|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 the numbers used for ordinary measurement of physical things like length, area, weight, charge, etc. 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 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, complete, and Archimedean. The reals can be constructed from the rationals by means of Dedekind cuts or Cauchy sequences, i.e. it is the completion of the metric space of rational numbers.
The real numbers can be thought of as a line, called the real line. Each real number represents a point on the real line. However, it is a mistake to think of the real line as a row of individual points, like beads. There is no real number “just to the right” of a given real number. This is because the real numbers, like the rational numbers, are a dense set, so points accumulate around each other.
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.
- 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.
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 caclulus and analysis. It is essential for many theorems, notably the mean value theorem and the intermediate value theorem.
The real numbers do not include or (infinity and minus infinity). 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 no smallest real number, because you can always make a real number smaller by subtracting from it.
Every real number is finite. One way to see this is to observe that you cannot subtract infinity from itself—the result is indeterminate—but, for any real number x, then x - x = 0, exactly.
It is sometimes convenient to have a set of numbers that does include infinity. For example, in computer programming, "real arithmetic" is often done by a specific system defined by standard IEEE 754-1985; this system is built in to modern processor chips. It provides for values which print out as INF and -INF and which participate in arithmetic as if they were numbers. Thus, division by zero, which was often an error that stopped calculation on older machines, can be a legal operation which simply produces a +INF or -INF result. The system of numbers implemented in IEEE 754 is known in mathematics as the "affinely extended real numbers."
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.