|This article/section deals with mathematical concepts appropriate for a student in mid to late high school.|
The real numbers are a set of numbers with extremely important theoretical and practical properties. They can be considered to be the numbers used for ordinary measurement of physical things like length, area, weight, charge, etc. Mathematicians denote the set of real numbers with an ornate capital letter: . They are the 4th item in this hierarchy of types of numbers:
- The "natural numbers", 1, 2, 3, ... (There is controversy about whether zero should be included. It doesn't matter.)
- The "integers"—positive, negative, and zero
- The "rational numbers", or fractions, like 355/113
- The "real numbers", including irrational numbers
- The "complex numbers", which give solutions to polynomial equations
Real numbers are typically represented by a decimal (or any other base) representation, as in 3.1416. It can be shown (see below) that any decimal representation that either terminates or gets into an endless repeating pattern is rational. The other numbers are real numbers that are irrational. Examples are and . These decimal representations neither repeat nor terminate.
Formally, real numbers are defined as the unique field which is ordered, metrically complete, and Archimedean. The reals can be constructed from the rationals by means of Dedekind cuts or Cauchy sequences, as outlined below.
The real line is useful as a coordinate system for graphing functions. Thus, the x-axis and y-axis are both instances of the real line. The real line is the basis for geometric measurements, and more generally for ideas in metric topology.
What is the problem? Aren't rational numbers good enough?
Any real-world measurement that anyone could possibly make, one can make as accurately as one wants with rational numbers. For example, one can calculate the ratio of the circumference of a circle to its diameter to within one part is a trillion using the number 3.1415926535898 ( itself is irrational.) Put another way, you never have to worry about the difference between the rationals and the reals in a lumber yard or a laboratory. The technical term that topologists use for this state of affairs is that the rationals are dense.
The shortcoming of the rationals, that is overcome by defining the reals, is a somewhat subtle theoretical point. The most direct example is that, if one lived in a world with only rational numbers, 2 has no square root, even though it obviously should have one.
- One can easily prove that there is no rational number m/n such that (m/n)2 = 2. The factors of m2 all come in pairs, as do the factors of n2. But the factors of m2 must be the same as the factors of n2 except for a single extra factor of 2.
The theoretical property that the rational numbers lack is called the least upper bound property.
- Definition: A number B is an upper bound for a set of numbers if no element of the set is greater than B. (There is also the notion of a lower bound.)
For example, 10 is an upper bound for the open interval . 7 is also an upper bound, as is 6. 5 is not. 2 is a lower bound.
Some sets do not have upper bounds. For example, all rational or real numbers, or all odd integers.
- Definition: A number L is a least upper bound (often abbreviated "lub") if it is an upper bound and no other upper bound is smaller. (There is also the notion of a greatest lower bound, abbreviated "glb".) 6 is the lub of the open interval . 3 is its glb. 6 and 3 are also the lub and glb of the closed interval —the inclusion of the endpoints makes no difference.
- The least upper bound is also sometimes called the "supremum", abbreviated "sup". The greatest lower bound is also sometimes called the "infimum", abbreviated "inf".
A set has the least upper bound property if every set that has an upper bound has a least upper bound. There is also a greatest lower bound property, and any reasonable set having one property has the other.
The least upper bound property is extremely important in calculus and analysis. It is essential for many theorems, notably the mean value theorem and the intermediate value theorem.
- The rational numbers do not satisfy the least upper bound property.
For example, if we can only use rational numbers, the set of numbers that have squares less than 2 has no rational least upper bound. 1.4142136 is an upper bound, but 1.41421357 is a smaller one. The exact square root of 2 is the least upper bound that we need, but it isn't rational.
Two ways to define the reals formally
There are two ways of formally constructing the reals from the rationals. The simpler way is as Dedekind cuts, which see. A Dedekind cut could be thought of as a formal least upper bound. That is, the real number is, in effect, defined as "the least upper bound of the set of numbers whose squares are less than 2".
- (This is a common motif in theoretical mathematics—you define something as the abstract set of things that have the properties that you want, and then show that they obey all the familiar properties of the original set.)
The set thus created is "Dedekind complete", which is the same as having the least upper bound and greatest lower bound properties.
The second way is as Cauchy sequences, which see. The rationals are not "metrically complete" or "Cauchy complete", in that Cauchy sequences do not necessarily converge. The reals can be, in effect, defined as "the things that Cauchy sequences would converge to".
The reals are both Dedekind complete and metrically complete. The rationals are neither. (In general, the two properties are not the same—the complex numbers are metrically complete but not Dedekind complete.)
The real numbers do not include infinity. Every real number is finite, though the set of reals is an infinite set.
- INFINITY IS NOT A NUMBER!
However, there are non-standard models of real numbers which include or include both and .
There is no largest real number, because you can always make a real number larger by adding 1 (or 137.035 or 6.023·1023) to it, and, similarly, no smallest real number.
How do we know that irrational numbers never get into an endless repetition of digits, or, equivalently, that any number that gets into an endless repetition is rational?
Suppose we know that a number is .0039571428571428571428571428.... with the sequence "571428" repeating forever. (Astute readers will recognize instantly where this construction is going.) Take the number .99999, that is, with a number of "9" digits equal to the length of the repeating pattern. Take its reciprocal, that is (1/.999999). When working out the long division, it will be seen that the result is 1.000001000001000001... repeating forever. Multiply that by the pattern, getting 571428 * (1/.999999) = 571428.571428571428571428571428571428.... Multiply by the appropriate power of 10 (in this case 10^(-10) ) and add the necessary digits for the initial non-repeating part, in this case .0039. All of these operations were plain arithmetical operations, so the result is rational.
|This article/section deals with mathematical concepts appropriate for a student in late university or graduate level.|
In the field of topology, an important difference between the rational and real numbers is that the rational numbers are "totally disconnected", whereas the real numbers are connected. To see this, observe that the set of numbers strictly less than is open in both sets, but it is also closed in the rationals but not in the reals. In each case, it has a limit point, namely, , which it clearly does not contain. In the reals, the fact that is not contained makes the set not closed, whereas in the rationals, it doesn't matter because that point doesn't exist. In the rationals, the fact that the set is both open and closed makes it disconnected from its complement. Any open set in the rationals contains situations like this, which makes the rationals totally disconnected.
The ancient Greek mathematicians (Archimedes, Euclid, Pappus, Pythagoras and Zeno) are perhaps the first people to have created the abstract notion of a "number" (a real number; not an integer) to represent a geometrical measurement. They developed the correspondence between numbers and measurements such as distances, areas, and angles. To honor Archimedes' contribution, real analysts have named a property of the real numbers the Archimedean property. Real analysis remained in geometry's shadow until the development of the subfield of calculus. This subject subsumed all geometry known at the time, creating the field of analytic geometry.