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CS131
Part I, Number Systems
CS131 Mathematics for Computer Scientists II
Note 2
REAL NUMBERS
The real numbers can be thought of as corresponding to points on an infinite
straight line. The set of all real numbers will be denoted by R.
−2
−1 − 21 0
2
3
1
√
2 2
e 3π
An important subset of R is the set Q of rational numbers. A rational
number has the form m
n where m and n are integers and n 6= 0. We can
always choose m and n so that n ≥ 1 and gcd(m, n) = 1. Every non-zero
rational number q has an inverse q −1 with qq −1 = 1.
If q = m/n (m 6= 0), then q −1 = n/m.
That not all numbers are rational has been known since the time of Pythagoras (about 550 BC) and greatly affected Greek mathematics.
Theorem. There is no rational number x with x2 = 2.
Proof. Suppose that there is a rational number x with x2 = 2. Then we
can write x = m/n where m and n are integers with gcd(m, n) = 1.
This gives
m2
=2
or
m2 = 2n2 .
2
n
2
Hence m is even and it follows that m is even. Now we can write m = 2k
for some integer k which gives 4k 2 = 2n2 or n2 = 2k 2 . Hence n2 is even so
n must be even.
Now we have shown that both m and n are even i.e. they have a
common factor 2 which is impossible since gcd(m, n) = 1. Hence the
original assumption that there is a rational x with x2 = 2 must be false. The system R of real
√ numbers
√ contains solutions of the equation
2
x = 2 (the numbers 2 and − 2). Real numbers of this form which are
solutions of polynomial equations with rational coefficients are called
algebraic numbers.
The set R also contains other numbers (called
transcendental numbers) such as π and e, which cannot be the solutions
of polynomial equations with rational coefficients.
2–1
A real number can be thought of as a sequence of rational numbers. Every
real number x has a decimal expansion which gives us a sequence of rational
numbers converging to x.
For example π can be thought of as the limit of the sequence
3, 3.1, 3.14, 3.141, 3.1415, 3.14159, . . .
of rational numbers (we deal with sequences and their convergence later in
this module).
Basic Arithmetic Properties of R.
All the properties of the real number system can be derived from thirteen
axioms. Below we look at some elementary results which are provable in
terms of these axioms.
Note: although we often use the symbol ∞, it does not represent a real
number and should not be treated like one. Usually, it represents the impossibility of carrying out a particular arithmetic operation, such as division.
Basic Order Properties of R.
Let a, b and c be real numbers.
(1) Exactly one of the following three properties holds:
a < b,
(2)
(3)
(4)
(5)
if
if
if
if
a<b
a<b
a<b
a<b
a = b,
a > b.
and b < c then a < c,
then a + c < b + c
and c > 0 then ac < bc
and c < 0 then ac > bc
Further properties can now be derived from these basic ones as we now
show.
1
>0
a
Solution. Let a > 0. If 1/a = 0 then 1 = a.(1/a) = a.0 = 0 a
contradiction. If 1/a < 0 then using property (4) to multiply both sides by
the positive number a we have a.(1/a) < a.0 or 1 < 0, again a
contradiction. Now by property (1) we must have a1 > 0
Problem. Show that if a > 0, then
Problem. Show that if x and y are positive then
x < y ⇐⇒ x2 < y 2
2–2
Solution. First we show that x < y =⇒ x2 < y 2 . If x < y then by
property (4) we can multiply by x to get x2 < xy and similarly we can
multiply by y to get xy < y 2 . Now by property (2) we have x2 < y 2 .
Now we show the reverse implication x2 < y 2
=⇒
x < y.
2
2
2
Suppose x < y . Then using property (3) to add −x to each side we
have 0 < y 2 − x2 or 0 < (y − x)(y + x). Now by property (4) we can
1
to get 0 < y − x i.e. x < y.
multiply both sides by y+x
Intervals.
A set of real numbers is called an interval if it has one of the following forms
for some a, b ∈ R:
[a, b] = {x|a ≤ x ≤ b}
(a, ∞) = {x|x > a}
(a, b] = {x|a < x ≤ b}
[a, ∞) = {x|x ≥ a}
[a, b) = {x|a ≤ x < b}
(−∞, a) = {x|x < a}
(a, b) = {x|a < x < b}
(−∞, a] = {x|x ≤ a}
Intervals of the form (a, b), (−∞, a), (a, ∞) are called open.
Intervals of the form [a, b], [−∞, a], [a, ∞] are called closed.
Roots.
Let n be an integer with n ≥ 2. For any real number a ≥ 0 there is exactly
one x ≥ 0 with xn = a. This number x is called the nth root of a and is
1
denoted by a n . For any a, b ∈ R with a, b ≥ 0 and any integer n ≥ 2 we
have
a < b ⇐⇒ a1/n < b1/n .
√
√
1
When n = 2 we write a 2 as a. Note that a is the unique positive square
root of a. So for a ≥ 0 we have
√
√
x2 = a ⇐⇒ x √
= a or √x = − a
x2 < a ⇐⇒ − a < x < a.
Modulus.
The modulus (or absolute value) |x| of a real number x is defined by
x if x ≥ 0
|x| =
−x if x < 0
√
Note that |x| = x2 for every real number x.
y = |x|
y
6
@
@
2–3
-
x
Properties of the modulus. For any real numbers x and y:
(1)
(2)
(3)
(4)
−|x| ≤ x ≤ |x|,
|xy| = |x||y|,
|x
+ y| ≤ |x| + |y|,
|x| − |y| ≤ |x − y|.
To prove property (1) note that x is equal to one of the numbers |x| or −|x|.
The other properties follow from similar results concerning the modulus of
a complex number which will be treated in the next Note.
Upper bound, lower bound, supremum and infimum. Let S be a
set of real numbers. A real number u is called an upper bound of S if x ≤ u
for all x ∈ S. A real number l is called a lower bound of S if l ≤ x for every
x ∈ S. A real number U is called the least upper bound (supremum) of S if
U is an upper bound of S and U ≤ u for every upper bound u of S. A real
number L is called the greatest lower bound (infimum) of S if L is a lower
bound of S and l ≤ L for every lower bound l of S.
Completeness Property of R.
Every non-empty set of real numbers which has an upper bound has a least
upper bound. Every non-empty set of real numbers which has a lower bound
has a greatest lower bound.
The real numbers are completely characterized by twelve basic arithmetic
and order properties and the Completeness Property. Any theorem about
real numbers can (eventually) be derived from these. Also any structure
satisfying these properties can be shown to be essentially identical to R.
In particular the Completeness Property implies that the set
{x ∈ R|x2 < 2} has a least upper bound and it follows from this that there
is a real number x with x2 = 2.
2–4
Important consequence of the Completeness Property
• The Archimedean Property of R
If is a real number with > 0 then there is an integer n > 0
with 1/n < • Between any two distinct real numbers there are both rational
and irrational numbers
• Every real number can be represented by a (possibly infinite)
decimal expansion
ABSTRACT
Content Real numbers, order properties, intervals, modulus function.
In this Note the real number system and its properties are introduced. We also study a further important
method of mathematical proof - Proof by Contradiction. This approach was also created by the Greeks
and is sometimes known as Reductio ad absurdum.
Real intervals and the modulus function are also defined.
History
For many years the concept of a negative number was rejected although the Chinese had been using it as
early as 100 BC.
One of the first to give them open-minded consideration was the Italian Leonardo da Pisa (1170 - 1250
AD), often referred to as ’Fibonacci’. He accepted that negative numbers were needed to complete the
solution of algebraic equations.
Nevertheless, negative numbers were not fully accepted until the 16th century when another Italian,
Girolamo Cardano, published the laws which govern them. He also proposed yet another kind of number the ’fictitious’ number - which met yet further resistance for many years but ultimately was accepted and
called a ’complex’ number.
In modern mathematics, negative numbers (indeed all numbers) are best accepted as abstractions and
need have no physical meaning.
2–5