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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.
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.
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
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
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.
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.
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.
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)
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)
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 +∞,−∞
The Real FieldThe Extended Real Number System
Euclidean Spaces1
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
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).
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.
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.
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|
The Real FieldThe Extended Real Number System
Euclidean Spaces1
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.
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.
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.
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.
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
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