<Li> Since y is an integer, and 2y = b, b is divisible by 2, and therefore even . </Li> <Li> Since b is even, b must be even . </Li> <Li> We have just shown that both b and c must be even . Hence they have a common factor of 2 . However this contradicts the assumption that they have no common factors . This contradiction proves that c and b cannot both be integers, and thus the existence of a number that cannot be expressed as a ratio of two integers . </Li> <P> Greek mathematicians termed this ratio of incommensurable magnitudes alogos, or inexpressible . Hippasus, however, was not lauded for his efforts: according to one legend, he made his discovery while out at sea, and was subsequently thrown overboard by his fellow Pythagoreans "...for having produced an element in the universe which denied the...doctrine that all phenomena in the universe can be reduced to whole numbers and their ratios ." Another legend states that Hippasus was merely exiled for this revelation . Whatever the consequence to Hippasus himself, his discovery posed a very serious problem to Pythagorean mathematics, since it shattered the assumption that number and geometry were inseparable--a foundation of their theory . </P>

What is a real number that is irrational