Mathematical Induction on rationals

In summary, a high school student presented a proof for a theorem involving statements about rational numbers. Some professors deemed it trivial and suggested using double induction instead, but the student questions whether it is redundant. The theorem states that if P(r) is true for r = 1 and for all m,n in natural numbers where m >= n, P(m/n) implies P((m+1)/n), then P(r) is true for all rational numbers r greater than or equal to 1. The student apologizes for any mistakes and acknowledges that as a high-school student, extending induction to the rationals is not common.
  • #1
mumatics
2
0
Hi
I'm a high school student. I gave a proof for the following theorem, but I was told by some professors that this is trivial and using natural induction twice for the rationals will do the same thing. What do you think? Is it just redundant?

Theorem:


Let P(r) be a statement about r, then if :
1) P(1) is true and,
2) [tex]\forall[/tex] m,n [tex]\in[/tex] [tex]N[/tex] , m[tex]\geq[/tex]n ; P([tex]\frac{m}{n}[/tex])[tex]\rightarrow[/tex] P([tex]\frac{m+1}{n}[/tex])

Then [tex]\forall[/tex] r[tex]\in[/tex] Q, r[tex]\geq[/tex]1 ; P(r).



(PS: I apologize for my (probable) mistakes, because I'm neither an English speaker nor familiar with Latex.)
 
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  • #2
It's not quite true as stated; take P(r) = true if r > 0 and false otherwise. But the basic idea is right.

If you were a college student, I would agree that this is a trivial result by double induction. But as a high-school student I actually think it's pretty good. Most wouldn't think to extend induction to the rationals at all.
 
  • #3
I can't get why it's not true. Did you notice r[tex]\geq[/tex]1 in the last sentence?

And thank you for the reply.
 

1. What is mathematical induction on rationals?

Mathematical induction on rationals is a proof technique used to prove statements about all rational numbers. It involves three steps: the base case, the inductive hypothesis, and the inductive step.

2. How does mathematical induction on rationals differ from induction on integers?

Mathematical induction on rationals is similar to induction on integers, but it involves proving statements about all rational numbers instead of just integers. It also requires a different approach to the inductive step, as it involves dealing with fractions instead of just whole numbers.

3. What is the importance of mathematical induction on rationals?

Mathematical induction on rationals is an important proof technique in mathematics. It allows us to prove statements about all rational numbers, which is useful in many areas of mathematics, including number theory and algebra.

4. Can mathematical induction on rationals be used to prove statements about irrational numbers?

No, mathematical induction on rationals cannot be used to prove statements about irrational numbers. This is because irrational numbers cannot be expressed as fractions and therefore do not follow the rules of mathematical induction on rationals.

5. Are there any limitations to using mathematical induction on rationals?

One limitation to using mathematical induction on rationals is that it only works for statements that can be expressed as a fraction. It also cannot be used to prove statements about real numbers, as they include both rational and irrational numbers.

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