1 < 0.
(38) If a < b and a, b are both greater than 0, then b−1 < a−1 .
(39) If a < b and c < d, then a + c < b + d.
(40) If a < b and c < 0, then bc > ac.
(41) If a < b and c < d and all are nonnegative, then ac < bd (a number
is negative if it is less than 0, and positive if it is greater than 0).
(42) For all a, b ∈ R, ab > 0 if and only if a, b are either both positive or
both negative.
(43) For all a, b ∈ R, ab < 0 if and only if one of them is negative and the
other is positive.