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Real number

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 $a\in\alpha$, $p\in\alpha\forall p<a$

(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.

Field Axioms

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

See Also

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