Irrational Numbers Proof
What's the Cliff Note's on this for non-mathematicians??
2 Replies
What's the Cliff Note's on this for non-mathematicians??
It's easy to be irrational but hard to prove it.
PairTheBoard
The article is discussing proof of irrationality via the idea that if a given value is rational then it can differ from another rational number by a minimum amount. For example suppose x=1/13. We know that is a rational number, but suppose we did not. We take another value we know to be rational, say 1/7. We calculate the difference between x and 1/7. Since x is rational, we know that this difference must be m/91 for some m. Therefore if we try to approximate x by a rational number that has 7 in the denominator, we cannot do better than a difference of 1/91. If we want a closer approximation we must use a new rational with a larger denominator. We can make a sequence of rational approximations with ever increasing denominator. The essence of the proof in the article is that if the difference between x and each successive term of the sequence decreases rapidly enough, we can conclude that x is irrational.
