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The Real FieldThe Extended Real Number System
Euclidean Spaces1
Existence of Reals
Theorem
There exists an ordered field R which has the least upper
boundproperty. Moreover R contains Q as a subfield.
The elements of R are called real numbers.
Theorem
Any two ordered fields with the least upper bound property
areisomorphic.
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The Real FieldThe Extended Real Number System
Euclidean Spaces1
Archimedean Property
Theorem
(a) If x , y ∈ R and x > 0 then there is a positive integer n
suchthat nx > y
(b) If x , y ∈ R and x < y then there exists a p ∈ Q such
thatx < p < y
Part (a) is the called the archimedean property of R. Part (b)
maysays Q is dense in R, i.e. between any two real numbers there is
arational one.
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The Real FieldThe Extended Real Number System
Euclidean Spaces1
nth Root
Theorem
For every real x > 0 and every integer n > 0 there is one
and onlyone positive real y such that yn = x
The number y is written y = n√
x or y = x1/n
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The Real FieldThe Extended Real Number System
Euclidean Spaces1
nth Root
Corollary
If a and b are positive real numbers and n is a positive integer
then
(ab)1/n = a1/nb1/n
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The Real FieldThe Extended Real Number System
Euclidean Spaces1
Decimals
Let x > 0 be a real number. Then let n0 be the largest
integersuch that n0 ≤ x (such an integer exists by the
archimedeanproperty of R).
Suppose we have chosen n0, n1, . . . , nk−1. Then let nk be
thelargest integer such that
n0 +n1
101+ · · ·+ nk
10k≤ x
If we let E = {n0 + n1101 + · · ·+nk10k
: k ∈ N} then x = sup E andthe decimal expansion of x is n0.n1n2
· · ·
Conversely for any decimal expansion n0.n1n2 · · · , the
set{n0.n1n2 · · · nk : k ∈ N} is bounded above and hence must have
aleast upper bound.
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The Real FieldThe Extended Real Number System
Euclidean Spaces1
Decimals
Let x > 0 be a real number. Then let n0 be the largest
integersuch that n0 ≤ x (such an integer exists by the
archimedeanproperty of R).
Suppose we have chosen n0, n1, . . . , nk−1. Then let nk be
thelargest integer such that
n0 +n1
101+ · · ·+ nk
10k≤ x
If we let E = {n0 + n1101 + · · ·+nk10k
: k ∈ N} then x = sup E andthe decimal expansion of x is n0.n1n2
· · ·
Conversely for any decimal expansion n0.n1n2 · · · , the
set{n0.n1n2 · · · nk : k ∈ N} is bounded above and hence must have
aleast upper bound.
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The Real FieldThe Extended Real Number System
Euclidean Spaces1
Decimals
Let x > 0 be a real number. Then let n0 be the largest
integersuch that n0 ≤ x (such an integer exists by the
archimedeanproperty of R).
Suppose we have chosen n0, n1, . . . , nk−1. Then let nk be
thelargest integer such that
n0 +n1
101+ · · ·+ nk
10k≤ x
If we let E = {n0 + n1101 + · · ·+nk10k
: k ∈ N} then x = sup E andthe decimal expansion of x is n0.n1n2
· · ·
Conversely for any decimal expansion n0.n1n2 · · · , the
set{n0.n1n2 · · · nk : k ∈ N} is bounded above and hence must have
aleast upper bound.
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The Real FieldThe Extended Real Number System
Euclidean Spaces1
The Extended Real Number System
Definition
The extended real number system consists of the real field R
andtwo symbols, +∞ and −∞.
We preserve the original order from R and define
−∞ < x < +∞
for all x ∈ R.
Theorem
Every subset of the extended real numbers has a least upper
bound(as well as a greatest lower bound)
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The Real FieldThe Extended Real Number System
Euclidean Spaces1
The Extended Real Number System
Definition
The extended real number system consists of the real field R
andtwo symbols, +∞ and −∞.
We preserve the original order from R and define
−∞ < x < +∞
for all x ∈ R.
Theorem
Every subset of the extended real numbers has a least upper
bound(as well as a greatest lower bound)
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The Real FieldThe Extended Real Number System
Euclidean Spaces1
The Extended Real Number System
Definition
The extended real number system is not a field. However it
iscustomary to make the following conventions
(a) If x is real then
x +∞ = +∞x −∞ = −∞
x+∞ =
x−∞ = 0
(b) If x > 0 then x · (+∞) = +∞ and x · (−∞) = −∞(c) If x
< 0 then x · (+∞) = −∞ and x · (−∞) = +∞
We also call elements of R finite when we want to
distinguishthem from +∞,−∞
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The Real FieldThe Extended Real Number System
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Real Vector Spaces
Definition
For each positive integer k let Rk be the set of all ordered
k-tuples
x = 〈x1, . . . , xk〉
where x1, . . . , xk are real numbers called the coordinates of
x. Theelements of Rk are called points or vectors (especially if k
> 1).
R1 is often called the real line and R2 is often called the real
plane
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The Real FieldThe Extended Real Number System
Euclidean Spaces1
Real Vector Spaces
Definition
If x = 〈x1, . . . , xk〉 and y = 〈y1, . . . , yk〉 are elements of
Rk andα ∈ R then we define
x + y = 〈x1 + y1, . . . , xk + yk〉αx = 〈αx1, . . . , αxk〉
This defines addition of vectors and multiplication of a vector
by areal number (called a scalar).
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The Real FieldThe Extended Real Number System
Euclidean Spaces1
Real Vector Spaces
Theorem
Vector addition and scaler multiplication satisfy the
commutative,associative and distributive laws. Hence Rk is a vector
space overthe real field.
Definition
The zero element of Rk is 0 = 〈0, . . . , 0〉 and is sometimes
calledthe origin or null vector.
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The Real FieldThe Extended Real Number System
Euclidean Spaces1
Inner Product
Definition
If x = 〈x1, . . . , xk〉 and y = 〈y1, . . . , yk〉 are elements of
Rk thenwe define the inner product (or scalar product) of x and y
as
x · y =k∑
i=1
xiyi
we also define the norm of x to be
|x| = (x · x)1/2 =
√√√√ k∑i=1
x2i
The above structure (the vector space Rk with the above
innerproduct and norm) is called Euclidean k-space.
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The Real FieldThe Extended Real Number System
Euclidean Spaces1
Theorems
Theorem
Suppose x, y, z ∈ Rk and α ∈ R. Then(a) |x| ≥ 0(b) |x| = 0 if
and only if x = 0(c) |αx| = |α||x|(d) |x · y| ≤ |x||y|(e) |x + y| ≤
|x|+ |y|(f) |x− z| ≤ |x− y|+ |y− z|
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The Real FieldThe Extended Real Number System
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Misc. Theorems
Theorem
Any terminated decimal represents a rational number
whosedenominator contains no prime factors other than 2 or
5.Conversely, any such rational number can be expressed, as
aterminated decimal.
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The Real FieldThe Extended Real Number System
Euclidean Spaces1
Misc. Theorems
Theorem
Show that there is a one-to-one correspondence between the set
Nof integers and the set Q of rational numbers, but that there is
noone-to-one correspondence between N and the set R of
realnumbers.
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The Real FieldThe Extended Real Number System
Euclidean Spaces1
Misc. Theorems
Theorem
Let xn + a1xn−1 + a2x
n−2 + ...+ an = 0 be a polynomial equationwith integer
coefficients (note that the leading coefficient is 1).Then the only
possible rational roots are integers.
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The Real FieldThe Extended Real Number System
Euclidean Spaces1
Misc. Theorems
Theorem
If a and b are both rational, then√
a +√
b is not rational unless√a and
√b are both rational.
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The Real FieldThe Extended Real Number System
Euclidean Spaces1
Misc. Theorems
Theorem
Let a and b denote positive real numbers and n a positive
integer.Then
1
2(an + bn) ≥ [1
2(a + b)]n
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The Real FieldThe Extended Real Number System
Euclidean Spaces1
Misc. Theorems
Theorem
Let a1, a2, . . . , an be positive real numbers. Then
(a1 + a2 + · · ·+ an)(a−11 + a−12 + · · ·+ a
−1n ) ≥ n2
The Real FieldThe Extended Real Number SystemEuclidean
SpacesMisc. Results