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1. The real numbers
Contents
1.1 Introduction 1.2 The order relation 1.3 The natural numbers 1.4 The integers 1.5 The rational numbers 1.6 Absolute value, intervals, bounds 1.7 Polynomials
1.1 Introduction
The set R of the real numbers is constructed in such a way that each point of a straight line corresponds to a real number, and conversely.
Independently of the formal construction, we summarize the properties of R. On R is defined a sum that assigns to each pair of real numbers a, b another real number a + b. This operation has the following properties:
- (Associative) a + (b + c) = (a + b) + c for all a, b, c ∈ R.
- (Commutative) a + b = b + a for all a, b ∈ R.
- (Existence of neutral element) There exists a real number 0 such that a + 0 = a for all a ∈ R.
- (Existence of opposite elements) For each real number a, there exists a real number −a such that a + (−a) = 0.
On R is defined a second operation, the product, which assigns to each pair of real numbers a, b another real number ab. This operation has the following properties:
- (Associative) a(bc) = (ab)c for all a, b, c ∈ R.
- (Commutative) ab = ba for all a, b ∈ R.
- (Existence of neutral element) There exists a real number 1 such that a · 1 = a for all a ∈ R.
- (Existence of inverse elements) For each a ∈ R \ { 0 }, there exists a real number a−^1 (denoted also 1/a) such that a · a−^1 = 1.
Finally, the distributive property relates both operations:
- a(b + c) = ab + ac for all a, b, c ∈ R.
Any set with two operations satisfying all the properties above is called a field. These properties justify usual conventions, such as not writing parenthesis when there are more than two addends or factors, writing a − b instead of a + (−b) and so on. From these properties we can deduce also another well known ones, as −(a + b) = −a − b, the laws for simplification in sums (a + c = b + c ⇔ a = b) and products (ac = bc ⇔ a = b, if c 6 = 0), etc.
1.2 The order relation
On R is defined an order relation ≤. The notation a < b means a ≤ b and a 6 = b. This relation is total, which means that, given two real numbers a, b, it holds exactly one of the three properties a < b, a = b or a > b.
The behavior of the order relation with respect to the operations is the following. For all a, b, c ∈ R,
- a < b ⇔ a + c < b + c.
- If c > 0, then a < b ⇔ ac < bc.
- If c < 0, then a < b ⇔ ac > bc.
1.3 The natural numbers
The set of natural numbers is the subset N of R formed by the numbers obtained summing up 1 to itself repeatedly: N = { 1 , 2 , 3 ,.. .}.
The sum and product of two natural numbers is also a natural number. Certainly, the valid properties for the real numbers (associative, commutative, distributive, etc.) are particularly valid in N. Note, however, that the neutral of the sum 0 is not natural, that the opposite of a natural is not natural and the inverse of a natural is not a natural except for 1, which is its own inverse.
The restriction to N of the order relation has at least two particular relevant properties. The first one is that, if A is a nonempty subset of N, then there exists an element m ∈ A such that m ≤ a for all a ∈ A; this element is called minimum or first element of A. The second one is the following.
Induction principle. Let A be a nonmempty subset of N and m its minimum. If n ∈ A implies n + 1 ∈ A, then A = {n ∈ N : n ≥ m}.
1.4 The integers
The set Z of the integers is constituted by the natural ones, the opposite of them and the number 0: Z = {... , − 3 , − 2 , − 1 , 0 , 1 , 2 , 3 ,.. .}. The sum and product of two integers are integers. On Z, these operations have the same properties that the sum and product of real numbers, except for the fact that it is not true that every integer a 6 = 0 has an inverse in the set of integers (only 1 and −1 have inverses). The definition commutative
The distance betweeen two real numbers a and b is the number |a − b|.
The intervals are relevant subsets of R. Let a, b ∈ R with a < b. The set [a, b] = {x ∈ R | a ≤ x ≤ b} is called closed interval with endpoints a and b; the set (a, b) = {x ∈ R | a < x < b} is called open interval with endpoints a and b; the sets (a, b] = {x ∈ R | a < x ≤ b} and [a, b) = {x ∈ R | a ≤ x < b} are half–closed or half-open intervals. Finally, the sets
(−∞, a] = {x ∈ R | x ≤ a}, (−∞, a) = {x ∈ R | x < a}, [a, +∞) = {x ∈ R | x ≥ a}, (a, +∞) = {x ∈ R | x > a},
are called unbounded intervals. Sometimes we will write R = (−∞, +∞).
Let a and δ > 0 be real numbers. The δ–neighbourhood of a is the open interval (a − δ, a + δ). Note the equivalence
x ∈ (a − δ, a + δ) ⇔ |x − a| < δ.
The open intervals (a, a + δ) and (a − δ, a) will be called, respectively, right–half–δ– neighbourhood and left–half–δ–neighbourhood.
A neighbourhood of +∞ is an interval of the form (a, +∞) and a neighbourhood of −∞ is an interval of the form (−∞, a).
Let A be a subset of R. The number k is an upper bound of A if a ≤ k for all a ∈ A. If A has upper bounds we will say that A is bounded from above, and the smallest upper bound is called the supremum of A. The number h is a lower bound of A if h ≤ a for all a ∈ A. If A has lower bounds we will say that A is bounded from below, and the greatest lower bound is called the infimum of A. The following property holds in R.
Supremum (infimum) theorem. Any nonempty subset of R bounded from above (resp. below) has supremum (resp. infimum).
If A has supremum M and M belongs to A, then M is called maximum of A, denoted max A; if A has infimum m and m belongs to A, then m is called minimum of A, denoted min A.
Let x be a real number. The lower integer part of x (also called floor of x) is bxc = max{m ∈ Z : m ≤ x}.
The upper integer part of x (also called ceiling of x) is dxe = min{m ∈ Z : m ≤ x}.
If x is an integer, then x = bxc = dxe. If x is not an integer, then bxc and dxe are, respectively, the first integer at left and the first integer at right of x when the real numbers are represented on a straight line.
1.7 Polynomials
A polynomial with coefficients in R is a mapping p : R → R of the form
p(x) = anxn^ + an− 1 xn−^1 + · · · + a 1 x + a 0
where n ≥ 0 is a natural number and a 0 ,... , an are real numbers, called coefficients of the polynomial. If an 6 = 0, the number n is called degree of the polynomial p(x) and is denoted by deg p(x). The set of all the polynomials with coefficients in R is denoted by R[x].
Note that if p(x) = anxn^ + · · · + a 0 is a polynomial and m > n, it can also be written as p(x) = amxm^ + · · · + anxn^ + · · · + a 0 by taking am = am− 1 = · · · = an+1 = 0.
The sum of two polynomials p(x) = anxn^ + · · · + a 0 and q(x) = bnxn^ + · · · + b 0 is the polynomial defined by
(p + q)(x) = p(x) + q(x) = (an + bn)xn^ + (an− 1 + bn− 1 )xn−^1 + · · · + (a 0 + b 0 ).
This operation is associative and commutative; it admits neutral element, which is the polynomial defined by e(x) = 0; and each polynomial p(x) has an opposite, which is the polynomial −p(x), obtained by changing the sign of all the coefficients of p(x).
The product of two polynomials p(x) = anxn^ + · · · + a 0 and q(x) = bnxn^ + · · · + b 0 is the polynomial defined by
(p · q)(x) = p(x)q(x) = cn+mxn+m^ + · · · + c 0 , with ck = akb 0 + ak− 1 b 1 + · · · + a 0 bk,
for all k ∈ { 0 ,... , n + m}. This product is associative and commutative; it admits neutral element, which is the polynomial defined by u(x) = 1; and it is distributive with respect to the sum. So, R[x], with this two operations, is a commutative ring.
Let a ∈ R. The constant polynomial a is the polynomial defined by p(x) = a. The sum and product of constant polynomials have the same properties as the sum and product of real numbers, so every constant polynomial p(x) = a is identified with the real number a.
R[x] is not a field, because it is not true that every polynomial has inverse for the product. The only polynomials having inverses are the nonzero constants.
Division theorem. If p(x), q(x) ∈ R[x], with q(x) 6 = 0, then there exist unique polynomials c(x), r(x) ∈ R[x] such that
p(x) = q(x)c(x) + r(x), with r(x) = 0 or deg r(x) < deg g(x).
The polynomials c(x) and r(x) are called, respectively, quotient and remainder of the division of p(x) by g(x). If r(x) = 0, we say that p(x) is divisible by q(x) and that p(x) can be decomposed into product of the two factors q(x) and c(x).
Remainder theorem. Let p(x) be a polynomial and a ∈ R. The value p(a) is the remainder of the division of p(x) by x − a.
n 0
n n
= 1, (n ≥ 0).
n 1
n n − 1
= n, (n ≥ 1).
n k
n n − k
, (0 ≤ k ≤ n).
n − 1 k − 1
n − 1 k
n k
, (1 ≤ k ≤ n).
The binomial numbers appear in the Newton binomial formula. We will apply it essentially to polynomials, but it is valid in any commutative ring. If n ≥ 0 is a natural number and a, b belong to a commutative ring (for example, polynomials), then
(a + b)n^ =
n 0
an^ +
n 1
an−^1 b +
n 2
an−^2 b^2 + · · · +
n n − 1
abn−^1 +
n n
bn.
In the polynomials case, this formula can be seen as a factorization of the right–hand side:
( n 0
xn^ +
n 1
xn−^1 a +
n 2
xn−^2 a^2 + · · · +
n n − 1
xan−^1 +
n n
an^ = (x + a)n
Other useful factorizations are the following:
- xn^ − an^ = (x − a)(xn−^1 + xn−^2 a + · · · xan−^1 + an−^1 ).
- For n odd, xn^ + an^ = (x + a)(xn−^1 − xn−^2 a + · · · − xan−^2 + an−^1 ).
As particular cases:
- x^2 − a^2 = (x + x)(x − a).
- xn^ − 1 = (x − 1)(xn−^1 + xn−^2 + · · · + x + 1).
- For n odd, xn^ + 1 = (x + 1)(xn−^1 − xn−^2 + · · · − x + 1).
