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gotmilk04
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Homework Statement
Prove the set of irrational numbers is uncountable.
Homework Equations
The Attempt at a Solution
We proved that the set [0,1] is uncountable, but I'm not sure how to do it for the irrational numbers.
An irrational number is a real number that cannot be written as a ratio of two integers. It is a non-repeating, non-terminating decimal and has an infinite number of decimal places.
An uncountable set is a set that cannot be put into a one-to-one correspondence with the set of natural numbers (countably infinite). This means that there is no way to list out all the elements in the set in a systematic way.
The proof for the uncountability of the set of irrational numbers is based on a proof by contradiction. It involves assuming that the set of irrational numbers is countable and then showing that this leads to a contradiction. This contradiction proves that the set is actually uncountable.
One example of a proof is the famous "diagonalization argument" by Georg Cantor. This proof involves creating a list of all the irrational numbers and then constructing a new irrational number by changing the nth decimal place of each number in the list. This new number is then shown to be different from every number in the list, proving that the list is incomplete and therefore the set is uncountable.
The uncountability of the set of irrational numbers has significant implications in mathematics, as it shows that there are more real numbers than just the rational numbers. This discovery was groundbreaking and led to the development of new mathematical concepts such as infinity and the continuum, which have applications in various fields of science and technology.