Difference between revisions of "Proofs of irrationality"
(Created page with "One of the classic proofs in mathematics is that the square root of 2 is irrational. The classic proof of this is the proof by "infinite descent" that is included on [https://...") |
(→Assignment) |
||
| Line 6: | Line 6: | ||
You will be asked to prove that a number is irrational using the same basic strategy as the proof that the square root of 2 is irrational. In general, you will be assigned integers m and n, and asked to show that the m-th root of n is irrational. | You will be asked to prove that a number is irrational using the same basic strategy as the proof that the square root of 2 is irrational. In general, you will be assigned integers m and n, and asked to show that the m-th root of n is irrational. | ||
| − | ''Pass rating check'' Your proof must satisfy the following. | + | '''Pass rating check''' Your proof must satisfy the following. |
* State your claim (e.g., that sqrt(2) is irrational). | * State your claim (e.g., that sqrt(2) is irrational). | ||
* Proof is by contradiction, and the assumption is given in generality (e.g., if the number is rational, then let it be i/j for some integers i and j). | * Proof is by contradiction, and the assumption is given in generality (e.g., if the number is rational, then let it be i/j for some integers i and j). | ||
* Correct argument is given that the assumption leads to a contradiction, including working through the formulae. | * Correct argument is given that the assumption leads to a contradiction, including working through the formulae. | ||
* Your claim and proof are complete - someone who is not in our class could follow through your argument and validate your proof. | * Your claim and proof are complete - someone who is not in our class could follow through your argument and validate your proof. | ||
Revision as of 15:42, 29 August 2022
One of the classic proofs in mathematics is that the square root of 2 is irrational. The classic proof of this is the proof by "infinite descent" that is included on wikipedia's page on sqrt(2). CS students who have taken CS 303 or CS 600 should be able to demonstrate this proof on the board, or write it out on paper.
The same basic proof technique can be used to prove that other numbers are irrational - the square root of any number that is not a perfect square, the cube root of any number that is not a perfect cube, etc.
Assignment
You will be asked to prove that a number is irrational using the same basic strategy as the proof that the square root of 2 is irrational. In general, you will be assigned integers m and n, and asked to show that the m-th root of n is irrational.
Pass rating check Your proof must satisfy the following.
- State your claim (e.g., that sqrt(2) is irrational).
- Proof is by contradiction, and the assumption is given in generality (e.g., if the number is rational, then let it be i/j for some integers i and j).
- Correct argument is given that the assumption leads to a contradiction, including working through the formulae.
- Your claim and proof are complete - someone who is not in our class could follow through your argument and validate your proof.
