Real numbers can be thought of as numbers which can be represented by some infinite or finite decimal representation, such as 0.707106781187...
They are called "real" because they actually exist in the real world, i.e. measures, weights, temperatures and so on are all real numbers, as opposed to the imaginary numbers, which are just an abstract concept, but do not really exist.
In a number line representation, the real numbers correspond to all the points on a geometric line. The distance between any two points on a line is a real number.
The real numbers include within them all of these other kinds of numbers:
- The "natural numbers" or positive integers, 1, 2, 3, ...
- Zero and the negative integers
- Fractions, like 355/113
- Any decimal representation which terminates (comes to an end), like 6.023, because this is just a way of writing a fraction (in this case, 6023/1000)
- Any decimal representation which repeats or recurs, like 1.86292929292929..., because these can be shown to be fractionsTemplate:Prove
- Irrational numbers, like π = 3.1415926525..., whose decimal representations never repeat or 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 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."
Notes and references
- In reality, the actual values the computer uses are very-high-precision fractions which can equal or approximate real numbers
The introduction is actually a blatant misrepresentation of the truth. Real numbers are no more or less "abstract" than imaginary numbers. The author, one must conclude, was trying to allude to the fact that there cannot be an imaginary number of objects, but there cannot be a negative number of objects either. Or perhaps s/he was trying to say that the imaginary numbers cannot be represented on the number line, which is irrelevant, since they can be represented on the complex plane, an equally valid, and more complete, numerical device. Finally, perhaps the author simply meant that the square root of a negative number does not exist, which is wrong. It does exist, in the guise of an imaginary number. While this may seem cyclical, simply because a number is "unusual" does not mean that it is abstract and non-existent.
Illustrations of this fact, and the incorrectness of the introduction, can be seen when solving differential equations which contain a bifurcation parameter, where the method of eigenvalues must be used. There, an complex, "imaginary" solution to one of several equations often contains eigenvalues and thus a corresponding eigenfunction. Therefore, the real-life existent solution to a real-life existent problem depends on an imaginary number. This exemplifies why complex numbers are just as valid as real numbers, and are not merely an "abstract concept," as the introduction erroneously points out. Or to be more precise, ALL of mathematics is an abstract concept, but there exists an isomorphism to the "real world"; an isomorphism that holds just as strongly in the domain of complex numbers as in the subset of complex numbers, the reals.