An irrational number is a number that cannot be expressed as a ratio of two integers. These numbers have infinite non-repeating decimal representations and cannot be written as fractions.
One way to prove that a number is irrational is by contradiction. Assume that the number can be expressed as a ratio of two integers, then show that this leads to a contradiction. Another method is to show that the number has an infinite non-repeating decimal representation.
Some examples of irrational numbers are pi (3.141592...), the square root of 2 (1.414213...), and the golden ratio (1.618033...).
No, by definition, irrational numbers cannot be expressed as fractions. They are non-terminating and non-repeating decimals.
Irrational numbers play a crucial role in mathematics, especially in geometry and physics. They help us understand and model real-world phenomena, and they are essential in many mathematical proofs and calculations.