The Real Field The Extended Real Number System...

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.


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.


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


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


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.


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)


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 +∞,−∞


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


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).


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.


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.


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|


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.


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.


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.


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.


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


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

The Real Field The Extended Real Number System...

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