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Note 2 : REAL NUMBERS
01
Note 2 : REAL
NUMBERS
CS131: Mathematics for Computer Science II
Part I, Number Systems
Note 2 : REAL NUMBERS
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Real Numbers
The real numbers can be thought of as corresponding to points on an infinite
straight line.
−2
The integers Z
−1
0
1
2
3
Note 2 : REAL NUMBERS
03
Real Numbers
The real numbers can be thought of as corresponding to points on an infinite
straight line.
−2
The rationals Q
−1 − 12
0
2
3
1
2
3
Note 2 : REAL NUMBERS
04
Real Numbers
The real numbers can be thought of as corresponding to points on an infinite
straight line.
−2
“algebraic numbers”
−1
− 12
0
2
3
1
√
2
2
3
Note 2 : REAL NUMBERS
05
Real Numbers
The real numbers can be thought of as corresponding to points on an infinite
straight line.
−2
−1
“transcendental numbers”
− 12
0
2
3
1
√
2
2
e
3 π
Note 2 : REAL NUMBERS
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Real Numbers
The real numbers can be thought of as corresponding to points on an infinite
straight line.
−2
−1
− 12
0
2
3
1
The set of all real numbers will be denoted by R.
√
2
2
e
3 π
Note 2 : REAL NUMBERS
Rational Numbers
An important subset of R is the set Q of rational numbers.
07
Note 2 : REAL NUMBERS
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Rational Numbers
An important subset of R is the set Q of rational numbers.
• A rational number has the form
m
n
where m, n ∈ Z and n 6= 0.
Note 2 : REAL NUMBERS
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Rational Numbers
An important subset of R is the set Q of rational numbers.
• A rational number has the form
m
n
where m, n ∈ Z and n 6= 0.
• We can always choose m and n so that n ≥ 1 and gcd(m, n) = 1.
Note 2 : REAL NUMBERS
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Rational Numbers
An important subset of R is the set Q of rational numbers.
• A rational number has the form
m
n
where m, n ∈ Z 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 the product
q q −1 = 1.
Note 2 : REAL NUMBERS
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Rational Numbers
An important subset of R is the set Q of rational numbers.
• A rational number has the form
m
n
where m, n ∈ Z 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 the product
q q −1 = 1.
We have known about irrational numbers since the time of Pythagoras (circa
550BC).
Note 2 : REAL NUMBERS
Theorem There is no rational number x with x 2 = 2.
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Note 2 : REAL NUMBERS
Theorem There is no rational number x with x 2 = 2.
Proof:
• Suppose for contradiction that there exists x ∈ Q s.t. x 2 = 2.
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Note 2 : REAL NUMBERS
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Theorem There is no rational number x with x 2 = 2.
Proof:
• Suppose for contradiction that there exists x ∈ Q s.t. x 2 = 2.
• Then we can write x =
m
n
where m, n ∈ Z with gcd(m, n) = 1.
Note 2 : REAL NUMBERS
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Theorem There is no rational number x with x 2 = 2.
Proof:
• Suppose for contradiction that there exists x ∈ Q s.t. x 2 = 2.
• Then we can write x =
• This gives
m2
n2
m
n
where m, n ∈ Z with gcd(m, n) = 1.
= 2 or equivalently m 2 = 2n 2 .
Note 2 : REAL NUMBERS
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Theorem There is no rational number x with x 2 = 2.
Proof:
• Suppose for contradiction that there exists x ∈ Q s.t. x 2 = 2.
• Then we can write x =
• This gives
m2
n2
m
n
where m, n ∈ Z with gcd(m, n) = 1.
= 2 or equivalently m 2 = 2n 2 .
• It follows that m 2 is even and hence that m is also even.
Note 2 : REAL NUMBERS
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Theorem There is no rational number x with x 2 = 2.
Proof:
• Suppose for contradiction that there exists x ∈ Q s.t. x 2 = 2.
• Then we can write x =
• This gives
m2
n2
m
n
where m, n ∈ Z with gcd(m, n) = 1.
= 2 or equivalently m 2 = 2n 2 .
• It follows that m 2 is even and hence that m is also even.
• Thus m = 2k for some k ∈ Z. Hence 4k 2 = 2n 2 or n 2 = 2k 2 .
Note 2 : REAL NUMBERS
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Theorem There is no rational number x with x 2 = 2.
Proof:
• Suppose for contradiction that there exists x ∈ Q s.t. x 2 = 2.
• Then we can write x =
• This gives
m2
n2
m
n
where m, n ∈ Z with gcd(m, n) = 1.
= 2 or equivalently m 2 = 2n 2 .
• It follows that m 2 is even and hence that m is also even.
• Thus m = 2k for some k ∈ Z. Hence 4k 2 = 2n 2 or n 2 = 2k 2 .
• It follows that n 2 is even and hence n is even too!
Note 2 : REAL NUMBERS
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Theorem There is no rational number x with x 2 = 2.
Proof:
• Suppose for contradiction that there exists x ∈ Q s.t. x 2 = 2.
• Then we can write x =
• This gives
m2
n2
m
n
where m, n ∈ Z with gcd(m, n) = 1.
= 2 or equivalently m 2 = 2n 2 .
• It follows that m 2 is even and hence that m is also even.
• Thus m = 2k for some k ∈ Z. Hence 4k 2 = 2n 2 or n 2 = 2k 2 .
• It follows that n 2 is even and hence n is even too!
• We have shown that both m and n are even contradicting gcd(m, n) = 1.
Note 2 : REAL NUMBERS
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Theorem There is no rational number x with x 2 = 2.
Proof:
• Suppose for contradiction that there exists x ∈ Q s.t. x 2 = 2.
• Then we can write x =
• This gives
m2
n2
m
n
where m, n ∈ Z with gcd(m, n) = 1.
= 2 or equivalently m 2 = 2n 2 .
• It follows that m 2 is even and hence that m is also even.
• Thus m = 2k for some k ∈ Z. Hence 4k 2 = 2n 2 or n 2 = 2k 2 .
• It follows that n 2 is even and hence n is even too!
• We have shown that both m and n are even contradicting gcd(m, n) = 1.
• Hence our original assumption must be false.
2
Note 2 : REAL NUMBERS
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Irrational Numbers: Algebraic and Transendental
We have shown that there is no rational number x such that x 2 = 2. However, the
system R of real numbers contains two solutions of the equation x 2 = 2 (the
√
√
numbers 2 and − 2).
Note 2 : REAL NUMBERS
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Irrational Numbers: Algebraic and Transendental
We have shown that there is no rational number x such that x 2 = 2. However, the
system R of real numbers contains two solutions of the equation x 2 = 2 (the
√
√
numbers 2 and − 2).
√
√
An algebraic number is a real number (like 2 and − 2) that is the solution of a
polynomial equation with rational coefficients.
Note 2 : REAL NUMBERS
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Irrational Numbers: Algebraic and Transendental
We have shown that there is no rational number x such that x 2 = 2. However, the
system R of real numbers contains two solutions of the equation x 2 = 2 (the
√
√
numbers 2 and − 2).
√
√
An algebraic number is a real number (like 2 and − 2) that is the solution of a
polynomial equation with rational coefficients.
Transcendental numbers are real numbers which cannot be the solutions of
polynomial equations with rational coefficients. Examples include π and e.
Note 2 : REAL NUMBERS
Reals as Converging Sequences of Rationals
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 .
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Note 2 : REAL NUMBERS
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Reals as Converging Sequences of Rationals
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 following sequence of rational
numbers.
3,
3.1,
3.14,
3.141,
3.1415,
3.14159,
...
Note 2 : REAL NUMBERS
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Reals as Converging Sequences of Rationals
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 following sequence of rational
numbers.
3,
3.1,
3.14,
3.141,
3.1415,
3.14159,
...
(We deal with sequences and their convergence later in the module).
Note 2 : REAL NUMBERS
Basic Arithmetic Properties of Reals
Before we discuss the basic properties of real numbers we note the following:
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Note 2 : REAL NUMBERS
Basic Arithmetic Properties of Reals
Before we discuss the basic properties of real numbers we note the following:
1. All the properties of the real number system can be derived from thirteen
axioms.
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Note 2 : REAL NUMBERS
Basic Arithmetic Properties of Reals
Before we discuss the basic properties of real numbers we note the following:
1. All the properties of the real number system can be derived from thirteen
axioms.
2. Although we often use the symbol ∞, it does not represent a real number
and should not be treated like one.
• The symbol ∞ is often used to represent the impossibility of carrying out
a particular arithmetic operation, such as division by zero.
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Note 2 : REAL NUMBERS
Basic Order Properties of Reals
Given a, b, c ∈ R
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Note 2 : REAL NUMBERS
31
Basic Order Properties of Reals
Given a, b, c ∈ R
1. Exactly one of the following three properties holds:
• a < b,
a = b,
a > b.
Note 2 : REAL NUMBERS
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Basic Order Properties of Reals
Given a, b, c ∈ R
1. Exactly one of the following three properties holds:
• a < b,
a = b,
a > b.
2. If a < b and b < c then it follows that a < c.
Note 2 : REAL NUMBERS
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Basic Order Properties of Reals
Given a, b, c ∈ R
1. Exactly one of the following three properties holds:
• a < b,
a = b,
a > b.
2. If a < b and b < c then it follows that a < c.
3. If a < b then it follows that a + c < b + c.
Note 2 : REAL NUMBERS
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Basic Order Properties of Reals
Given a, b, c ∈ R
1. Exactly one of the following three properties holds:
• a < b,
a = b,
a > b.
2. If a < b and b < c then it follows that a < c.
3. If a < b then it follows that a + c < b + c.
4. If a < b and c > 0 then it follows that ac < bc.
Note 2 : REAL NUMBERS
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Basic Order Properties of Reals
Given a, b, c ∈ R
1. Exactly one of the following three properties holds:
• a < b,
a = b,
a > b.
2. If a < b and b < c then it follows that a < c.
3. If a < b then it follows that a + c < b + c.
4. If a < b and c > 0 then it follows that ac < bc.
5. If a < b and c < 0 then it follows that ac > bc.
Note 2 : REAL NUMBERS
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Basic Order Properties of Reals
Given a, b, c ∈ R
1. Exactly one of the following three properties holds:
• a < b,
a = b,
a > b.
2. If a < b and b < c then it follows that a < c.
3. If a < b then it follows that a + c < b + c.
4. If a < b and c > 0 then it follows that ac < bc.
5. If a < b and c < 0 then it follows that ac > bc.
Further properties can be derived from these . . .
Note 2 : REAL NUMBERS
Question: Show that if a > 0, then
37
1
a
> 0.
Note 2 : REAL NUMBERS
Question: Show that if a > 0, then
Answer:
Suppose a > 0.
38
1
a
> 0.
Note 2 : REAL NUMBERS
39
Question: Show that if a > 0, then
1
a
> 0.
Answer:
Suppose a > 0.
There are three possibilities:
1
a
< 0,
1
a
= 0 or
1
a
> 0.
Note 2 : REAL NUMBERS
40
Question: Show that if a > 0, then
1
a
> 0.
Answer:
Suppose a > 0.
There are three possibilities:
• If
1
a
1
a
< 0,
= 0 then a . a1 = a . 0 = 0.
1
a
= 0 or
1
a
> 0.
Note 2 : REAL NUMBERS
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Question: Show that if a > 0, then
1
a
> 0.
Answer:
Suppose a > 0.
There are three possibilities:
1
a
< 0,
1
a
= 0 or
• If a1 = 0 then a . a1 = a . 0 = 0.
This is a contradiction since a . a1 = 1.
1
a
> 0.
Note 2 : REAL NUMBERS
42
Question: Show that if a > 0, then
1
a
> 0.
Answer:
Suppose a > 0.
There are three possibilities:
1
a
< 0,
1
a
= 0 or
1
a
> 0.
• If a1 = 0 then a . a1 = a . 0 = 0.
This is a contradiction since a . a1 = 1.
• If
1
a
< 0 then it follows that a . (1/a) < a . 0.
– (We have multiplied a positive and a negative number.)
Note 2 : REAL NUMBERS
43
Question: Show that if a > 0, then
1
a
> 0.
Answer:
Suppose a > 0.
There are three possibilities:
1
a
< 0,
1
a
= 0 or
1
a
> 0.
• If a1 = 0 then a . a1 = a . 0 = 0.
This is a contradiction since a . a1 = 1.
• If
1
a
< 0 then it follows that a . (1/a) < a . 0.
– (We have multiplied a positive and a negative number.)
This tells us that 1 < 0 : we have another contradiction.
Note 2 : REAL NUMBERS
44
Question: Show that if a > 0, then
1
a
> 0.
Answer:
Suppose a > 0.
There are three possibilities:
1
a
< 0,
1
a
= 0 or
1
a
> 0.
• If a1 = 0 then a . a1 = a . 0 = 0.
This is a contradiction since a . a1 = 1.
• If
1
a
< 0 then it follows that a . (1/a) < a . 0.
– (We have multiplied a positive and a negative number.)
This tells us that 1 < 0 : we have another contradiction.
• It must therefore be the case that
1
a
> 0.
2
Note 2 : REAL NUMBERS
Question:
Show that if x and y are both positive then x < y ⇔ x 2 < y 2 .
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Note 2 : REAL NUMBERS
Question:
Show that if x and y are both positive then x < y ⇔ x 2 < y 2 .
Answer:
First we show that x < y ⇒ x2 < y2 .
46
Note 2 : REAL NUMBERS
Question:
Show that if x and y are both positive then x < y ⇔ x 2 < y 2 .
Answer:
First we show that x < y ⇒ x2 < y2 .
Since x < y and x > 0 it follows that x .x < y.x i.e. x 2 < xy.
47
Note 2 : REAL NUMBERS
Question:
Show that if x and y are both positive then x < y ⇔ x 2 < y 2 .
Answer:
First we show that x < y ⇒ x2 < y2 .
Since x < y and x > 0 it follows that x .x < y.x i.e. x 2 < xy.
Similarly, since x < y and y > 0 it follows that xy < y 2 .
48
Note 2 : REAL NUMBERS
Question:
Show that if x and y are both positive then x < y ⇔ x 2 < y 2 .
Answer:
First we show that x < y ⇒ x2 < y2 .
Since x < y and x > 0 it follows that x .x < y.x i.e. x 2 < xy.
Similarly, since x < y and y > 0 it follows that xy < y 2 .
Hence, by transitivity, x 2 < y 2 .
49
Note 2 : REAL NUMBERS
Question:
Show that if x and y are both positive then x < y ⇔ x 2 < y 2 .
Answer:
First we show that x < y ⇒ x2 < y2 .
Since x < y and x > 0 it follows that x .x < y.x i.e. x 2 < xy.
Similarly, since x < y and y > 0 it follows that xy < y 2 .
Hence, by transitivity, x 2 < y 2 .
Next we show that x < y ⇐ x2 < y2 .
50
Note 2 : REAL NUMBERS
Question:
Show that if x and y are both positive then x < y ⇔ x 2 < y 2 .
Answer:
First we show that x < y ⇒ x2 < y2 .
Since x < y and x > 0 it follows that x .x < y.x i.e. x 2 < xy.
Similarly, since x < y and y > 0 it follows that xy < y 2 .
Hence, by transitivity, x 2 < y 2 .
Next we show that x < y ⇐ x2 < y2 .
From x 2 < y 2 it follows that 0 < y 2 − x 2 .
51
Note 2 : REAL NUMBERS
Question:
Show that if x and y are both positive then x < y ⇔ x 2 < y 2 .
Answer:
First we show that x < y ⇒ x2 < y2 .
Since x < y and x > 0 it follows that x .x < y.x i.e. x 2 < xy.
Similarly, since x < y and y > 0 it follows that xy < y 2 .
Hence, by transitivity, x 2 < y 2 .
Next we show that x < y ⇐ x2 < y2 .
From x 2 < y 2 it follows that 0 < y 2 − x 2 .
From which we deduce that 0 < (y − x )(y + x ).
52
Note 2 : REAL NUMBERS
Question:
Show that if x and y are both positive then x < y ⇔ x 2 < y 2 .
Answer:
First we show that x < y ⇒ x2 < y2 .
Since x < y and x > 0 it follows that x .x < y.x i.e. x 2 < xy.
Similarly, since x < y and y > 0 it follows that xy < y 2 .
Hence, by transitivity, x 2 < y 2 .
Next we show that x < y ⇐ x2 < y2 .
From x 2 < y 2 it follows that 0 < y 2 − x 2 .
From which we deduce that 0 < (y − x )(y + x ).
1
Multiplying both sides by y+x
we get 0 < y − x i.e. x < y.
53
Note 2 : REAL NUMBERS
Question:
Show that if x and y are both positive then x < y ⇔ x 2 < y 2 .
Answer:
First we show that x < y ⇒ x2 < y2 .
Since x < y and x > 0 it follows that x .x < y.x i.e. x 2 < xy.
Similarly, since x < y and y > 0 it follows that xy < y 2 .
Hence, by transitivity, x 2 < y 2 .
Next we show that x < y ⇐ x2 < y2 .
From x 2 < y 2 it follows that 0 < y 2 − x 2 .
From which we deduce that 0 < (y − x )(y + x ).
1
Multiplying both sides by y+x
we get 0 < y − x i.e. x < y.
The implication holds in both directions. Hence x < y ⇔ x 2 < y 2 .
54
Note 2 : REAL NUMBERS
Roots
Suppose that n ∈ N with n ≥ 2, and that a ∈ R with a ≥ 0.
55
Note 2 : REAL NUMBERS
Roots
Suppose that n ∈ N with n ≥ 2, and that a ∈ R with a ≥ 0.
• Then there is precisely one x ∈ R with x ≥ 0 such that x n = a.
56
Note 2 : REAL NUMBERS
Roots
Suppose that n ∈ N with n ≥ 2, and that a ∈ R with a ≥ 0.
• Then there is precisely one x ∈ R with x ≥ 0 such that x n = a.
• This number is called the n th root of a and is denoted by a 1/n .
57
Note 2 : REAL NUMBERS
58
Roots
Suppose that n ∈ N with n ≥ 2, and that a ∈ R with a ≥ 0.
• Then there is precisely one x ∈ R with x ≥ 0 such that x n = a.
• This number is called the n th root of a and is denoted by a 1/n .
• When n = 2 we write use
√
a to denote a 1/2 .
Note 2 : REAL NUMBERS
Roots
Suppose that n ∈ N with n ≥ 2, and that a ∈ R with a ≥ 0.
• Then there is precisely one x ∈ R with x ≥ 0 such that x n = a.
• This number is called the n th root of a and is denoted by a 1/n .
√
• When n = 2 we write use a to denote a 1/2 .
√
– Note that a represents the unique positive square root of a.
59
Note 2 : REAL NUMBERS
Roots
Suppose that n ∈ N with n ≥ 2, and that a ∈ R with a ≥ 0.
• Then there is precisely one x ∈ R with x ≥ 0 such that x n = a.
• This number is called the n th root of a and is denoted by a 1/n .
√
• When n = 2 we write use a to denote a 1/2 .
√
– Note that a represents the unique positive square root of a.
– For any a ∈ R with a ≥ 0 we have
√
√
2
x = a ⇔ x = a or x = − a
√
√
2
x < a ⇔ − a < x < a.
60
Note 2 : REAL NUMBERS
61
Roots
Suppose that n ∈ N with n ≥ 2, and that a ∈ R with a ≥ 0.
• Then there is precisely one x ∈ R with x ≥ 0 such that x n = a.
• This number is called the n th root of a and is denoted by a 1/n .
√
• When n = 2 we write use a to denote a 1/2 .
√
– Note that a represents the unique positive square root of a.
– For any a ∈ R with a ≥ 0 we have
√
√
2
x = a ⇔ x = a or x = − a
√
√
2
x < a ⇔ − a < x < a.
• For any a, b ∈ R with a, b ≥ 0 and any n ∈ N with n ≥ 2:
a<b
⇔
a 1/n < b 1/n .
Note 2 : REAL NUMBERS
62
Puzzle:
Estimate the value of
√
20.
Note 2 : REAL NUMBERS
63
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 | =
√
x 2 for every real number x .
y = |x |
y
6
@ -
@
x
The Notes remind you about some properties of the modulus.
Note 2 : REAL NUMBERS
Reminder from Maths for CS Part I...
Upper and Lower Bounds, Supremum and Infimum
Let S be a set of real numbers.
• A real number u is an upper bound of S if u ≥ x for all x ∈ S .
64
Note 2 : REAL NUMBERS
Upper and Lower Bounds, Supremum and Infimum
Let S be a set of real numbers.
• A real number u is an upper bound of S if u ≥ x for all x ∈ S .
• A real number U is a least upper bound (supremum) of S if U is an upper
bound of S and U ≤ u for every upper bound u of S .
65
Note 2 : REAL NUMBERS
Upper and Lower Bounds, Supremum and Infimum
Let S be a set of real numbers.
• A real number u is an upper bound of S if u ≥ x for all x ∈ S .
• A real number U is a 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 a lower bound of S if l ≤ x for all x ∈ S .
66
Note 2 : REAL NUMBERS
Upper and Lower Bounds, Supremum and Infimum
Let S be a set of real numbers.
• A real number u is an upper bound of S if u ≥ x for all x ∈ S .
• A real number U is a 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 a lower bound of S if l ≤ x for all x ∈ S .
• A real number L is a greatest lower bound (infimum) of S if L is a lower
bound of S and L ≥ l for every lower bound l of S .
67
Note 2 : REAL NUMBERS
Completeness Property of the Reals
68
Note 2 : REAL NUMBERS
Completeness Property of the Reals
• 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.
69
Note 2 : REAL NUMBERS
Completeness Property of the Reals
• 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 be derived from these.
• Also any structure satisfying these properties can be shown to be essentially
identical to R.
70
Note 2 : REAL NUMBERS
Completeness Property of the Reals
• 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 be derived from these.
• Also any structure satisfying these properties can be shown to be essentially
identical to R.
• The Completeness Property implies that {x ∈ R | x 2 < 2} has a least upper
bound. It follows that there exists x ∈ R with x 2 = 2.
71
Note 2 : REAL NUMBERS
The Archimedean Property of the Reals
72
Note 2 : REAL NUMBERS
73
The Archimedean Property of the Reals
1
n
•
Given any ∈ R+ there exists n ∈ N s.t.
•
Between any two distinct reals there are both
rational and irrational numbers.
•
Every real number can be represented by a
(possibly infinite) decimal expansion.
< .
Note 2 : REAL NUMBERS
74
The Archimedean Property of the Reals
1
n
•
Given any ∈ R+ there exists n ∈ N s.t.
•
Between any two distinct reals there are both
< .
rational and irrational numbers.
•
Every real number can be represented by a
(possibly infinite) decimal expansion.
This is an important consequence of the Completeness Property.
Note 2 : REAL NUMBERS
75
The Archimedean Property of the Reals
1
n
•
Given any ∈ R+ there exists n ∈ N s.t.
•
Between any two distinct reals there are both
< .
rational and irrational numbers.
•
Every real number can be represented by a
(possibly infinite) decimal expansion.
• Suppose for contradiction there is no such n so ≤
1
n
for every n
• n ≤ 1 for every n so {n | n ∈ N} has an upper bound
• by completeness it has a least upper bound l .
• for every n, n = (n + 1) − ≤ l − so l − is also an upper bound
• But l − is smaller than the least upper bound l giving a contradiction.
Note 2 : REAL NUMBERS
Addendum: Axioms of the Real Number System
Axioms hold for all x , y, z ∈ R.
1. Commutativity: x + y = y + x and x .y = y.x .
2. Associativity: x + (y + z ) = (x + y) + z and x .(y.z ) = (x .y).z .
3. Distributivity of . over +: x .(y + z ) = x .y + x .z .
4. There is an additive identity: There exists 0 ∈ R s.t. x + 0 = x .
5. There is a multiplicative identity: There exists 1 ∈ R s.t. x .1 = x .
6. The multiplicative and additive identities are distinct: 1 6= 0.
7. Every element has an additive inverse:
There exists (−x ) ∈ R s.t. x + (−x ) = 0.
8. Every non-zero element has a multiplicative inverse:
If x 6= 0 then there exists x −1 ∈ R s.t. x .x −1 = 1.
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Note 2 : REAL NUMBERS
9. Transitivity of ordering:
If x < y and y < z then x < z .
10. The trichotomy law:
Exactly one of the following is true: x < y, y < x or x = y.
11. Preservation of ordering under addition:
If x < y then x + z < y + z .
12. Preservation of ordering under multiplication:
If 0 < z and x < y then x .z < y.z .
13. Completeness: Every non-empty subset of R that is bounded above has a
least upper bound.
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