Numbers and interval notationWe first look at sets of numbers and conclude with notation for certain kinds of sets of real numbers.Let Z, Q, and R denote the sets of integers, rational numbers, and real numbers, respectively.Let N denote the set of natural numbers. Let C denote the set of complex numbers (also known asthe complex plane).When every element of a set A is also an element of another set B,wesaythatA is a subsetof B and express this relationship in symbols as A ⊂ B. With this subset notation we can expressthe relationship between the various sets of numbers compactly asN ⊂ Z ⊂ Q ⊂ R ⊂ C .(1) The set N of natural numbers consists of the whole, or counting, numbers {1, 2, 3,...,n,...} .(2) We get the next larger set Z of (positive and negative) integers from the set of natural numbersN by adding to N the solutions of all equations like x + b = a,wherea and b are naturalnumbers. We can denote the solution x using subtraction as x = a−b. For example, x+3 = 5yields nothing new, namely x =5− 3 = 2, but x + 5 = 3 does, namely x =3− 5=−2.(3) The set Q of rational numbers is obtained by adding to the set of integers Z the solutions ofall equations like bx =a where b is not zero. For example, 3 x = 12 gives nothing new asx =12/3 = 4, but 3 x =11does,namelyx =11/3.(4) The set of real numbers R is larger still. Every rational number can be expressed in decimalform. For example, 3/8=0.375 or 1/3=0.333 333 .... That not all quantities were rationalnumbers (ratios of whole numbers) was known to the Pythagorean Greeks two millenniaago. Indeed there is a very simple proof, attibuted to Euclid, that the square root of two isirrational. In symbols, this means that the equation x2=2hasno solution in the set Q ofrational numbers. In a certain sense, almost all decimal numbers are not rational, but to seethis requires a fair amount of effort. For example, F. Lindemann, in 1882, was the first personto show that π =3.141 592 65 ... was not only not rational, but was not the solution of anypolynomial equation with integer coefficents.(5) The set C of complex numbers is obtained from the set R of reals numbers by adding allthe solutions to all polynomial equations with real coeeficients. For example, although theequation x2+ 2 = 3 gives us nothing new, namely x2=3− 2=1,sox = ±1, the equationx2+ 3 = 2 does. Indeed, in this case x2= −1 and we can denote the two (complex orimaginary) solutions as x =+√−1=0+1i = i and x = −√−1=0−1 i = −i.However, we have now reached the end of our journey. Considering polynomial equations withcomplex coefficients we have reached closure: no new numbers are required. Although much antic-ipated, both Leonhard Euler (1707–1783) and Nicolaus Bernoulli (1687–1759) realized that everyn-th degree polynomial equation with complex coefficents had n complex roots, a fully rigorousdemonstration of the Fundamental Theorem of Algebra had to wait for Carl Friedrich Gauss’ 1799proof.1Numbers and interval notationInterval notation is a compact way to identify uninterrupted portions of the set of real numbers.We consider the set of real numbers that are not negative and that are not bigger than one.(1) The set I of real numbers from zero to one, including both zero and one.(2) The set I is the set of numbers x that are real and such that zero is less than, or equal to, xand x is less than, or equal to, one.(3) I = {x ∈ R |0 ≤ x and x ≤ 1}.(4) The set I =[0, 1].More generally, we have the following two conventions for bounded intervals.(a, b)={x ∈ R |a<x<b} is the open interval from a to b.[ a, b ]={x ∈ R |a ≤ x ≤ b} is the close d interval from a to b.There are four kinds of unbounded intervals:(a, ∞)={x ∈ R |a<x}[ a, ∞)={x ∈ R |a ≤ x}(−∞,b)={x ∈ R |x<b}(−∞,b]={x ∈ R |x ≤ b}In the preceding, the (finite) number a is the left endpoint and the number b is the right endpoint.Note than in the case that a = b, the open interval (a, a)=(b, b)hasno elements, i.e., is empty.The empty set is denoted by ∅. On the other hand [ a, a ] just consists of the one number a.Notmuch of an interval, right?Question: What numbers belong to the set [ 0, 10)?W0L1Patrick Dale McCray, 2 September
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