Rational numbers

#matematik-discmath

We don't have division yet, but know what we are aiming for; we know that once we have division, we can represent a rational number as for some (where ). This representation, however, is not unique; more specifically we have . We have defined multiplication, so we can make equivalence classes from this!

Definition: Construction of the rational numbers

Let and define an equivalence relation so that . We then define the rational numbers as the set of equivalence classes of with respect to , and denote with respect to . We also define for any , so . We define addition as and multiplication as Finally, we define for every positive .

We now show that these definitions of operations on rational numbers are well-defined and sufficient. If and we get We want to show that , which is equivalent to which we know from the previous equalities. Hence, addition is well-defined. If and we get and , so multiplication is also well-defined.

Now, for addition we get so addition is sufficiently defined. For multiplication, we first note that meaning This means that in general we get so multiplication is also sufficiently defined.

We also see that

Theorem: Density of rational numbers

Let where . Then there exists a such that .

Proof:
Let . Then we have and .

Theorem

A decimal number terminates or repeats it corresponds to a rational number.

Definition: Irrational number, real number

A decimal number that does not terminate or repeat we say correspond to an irrational number. Any number that is rational or irrational we call real, and we let the set of all real numbers be informally defined as

When we introduced we added subtraction, when we introduced we added division, and when we have we have added for example roots.

We will now consider sequences of numbers. Let , and consider a sequence of numbers in .

Definition: Strictly increasing, strictly decreasing

A sequence of numbers is said to be strictly increasing if for every , or strictly decreasing if for every .

We see that if , there are no strictly decreasing infinite sequences, since it must reach and then can't continue. But if , there are of course strictly decreasing infinite sequences, such as

Definition: Greatest lower bound

For a set of numbers , is a greatest lower bound if it is a lower bound of and for every lower bound .

We see that for example the sequence of rational numbers is strictly decreasing yet still has a greatest lowest bound in , namely . However, we can construct a strictly decreasing sequence of rational numbers, that has a lower bound but not a greatest lower bound in . For example, the recursive sequence is strictly decreasing but approaches . So for any rational lower bound, we can always find a greater rational lower bound. With real numbers, the problem is solved.

Theorem

A bounded, strictly decreasing sequence of real numbers has a greatest lower bound in .

Theorem

The set of rational numbers is Countable.

Theorem

The set of real numbers is uncountable.

Proof:
It is enough to prove that the set is uncountable (the set of all nonnegative real numbers with integer part ). Suppose this set is countable. Then we have a map from to , so . We can then create a real number with integer part and with decimal being any decimal different from decimal of , for every . is thus different from every , but , which is a contradiction.

Lemma

For any such that , there exists an such that .

Proof:
Let . We know and that is a lower bound of , meaning has a minimal element (so but ). This means and , so .

Theorem

If such that , there exists an such that .

Proof:
Let . Then there exists an such that . This means that which by Lemma means there exists an so that . This means .

Theorem: Archimedean property

If where , there exists an such that .

Proof:
The set does not have an upper bound.