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what are some examples off irrational numbers?

Usually, irrationality is proven by reductio ad absurdum (a proof by contradiction). Here are some examples:

Theorem 1: √2 is irrational.

Proof:

Assume that √2 is rational and that in lowest terms, √2 = A/B.

Then we have 2=A2/B2 so A2=2B2.

We see that A2 is even, thus A must be even. Now let A=2K and substitute.

(2K)2=2B2 so 4K2=2B2 so 2K2=B2.

We see that B2 is even, thus B must be even.

Since A and B are both even, A/B is not in lowest terms, a contradiction.

Theorem 2: log2 3 is irrational.

Proof:

First observe that log2 3 > 0

Assume that log2 3 = A/B where A and B are positive integers.

Then we get 2A/B=3 so 2A=3B, a contradiction, because 2A is even while 3B is odd.

There are also proofs by contradiction that establish irrationality of e and π.

Two well known facts:

1. If an integer K>0 is not an Nth power of an integer, it's Nth root is irrational.

2. For two integers A>1 and B>1, logB A is rational if and only if A and B are powers of the same integer:

Ex: log4 8 is rational because 4 and 8 are both powers of 2 (4=22 and 8=23).

An irrational number is one that goes on and on and on and on forever and ever and ever and ever and doesn't stop and doesn't repeat.  The most common ones are pi and square roots of numbers.

The technical definition -- a rational one is one that can be written as a fraction.  An irrational number cannot.  So remember that a number like 1/3 is 0.3333333333333333333333 (where 3 has a line over it and it goes on forever) repeats, so it isn't rational.  Some reallly smart college student tried to figure out what pi is a long time ago and he started writing it out and eventually filled up all the walls in the room he worked on with numbers.