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 2 R and x > 0 then there is a positive integer n such that nx > y (b) If x; y 2 R and x < y then there exists a p 2 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 p 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 = fn0 + 101 + ··· + 10k : k 2 Ng then x = sup E and the decimal expansion of x is n0:n1n2 ··· Conversely for any decimal expansion n0:n1n2 ··· , the set fn0:n1n2 ··· nk : k 2 Ng 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 fn0:n1n2 ··· nk : k 2 Ng 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 = fn0 + 101 + ··· + 10k : k 2 Ng 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 = fn0 + 101 + ··· + 10k : k 2 Ng then x = sup E and the decimal expansion of x is n0:n1n2 ··· Conversely for any decimal expansion n0:n1n2 ··· , the set fn0:n1n2 ··· nk : k 2 Ng 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, +1 and −∞. We preserve the original order from R and define −∞ < x < +1 for all x 2 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, +1 and −∞. We preserve the original order from R and define −∞ < x < +1 for all x 2 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 + 1 = +1 x − 1 = −∞ x x +1 = −∞ = 0 (b) If x > 0 then x · (+1) = +1 and x · (−∞) = −∞ (c) If x < 0 then x · (+1) = −∞ and x · (−∞) = +1 We also call elements of R finite when we want to distinguish them from +1; −∞ 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 2 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 jxj = (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 2 R and α 2 R. Then (a) jxj ≥ 0 (b) jxj = 0 if and only if x = 0 (c) jαxj = jαjjxj (d) jx · yj ≤ jxjjyj (e) jx + yj ≤ jxj + jyj (f) jx − zj ≤ jx − yj + jy − zj 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 p p If a and b are both rational, then a + b is not rational unless p p 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.