Examples of irrational numbers:

• A square root of a number √(x), is irrational if the number x IS NOT a perfect square. √(8), √(10), √(24), and √(40) are all irrational numbers since 8, 10, 24, and 40 are not perfect squares.
• A decimal that never ends and never repeats a group of the same digits is called a non-terminating, non-repeating decimal. Decimals of this type are irrational because they can't be written as a ratio of two integers.
  
  Examples of non-repeating non-terminating decimals are:
  • .405375920721... there is no group of digits that repeats.
  • .10100100010000... even though you can see a pattern (just add one more 0 each time) there is no group of digits that repeats.
  • 27.23233233323333... even though you can see a pattern (just add one more 3 each time) there is no group of digits that repeats.
  • One irrational number you are probably familiar with is π. If you are familiar with π (3.14159...) you know that it never ends and there is no group of digits that repeats. Like all non-repeating non-terminating decimals π can be written out to thousands of decimal places and yet it still keeps on going. That is why the symbol π is used to represent it instead of the approximations 3.14 or 22/7 when perfect accuracy is required.