Rational Number

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A Rational Number is a number that can be expressed as a ratio. Examples: The rational numbers aren't all there is. It surprised Pythagoras to discover that some numbers cannot be written as a ratio. For example, Root Two is irrational. In fact, between any two rationals there are infinitely many irrationals, and between any two irrationals there are infinitely many rationals. The problem is that Cantor showed that there are more irrational numbers than rational numbers.


The symbol for the rationals is usually Q so the above says $\sqrt2\not\in{Q}$

The set of Rational numbers is a countable set with size $\aleph_0$ . (see countable sets)

Euler's number (e) is an irrational number. You can see the proof that e is irrational.

A consideration of the mechanics of long division should convince that rational numbers must be represented by decimals expansions than either terminate or repeat.

The converse is also true: decimal expansions that terminate or repeat can be expressed as a fraction.

For example: let p = 0.143272727272727... statement (1)
Multiply by 100
two zeros, because the repeat
is of length 2
100p = 14.3272727272727 ... statement (2)
Subtract, taking (2) - (1) 99p = 14.184
Three decimal places,
so multiply by 1000
99000p = 14184
Divide both sides by 99000 so $p=\frac{14184}{99000}$

Consequently, irrational numbers must have infinite and non-repeating decimal expansions and vice versa.

More technical stuff ...

The rationals can be constructed from the integers as follows: The same technique of equivalence classes of things can be used to create the real numbers from the rationals. In that case we use Cauchy Sequences.
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