# Real number

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A real number is a number that falls on the real number line. It can have any value. Some examples of real numbers are:$1, 2, -23.25, 0, \frac{\pi}{\phi}$, and so on. Numbers that are not real are $\ 3i$, $\ 3+2.5i$, $\ 3+2i+2j+k$, i.e. complex numbers, and quaternions.

The set of real numbers, denoted by $\mathbb{R}$, is a subset of complex numbers($\mathbb{C}$). Commonly used subsets of the real numbers are the rational numbers ($\mathbb{Q}$), integers ($\mathbb{Z}$), natural numbers ($\mathbb{N}$) and irrational numbers (sometimes, but not universally, denoted $\mathbb{J}$). In addition $\mathbb{Z}^{+}$ means positive integers and $\mathbb{Z}^{-}$ means negative integers. The real numbers can also be divided between the algebraic numbers and transcendental numbers, although these two classes are best understood as subsets of the complex numbers.

## The set $\mathbb{R}~$

The set of Real numbers is a complete, ordered field under addition and multiplication.

Dedekind developed a method to construct $\mathbb{R}$, the set of Real numbers from the set of rational numbers, using the very elegant idea of cuts.

### Cuts

Let $\mathbb{Q}$ be the set of rational numbers.

Let $\alpha\subset\mathbb{Q}$ be non-empty

We say that $\alpha$ is a cut if and only if

(1) $\alpha\neq\mathbb{Q}$ and $\alpha$ is bounded above

(2) If $a\in\alpha$, $p\in\alpha\forall p

(3) If $p$ is a rational and $p\in\alpha$ then there exists a rational $q>p$ such that $q\in\alpha$

We define $\mathbb{R}$ to be the set of all cuts $\alpha$

Using this definition, we can show that $\mathbb{R}$ possesess all the properties mentioned above.

### Order Relation

Let $\alpha$, $\beta$ be cuts

We say that $\alpha<\beta$ iff $\alpha\subset\beta$

### Completeness

Let $A\subset\mathbb{R}$

Let $M\in\mathbb{R}$ be an upper bound of $A$

Let $S$ be the set of all rationals $q$ such that $q\notin\alpha$ $\forall$ $\alpha\in A$.

As $A$ is bounded above, $S$ is non empty.

Consider the set $S'=\{-q|q\in S\}$

We see that $S'$ is a cut, say $\gamma$

Thus, the cut $-\gamma$ is the supremum of set $A$

QED