A Rational Number is a number that can be expressed as a ratio.

 The number on top of the line (or vinculum ) is called the "Numerator," and the number on the bottom is called the "Denominator."
Examples:

• 3 (which is $\frac{3}{1}$ or $\frac{6}{2}$ or $\frac{-15}{-5},$ etc. )
• $22/7$
• $-23/6$
For any rational number there are infinitely many representations of the same value, because we can multiply the numerator and denominator by any non-zero constant to get an equivalent form. There is a canonical form - see "Reducing Fractions To Lowest Terms" for more.

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.

Tricky.

The symbol for the rationals is usually $\mathbb{Q}$ so the above says $\sqrt{2}\not\in{\mathbb{Q}}$

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

This next section deserves a page on its own, but finding a good name for it is hard ...

A consideration of the mechanics of long division should convince that rational numbers must be represented by decimals expansions that 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:
• Let P be the collection of all pairs of integers with a non-zero second element:
• $P=\{(a,b):a,b\in{Z},b\ne{0}\}$
• Let (a,b) be equivalent to (c,d) if ad=bc.
• For any pair (a,b) we can consider the collection of all pairs equivalent to it.
• This collection is called the equivalence class of (a,b), and we write it as E(a,b)
• We can now define arithmetic operations on the equivalence classes:
• The sum is obtained as E(a,b)+E(c,d) = E(ad+bc,bd)
• The product is obtained as E(a,b)*E(c,d) = E(ac,bd)
• The equivalence classes can be thought of as the rationals.
• We think of E(a,b) as being "the same as" $a/b.$
• You can easily check that E(ka,kb)=E(a,b) for all $k\ne{0}.$
• Lots and lots of checking required to see that the arithmetic on the equivalence classes is "the same as" the arithmetic on the rationals.
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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