The Real Field The Extended Real Number System Euclidean Spaces 1 Existence of Reals

Theorem There exists an ordered field R which has the least upper bound property. 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 are isomorphic. The Real Field The Extended Real Number System Euclidean Spaces 1 Archimedean Property

Theorem (a) If x, y ∈ R and x > 0 then there is a positive integer n such that nx > y (b) If x, y ∈ R and x < y then there exists a p ∈ Q such that x < p < y

Part (a) is the called the archimedean property of R. Part (b) may says Q is dense in R, i.e. between any two real numbers there is a rational one. The Real Field The Extended Real Number System Euclidean Spaces 1 nth Root

Theorem For every real x > 0 and every integer n > 0 there is one and only one positive real y such that y n = x √ The number y is written y = n x or y = x1/n The Real Field The Extended Real Number System Euclidean Spaces 1 nth Root

Corollary If a and b are positive real numbers and n is a positive integer then

(ab)1/n = a1/nb1/n Suppose we have chosen n0, n1,..., nk−1. Then let nk be the largest integer such that n n n + 1 + ··· + k ≤ x 0 101 10k

n1 nk If we let E = {n0 + 101 + ··· + 10k : k ∈ N} then x = sup E and the 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 a least upper bound.

The Real Field The Extended Real Number System Euclidean Spaces 1 Decimals

Let x > 0 be a real number. Then let n0 be the largest integer such that n0 ≤ x (such an integer exists by the archimedean property of R). Conversely for any decimal expansion n0.n1n2 ··· , the set {n0.n1n2 ··· nk : k ∈ N} is bounded above and hence must have a least upper bound.

The Real Field The Extended Real Number System Euclidean Spaces 1 Decimals

Let x > 0 be a real number. Then let n0 be the largest integer such that n0 ≤ x (such an integer exists by the archimedean property of R).

Suppose we have chosen n0, n1,..., nk−1. Then let nk be the largest integer such that n n n + 1 + ··· + k ≤ x 0 101 10k

n1 nk If we let E = {n0 + 101 + ··· + 10k : k ∈ N} then x = sup E and the decimal expansion of x is n0.n1n2 ··· The Real Field The Extended Real Number System Euclidean Spaces 1 Decimals

Let x > 0 be a real number. Then let n0 be the largest integer such that n0 ≤ x (such an integer exists by the archimedean property of R).

Suppose we have chosen n0, n1,..., nk−1. Then let nk be the largest integer such that n n n + 1 + ··· + k ≤ x 0 101 10k

n1 nk If we let E = {n0 + 101 + ··· + 10k : k ∈ N} then x = sup E and the 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 a least upper bound. Theorem Every subset of the extended real numbers has a least upper bound (as well as a greatest lower bound)

The Real Field The Extended Real Number System Euclidean Spaces 1 The Extended Real Number System

Definition The extended real number system consists of the real field R and two symbols, +∞ and −∞.

We preserve the original order from R and define

−∞ < x < +∞

for all x ∈ R. The Real Field The Extended Real Number System Euclidean Spaces 1 The Extended Real Number System

Definition The extended real number system consists of the real field R and two 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 Field The Extended Real Number System Euclidean Spaces 1 The Extended Real Number System

Definition The extended real number system is not a field. However it is customary 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 distinguish them from +∞, −∞ The Real Field The Extended Real Number System Euclidean Spaces 1 Real Vector Spaces

Definition k For each positive integer k let R be the set of all ordered k-tuples

x = hx1,..., xk i

where x1,..., xk are real numbers called the coordinates of x. The k elements of R are called points or vectors (especially if k > 1).

1 2 R is often called the real line and R is often called the real plane The Real Field The Extended Real Number System Euclidean Spaces 1 Real Vector Spaces

Definition k If x = hx1,..., xk i and y = hy1,..., yk i are elements of R and α ∈ R then we define x + y = hx1 + y1,..., xk + yk i

αx = hαx1, . . . , αxk i This defines addition of vectors and multiplication of a vector by a real number (called a scalar). The Real Field The Extended Real Number System Euclidean Spaces 1 Real Vector Spaces

Theorem Vector addition and scaler multiplication satisfy the commutative, k associative and distributive laws. Hence R is a vector space over the real field.

Definition k The zero element of R is 0 = h0,..., 0i and is sometimes called the origin or null vector. The Real Field The Extended Real Number System Euclidean Spaces 1 Inner Product

Definition k If x = hx1,..., xk i and y = hy1,..., yk i are elements of R then we define the inner product (or scalar product) of x and y as

k X x · y = xi yi i=1 we also define the norm of x to be v u k 1/2 uX 2 |x| = (x · x) = t xi i=1

k The above structure (the vector space R with the above inner product and norm) is called Euclidean k-space. The Real Field The Extended Real Number System Euclidean Spaces 1 Theorems

Theorem k Suppose x, y, z ∈ R 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 Field The Extended Real Number System Euclidean Spaces 1 Misc. Theorems

Theorem Any terminated decimal represents a rational number whose denominator contains no prime factors other than 2 or 5. Conversely, any such rational number can be expressed, as a terminated decimal. The Real Field The Extended Real Number System Euclidean Spaces 1 Misc. Theorems

Theorem Show that there is a one-to-one correspondence between the set N of integers and the set Q of rational numbers, but that there is no one-to-one correspondence between N and the set R of real numbers. The Real Field The Extended Real Number System Euclidean Spaces 1 Misc. Theorems

Theorem n n−1 n−2 Let x + a1x + a2x + ... + an = 0 be a polynomial equation with integer coefficients (note that the leading coefficient is 1). Then the only possible rational roots are integers. The Real Field The Extended Real Number System Euclidean Spaces 1 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 Field The Extended Real Number System Euclidean Spaces 1 Misc. Theorems

Theorem Let a and b denote positive real numbers and n a positive integer. Then 1 1 (an + bn) ≥ [ (a + b)]n 2 2 The Real Field The Extended Real Number System Euclidean Spaces 1 Misc. Theorems

Theorem

Let a1, a2,..., an be positive real numbers. Then

−1 −1 −1 2 (a1 + a2 + ··· + an)(a1 + a2 + ··· + an ) ≥ n