A Rational Number is a number that can be expressed as a ratio.
Examples:
- 3 (which is $\frac{3}{1}$ or $\frac{6}{2}$ or $\frac{-15}{-5},$ etc. )
- 22/7
- -23/6
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 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:
- 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 equivance 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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