Check whether the number can be written as a fraction of two integers
If it cannot be written as a fraction, it is irrational
Look for non-terminating, non-repeating decimal expansions
Test common examples like √2, √3, π, and e
Use square roots of non-perfect squares as likely irrational numbers
Verify whether a decimal pattern repeats; if it does not, it may be irrational
Use algebraic proofs to show a number cannot be expressed as a ratio of integers
Compare the number against known irrational constants
Examine whether the number comes from operations that preserve irrationality
Check if the number is the root of a polynomial with no rational solution