Real Analysis 1-The Real Numbers

Introduction to Mathematical Thinking. Real Analysis 1. The Real Numbers The following material assumes knowledge of set theory. Integers, Rationals, Reals It would be helpful to review the material in the special course reader on elementary set theory before proceeding with this lecture. Numbers were created through the combination of two different human cognitive conceptions, counting and measurements. Anthropologists have found evidence that both numbers and mathematical concepts existed before the introduction of written numbers.The earliest surviving notched bones date to 35,000 years ago; they were probably used for keeping count of the cycles of the Moon or the seasons. It is possible that early humans used sticks or lengths of vine to measure distances. Numbers, however, which represent quantities by abstract symbols such as the number of notches on a bone or the length of a measuring device, seem to have first appeared much later: around 10,000 years ago in the case of counting collections. The activities of counting, measuring, and weighing led to two kinds of numbers: discrete counting numbers and continuous real numbers. The relationship between these two kinds of numbers was not understood until the 19th century, when the modern rail number system was constructed. The delay was due to the subtlety of the issues that had to be resolved. Let's make a quick look at the construction of the real numbers and explain what some of the problems were. The relation between the rationals and the integers was first demonstrated by using the integers to define the rationals and then showing how those can be used to define real numbers. The rational numbers can be defined as ratios of integers, starting with the integers. Rational numbers can be understood as ratios of two integers. Although it may seem simple, in reality it is quite difficult to construct the rational numbers from the integers. The rationals are represented by a quotient, a over b, which is defined for every pair ab of integers where b is non-zero. The question of how to define such a system is not a simple one. What is your response to the question, "What is the quotient of a over b?" The reason you can't answer in terms of actual quotients is because you don't have them until after you define the rationals. The point is, constructing the rational numbers from their predecessors—the integers—is quite simple. It is more difficult, however, to construct the reals from the rationals. In this section, we will consider some of the difficulties inherent in creating and using the real numbers. We begin by looking at some of the properties of the rational numbers. The rationals are a system of numbers that provide a complete set of solutions to all real-world measurement problems. The following property of rational numbers captures this idea. Let's look at the theorem. If r and s are rationals, with r less than s, then there is a rational t with r < t < s. The rational line is said to be dense, since every point on it is an integer. The rational line shows that whenever we have a rational number r and another rational number s, we can find a rational number t in between the two. Now let's prove that. Let's take t to mean r and s, since it is clear that r is less than t and t is less than s. The question is whether or not t is a rational number. The answer is obvious, but let's prove it. Let r = m/n and s = p/q, where m, n, p and q are integers. Then t equals 1/2(m/n+p/q) = mq+np / 2nq. It is clear that mq + np and 2nq are integers. Thus, the result shows that t is in q. Density makes the rational numbers especially useful for measurements. Density tells us that we can get rational numbers as close to any particular length as we wish. If a rational value is slightly smaller than one given length, and if another rational value is slightly larger, then it's always possible to find an even closer rational approximation. However, density does not mean that there are no gaps in the rational line. The end of the proof. There is some interesting information to be found in works published as late as the 19th and early 20th centuries. The irrational number known as root 2 serves as an example of the contradictions inherent in the rational system. Let's make this a little bit more precise. Let A be the set of all those rationals x for which x is either negative or nonpositive, or x squared < 2. We define B to be the set of all rational numbers x such that x is positive and x squared is greater than or equal to 2. Look at the diagram. Let us place 0. And {A, an infinite set} is defined as the set whose elements go out to infinity on the left and approach some point at infinity. That's going to be a. Let us now proceed to the next point. Let A represent a set, and let B represent another set. Also, let the sets continue up to infinity. When we put all elements of A to the left of all elements of B, we will see that every element (of A) is to the left of every element (of B). If root 2 existed in the rational line, it would appear here. The rational line is Q. We have the rational numbers, which can be split into two sets. A is the set of all negative elements and any number whose square is strictly less than 2. B is the set of all positive numbers and any number whose square is greater than or equal to 2. This divides the rational line into two segments. Clearly, a union of two sets A and B is a rational line, but neither set A nor set B has a greatest or a smallest member. The real number system had long been a mystery to mathematicians, until the late 19th and early 20th centuries, when advances in logic and set theory allowed these thinkers to discover a rigorous construction of the real number system. But it took a couple of thousand years for the Greeks to discover that the square root of 2 is irrational. What can we learn from this? Rational numbers are inadequate for many mathematical calculations. Because the squared term in the equation x2 − 2 = 0 is nonzero, we cannot solve for x. Rational numbers are useful for measuring, for doing carpentry, for doing various kinds of geometrical operations, and for building things. They are also useful for tracking stars and for astronomy. As long as we can achieve the accuracy we need, it's fine to use approximations. If we are looking for actual solutions to equations, then the rational numbers are not helpful because they cannot solve a simple quadratic equation such as x2 - 2 = 0. A solution to this problem was developed in the late 19th century; mathematicians realized that there were several different number systems. Mathematics includes the natural numbers and the integers. The integers include negative numbers or negative versions of the natural numbers. Rational numbers, which include quotients of integers, form another important class of numbers. These are useful for counting. These are useful in calculations involving negative numbers. These are useful for measuring. If you want to do advanced mathematics, then you have to work with real numbers. This progression from the natural numbers to the integers, from there to the rational numbers, and from there to the real numbers was largely a historical progression. It was done in order to increase the expressivity. The earliest form of arithmetic, approximately 10,000 years ago, was exclusively based on the natural numbers. It was developed to provide a monetary system in Sumeria, in what is known today as the Fertile Crescent region—primarily present-day Iraq. Rational numbers were invented to measure quantities in the world. Then, more abstract real numbers were needed for mathematical work. In the late 19th century, mathematicians developed a way to construct an infinite number of rational numbers by filling in the holes between them. The rationals were constructed by filling in the gaps between the rationals and the reals. The mathematicians who made this discovery were surprised by some of their findings. Some of the holes in a line, no matter how it is counted, cannot be counted as integers. Now, the rational numbers include infinitely many points. Nevertheless, an infinite number of holes vastly outnumbered an infinite number of rational numbers on an infinite scale. It was a large, almost infinite number of holes. Consequently, when the holes in the number system were filled in, the resulting real numbers contained a greater multitude of numbers than did the rational numbers. Mathematicians developed systems for counting infinite collections in order to cope with the fact that infinite sets exist. It was discovered that the real numbers as a set have infinitely many more numbers than the rational numbers. I will describe how the real numbers were constructed after I introduce or reintroduce the notion of intervals on the real line. Let a and b be real numbers with a < b. The open interval (a,b) is the set x in R such that x < a < b. The real line can be described with a simple diagram. We start with point a, then move to point b. The open interval (a, b) is the set of all numbers x such that a < x < b. But the distance between these two points is not included in this interval, so it's a segment of the line that excludes those two endpoints. The closed interval [a,b], written with square brackets like this, is the set x in R such that a < x < b. The corresponding diagram is a rectangle divided into two parts by a horizontal line segment labeled "a" and a vertical line segment labeled "b." The open interval a < x < b does not include its endpoints, whereas the closed interval [a, b] includes both a and b as elements.This may seem like a minor distinction, but it is important to note that this segment is part of the real line. There are infinitely many numbers in the real number system. In fact, the infinity of these numbers is larger than the infinity of all rational numbers combined. The rational numbers are already an infinite set. We have an infinite number of numbers in this interval and an infinite number of points on this line, yet we can distinguish between these two points at the end of the interval. This may seem like a minor distinction, but rest assured that it's actually a big deal. The difference between these two turns out to be very significant. It is closely tied to the reasons why rational numbers are not adequate for doing mathematics, and why real numbers are good for carrying out mathematical operations. Let`s quickly talk about the half-open or half-closed intervals. We have open and half-open intervals, which are essentially the same thing. For example, a, b where that's closed, and that's open, square brackets, parentheses. That is, a set of all x and R such that a ≤ x ≤ b. Another way is to take all the x in R that satisfy a < x < b. The first one is called left closed, right open. The second one is called left open, right closed. The term closed refers to including the endpoint, and open refers to excluding the endpoint. If left closed, the left endpoint is included. If right open, the right endpoint is excluded. And one more bit of notation: we sometimes include intervals that extend to infinity. We write a set of real numbers as negative infinity a to denote the set of all x in R such that x is straight left at a. This is everything to the left of a. While we might write something like negative infinity with a closed bracket here, we denote a closed part of the interval with a square bracket. Similarly, a going-out-to infinity could be the set of all real numbers x such that x is greater than a. We could have closed the set of all x in R such that x ≥ a, which is known as the infinity. We cannot have a concept of something together with infinity, and we cannot have infinity alone without a concept of something. The number infinity cannot be placed next to a square bracket because it is not a real number. Next, we will look at the property that the real numbers have, but the rational numbers do not. This is known as completeness. The completeness property of the real numbers is what makes them ideal for doing mathematics, while their lack of completeness makes the rational numbers inadequate for doing mathematics. The formulation of a system of numbers satisfying the completeness property was one of the crowning glories of late 19th century mathematics. This groundwork laid the foundation for 20th century mathematics, which is essential to modern science and technology. Given a set of real numbers, an element b with the property that for every a in A, a ≤ b is said to be an upper bound of A. For any upper bound c of set A, if b is greater than or equal to c, then we can say that b is at least an upper bound of set A. A least upper bound is the least element of a subset and also an upper bound. It is also the least one. By formulating the concept of the real number system, mathematicians were able to construct a system of numbers adequate for doing modern science, physics, technology, etc. In mathematics, often seemingly trivial concepts turn out to be fundamental and have far-reaching applications. Mathematicians found that a precise notation was the key to their success in finding the least upper bound of A. In mathematical notation, the least upper bound of a set A is denoted by lub(A). Let us make a note: We can make the same definitions for N, Z and Q. The completeness property of the real number system states that every nonempty set of reals that has an upper bound (greatest lower bound) has a least upper bound (greatest lower bound). And let us stress that this least upper bound is in the set of real numbers. One elegant statement is the key to most of modern mathematics. You need to be acquainted with the background material on upper bounds and least upper bounds before you can proceed.