Real Numbers

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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: $\mathbb {R}$. They are the 4^th item in this hierarchy of types of numbers:

• The "natural numbers"—$\mathbb {N}$, 0, 1, 2, 3, ... (There is controversy about whether zero should be included. It doesn't matter.)
• The "integers"—$\mathbb {Z}$, positive, negative, and zero
• The "rational numbers"—$\mathbb {Q}$, or fractions, like 355/113
• The "real numbers"—$\mathbb {R}$, including irrational numbers
• The "complex numbers"—$\mathbb {C}$, 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 ${\sqrt {10}}=3.162277660168...\,$ and $\pi =3.1415926535...\,$. These decimal representations neither repeat nor terminate.

Formal definition

Formally, real numbers are the extension of the rationals that is metrically complete, as explained below. They could also be 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.

Real line

The real numbers can be thought of as a line, called the real line. Each real number represents a point on the real line.

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 in a trillion using the number 3.1415926535898 ($\pi \,$ 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.

What do we mean when we say that it's intuitively obvious that the square root of 2 exists? What we are really saying is the the function $f(x)=x^{2}$ goes from being less than 2 at x=1.414 to being greater than 2 at x=1.415, and is continuous. So it must pass through the value of 2 exactly. This is the intermediate value theorem, and is actually rather subtle. In fact, it is the problem that is addressed by the real numbers. The fact is, if one is restricted to the rationals, there is no square root of 2, (and hence the intermediate value theorem isn't true.) This is the famous "Pythagorean catastrophe". (The ancient Greeks did all of their mathematics geometrically, and all manipulations involved ratios of line lengths that had to involve integers. This meant that they could only deal with rational numbers. The circumference of a circle, and the diagonal of a unit square, were quite troubling to them.)

There is no rational number that has a square of 2. That is, ${\sqrt {2}}$ is not a member of $\mathbb {Q}$. The proof is by contradiction.

Let us first assume that ${\sqrt {2}}$ is a rational number. Then, it can be written in the form ${\frac {m}{n}}$, where both $m\,$ and $n\neq 0$ are integers. Without loss of generality, assume that $m$ and $n$ are the smallest such numbers, in other words, that ${\frac {m}{n}}$ is written in lowest terms. Thus, we can write:

$\left({\frac {m}{n}}\right)^{2}={\frac {m^{2}}{n^{2}}}=2$.

Or, equivalently, $m^{2}=2n^{2}$, which shows that $m^{2}$ is an even number. Then m is an even number, since the square of an even number is even and the square of an odd number is odd (justification left to student). Then write $(2k)^{2}=4k^{2}=2n^{2}$ for some integer $k$, which shows that $n^{2}$ is even and so therefore is $n$. Now we have $m$ and $n$ both even numbers, contradicting the assumption that ${\frac {m}{n}}$ is written in lowest terms. Hence ${\sqrt {2}}$ is not a rational number.

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 $(3,6)\,$. 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 $(3,6)\,$. 3 is its glb. 6 and 3 are also the lub and glb of the closed interval $[3,6]\,$—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". For simple cases like the real numbers, the terms "maximum" and "minimum" may also be used.

A set has the least upper bound property if every subset of said set that has an upper bound also 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 then 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 ${\sqrt {2}}$ is, in effect, defined as "the least upper bound of the set of numbers whose squares are less than 2" or as "the Dedekind cut whose square is the Dedekind cut known as 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.)

Infinity

The real numbers do not include infinity. Every real number is finite, though the set of reals is an infinite set.

See also

For a discussion of the philosophical and historical aspects of the various types of numbers, see Our_Playground:_The_Real_Numbers_and_Their_Development.