Understanding Rational and Irrational Numbers
To determine whether numbers involving the square root of 5 are rational or irrational, we need to define these terms clearly.
Rational Numbers
- A rational number is any number that can be expressed as a fraction, where both the numerator and denominator are integers, and the denominator is not zero.
- Examples: 1/2, -3, 4.5
Irrational Numbers
- An irrational number cannot be expressed as a simple fraction. Its decimal representation is non-terminating and non-repeating.
- Examples: √2, π, e
Examining √5
- The square root of 5 (√5) is an irrational number.
- It cannot be simplified to a fraction of two integers, which can be shown using a proof by contradiction. If √5 were rational, it could be written as p/q (where p and q are integers). Squaring both sides leads to a contradiction regarding the properties of even and odd integers.
Conclusion
- Therefore, any number in the form of n√5 (where n is a non-zero rational number) is also irrational.
- This is because multiplying a rational number by an irrational number results in an irrational number.
Summary
- √5 is irrational.
- Any multiple of √5 (n√5, n ≠ 0) is also irrational.
Understanding these concepts helps clarify why numbers involving √5 are classified as irrational.

Absolutely irrational no hai