Question

How can I identify an irrational number

Answer 1

An irrational number is a real number that cannot be expressed as a simple fraction, meaning it cannot be written in the form \(\frac{p}{q}\), where \(p\) and \(q\) are integers, and \(q \neq 0\). Here are several ways to identify irrational numbers:

1. **Non-repeating and Non-terminating Decimals**: If a number has a decimal expansion that goes on forever without repeating, it is irrational. For example, \( \pi \approx 3.14159...\) and \(e \approx 2.71828...\) are both irrational.

2. **Square Roots of Non-perfect Squares**: The square root of any natural number that is not a perfect square is irrational. For example, \(\sqrt{2}\), \(\sqrt{3}\), and \(\sqrt{5}\) are all irrational.

3. **Certain Operations with Rational Numbers**: If you add, subtract, multiply, or divide (except by zero) a rational number with an irrational number, the result is irrational. For example, \( \sqrt{2} + 1 \) or \( \sqrt{2} \times 3 \) are both irrational.

4. **Algebraic Properties**: If a number is a solution to a polynomial equation with integer coefficients, then it is either rational or irrational. For instance, \(x^2 - 2 = 0\) has the solutions \(x = \sqrt{2}\) and \(x = -\sqrt{2}\), both of which are irrational.

5. **Transcendental Numbers**: Some numbers are not only irrational but also transcendental, meaning they are not the roots of any non-zero polynomial equation with rational coefficients. Examples include \(\pi\) and \(e\).

To summarize, if you suspect a number may be irrational, check for its representation in decimal form, evaluate its square root if applicable, and consider its properties in relation to rational numbers.