Real number

The real numbers are a fundamental structure in the study of mathematics. The real numbers are a mathematical set with the properties of a complete ordered field. While these properties identify a number of facts, not all of them are essential to completely define the real numbers.

The real numbers can either be defined axiomatically as a complete ordered field, or can be reduced by set theory as a set of all limits of cauchy sequences of rational numbers (a completion of a metric space). Either way, the constructions produce field-isomorphic sets.

Axioms

The minimum set of properties that must be given "by definition" so that all other properties may be proven from them is the set of axioms for the real numbers. We begin with a set $\R$. We call the elements of $\R$ the real numbers.

Field axioms

Main article: Field

The field axioms define how two operations, addition (symbolized by $+$) and multiplication (symbolized by $\times$, a dot $\cdot$ or, where no confusion exists, simple concatenation of objects without a symbol) interact with the set of real numbers. For these axioms, we assume the existence of two operations, $+$ and $\times$ on $\R$.

For any two real numbers $x$, $y$ and $z$, we may assume:

1. Commutivity:
   1. $x+y=y+x$;
   2. $xy=yx$ (or, $x\times y=y\times x$);
2. Associativity:
   1. $\left(x+y\right)+z=x+\left(y+z\right)$;
   2. $\left(xy\right)z=x\left(yz\right)$;
3. Distributive property: $x\left(y+z\right)=xy+xz$;
4. Identities:
   1. Additive: There exists a real number $0$ such that for any real number $x$, $x+0=x$;
   2. Multiplicative: There also exists a real number $1$, different from $0$ such that for any real number $x$, $1x=x$;
5. Inverses:
   1. Additive: There exists a real number $-x$ such that $x+\left(-x\right)=0$.;
   2. Multiplicative: If $x \neq 0$, there exists a real number $x^{-1}$ such that $xx^{-1}=1$.

Order axioms

Along with the field properties, there exists a total order $\le$ on the real numbers that make the real numbers an ordered field. That $\le$ is a total order is to satisfy the following properties for all $x$, $y$, and $z$ in $\R$:

1. Antisymmetry: If $x \le y$ and $y \leq x$, then $x = y$;
2. Transitivity: If $x \le y$ and $y \le z$, then $x \le z$;
3. Totality: if $x \le y$ and $y \leq x$, then $x = y$;

With the total order property, we can define other relations $\geq$, $<$, and $>$.

Furthermore, an ordered field must also satisfy the following properties:

1. Translation invariance: If $x \le y$, then $x+z\le y+z$;
2. Closure of multiplication on non-negative elements: if $x>0$ and $y>0$, then $xy>0$;

Least upper bound axiom

To understand the least upper bound axiom, we must first understand what an upper bound is. If $S$ is a subset of the real numbers, and $x$ is a real number, we define $x$ to be an upper bound for $S$ if $x\ge s$ for all $s$ in $S$.

The least upper bound property states:

1. If $S$ is a non-empty subset of the real numbers, and there exists some upper bound of $S$, then there exists an upper bound $M$ of $S$ that is the least upper bound. That is, if $N$ is any real number that is also an upper bound for $S$, then $M \le N$.

The immediate implications of the least upper bound property is that if $S$ is a non-empty subset of the reals that has an upper bound, then the least upper bound of $S$ is the maximum element in $S$ if it exists. Otherwise, the least upper bound is the minimum element of the set of real numbers that are all greater than every element in $S$.

Natural number axioms

The natural numbers ($\mathbb{N}$) can be axiomatized using the Peano axioms. However, if one wishes to consider $\mathbb{N}$ as a subset of $\R$, it may be necessary to include these axioms:

1. Zero is a natural number;
2. For any real number $n$, if $n$ is a natural number, then $n+1$ is also a natural number.
3. Given any predicate P on the natural numbers, if $P\left(0\right)$ is true and $P\left(k\right)$ implies $P\left(k+1\right)$ for any $k\in\mathbb{N}$, then $P\left(n\right)$ is true for all $n \in \N$. (Principle of mathematical induction.)

It can be then shown that the natural numbers as axiomatized above also model the Peano axioms. Indeed, two of these axioms are directly taken from the Peano axioms, while another is a minor restatement of another Peano axiom.

Definitions

As with any field, we may define subtraction and division as follows:

1. For all $x, y \in \R$, $x-y=x+\left(-y\right)$;
2. For all $x, y \in \R$ with $y \ne 0$, $\frac{x}{y}=xy^{-1}$

As with an ordered field, we may define:

1. Absolute value function: For all real numbers $x$, $\left|x\right|=x$ if $x \ge 0$ and $\left|x\right|=-x$ otherwise.

In addition to the natural numbers axiomatized as a subset, we may define these additional subsets of $\R$:

1. The set of integers: $\mathbb {Z} =\left\{m-n\mid m,n\in \mathbb {N} \right\}$, or alternatively, the union of the natural numbers and their additive inverses;
2. The set of rational numbers: $\Q=\left\{\frac{p}{q}\mid p,q\in\Z,q\ne0\right\}$;
3. The set of positive real numbers: $\mathbb {R} ^{+}=\left\{x\in \mathbb {R} \mid x>0\right\}$;
4. The set of negative real numbers: $\mathbb {R} ^{-}=\left\{x\in \mathbb {R} \mid x<0\right\}$;
5. The set of nonzero real numbers: $\mathbb {R} ^{*}=\left\{x\in \mathbb {R} \mid x\neq 0\right\}$;
6. The set of positive rational numbers: $\mathbb {Q} ^{+}=\left\{x\in \mathbb {Q} \mid x>0\right\}$;
7. The set of negative rational numbers: $\mathbb {Q} ^{-}=\left\{x\in \mathbb {Q} \mid x<0\right\}$;
8. The set of nonzero rational numbers: $\mathbb {Q} ^{*}=\left\{x\in \mathbb {Q} \mid x\neq 0\right\}$;
9. The set of positive integers: $\mathbb {Z} ^{+}=\left\{x\in \mathbb {Z} \mid x>0\right\}$;
10. The set of negative integers: $\mathbb {Z} ^{-}=\left\{x\in \mathbb {Z} \mid x<0\right\}$

Theorems

The following are true for any $x$, $y$, and $z$, and are a direct result of the real numbers being a field (as well as a ring).

1. $-\left(-x\right)=x$ (proof);
2. If $x \neq 0$, then $\left(x^{-1}\right)^{-1}=x$ (proof);
3. $0x = 0$ (this is why $0$ does not have a multiplicative inverse) (proof);
4. $\left(-x\right)y=x\left(-y\right)=-\left(xy\right)$ (proof);
5. $\left(-x\right)\left(-y\right)=xy$ (proof);
6. $\left(-1\right)\cdot x=-x$ (proof);
7. $\left(-1\right)\left(-1\right) = 1$ (proof);
8. $x\left(y-z\right)=xy-xz$ (proof).

This is a result of the Least Upper Bound axiom:

1. The set of real numbers is a complete metric space. That is, every Cauchy sequence converges.