1) Group as an algebraic structure have the following properties
For a group G there exist a function, say * here.
(Just recall that the function maps a single element from a set to a unique element in co-domain.)
-Closure. i.e., if
-Associativity:
-Existance of identity: exist
-Existance of inverse element: for every
Example: non-zero Rational number with multiplication is a group, identity 1.
2) Rational numbers can be written in forms of p/q where p is integer and q is natrual number. More precisely rational numbers have their own unique expressions if (|p|,|q|) = 1.
Inversely irrational numbers can't be expressed in fractions, this gives us the fundamental way to show that a number is irrational.
Example 2: Prove
Classic question. Assume
By squaring both sides and change of term, we have
Then following logic finishes the prove by contradiction.
Now we will introduce another method to show irrationality.
Assume
Corollary: Fractions, in decimal, are recurring.
Prove: Consider prime p, we are going to prove 1/p is recurring.
Consider Fermat's little theorem,
Therefore p is a divisor of
Therefore 1/p is recurring.
Note that recurring period may not be p-1 if it can be further simplified, but the general trend is rising.
Proof: For every
Then if the recurring decimal is infinity, it is irrational. That gives the idea to prove a number being irrational.
Example 3: Prove e is irrational.
Consider
Another approach is directly combining terms by definition, since the expressed forms in simpliest terms* monotonically increase, the sum has a infinitely large denominator, hence irrational.
4) Back to out main concern, we would like to show some properties of irrational numbers.
- Irrational numbers do not perform closure under most elementary functions.
Example 4: Proof the existance of infinite pairs of (x,y) where x,y are irrational and x+y, xy are rational. More precisely, show that for every irrational x show that there's infinitely many y satisfying the condition.
Proof: Let
5) Show that
This is a bit tricky. We show it by De Morive's theorem.
By expanding
Exercise:
1) Show that (Z,+) is a group.
2) Show that
3) Show that
4) Show that irrational numbers can be written in continued fraction of infinite length.
5) Show that 37 is a factor of 999 considering 1/37.
6) Provide a full mechanism such that there's infinite r to satisfy the considition "x,y, irrational and x+y, xy rational".
7) Discuss the countability of irrational numbers.
8) Discuss the countability of irrational pairs (x,y) such that x+y and xy are rational.
9) If x,y are irrational while
10) Show that
11) For
12) Show that
13) If x,y are rational numbers with x>y, show that there exist rational numbers a,b and irrational number c that x>a>c>b>y. Hence prove real number is dense.
Extra infomation: we prove the transcendency of a number by similar method but stricter bounding. You can visit the page talking about Liouville's numbers. Then finish this question:
CGMO 2007 day 2 Q3
If a,b,c are integers that
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