Understanding Irrational Number Properties
Irrational numbers are real numbers that cannot be expressed as a simple fraction or ratio of two integers. They have several unique properties that distinguish them from rational numbers.
Definition
An irrational number is defined as a number that cannot be written in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$.
Examples of Irrational Numbers
 Pi ($\pi$): Approximately equal to 3.14159.
 Euler's Number ($e$): Approximately equal to 2.71828.
 Square Roots of NonPerfect Squares: For example, $\sqrt{2}$, $\sqrt{3}$, etc.
Properties
NonTerminating and NonRepeating Decimals
One of the key properties of irrational numbers is that when expressed in decimal form, they do not terminate and do not repeat. For instance,
$\pi = 3.141592653589793238 \ldots$
Closure Properties

Addition: The sum of two irrational numbers can be either rational or irrational. For instance, $\pi + (\pi) = 0$ (rational) but $\pi + e$ is irrational.

Multiplication: The product of two irrational numbers can also be either rational or irrational. For example, $\pi \cdot \frac{1}{\pi} = 1$ (rational), but $\pi \cdot e$ is irrational.
Irrationality of Certain Operations
Certain operations involving irrational numbers retain their irrationality:
$\sqrt{2} + \sqrt{3} \neq \sqrt{5}$
Instead:
$\sqrt{2} + \sqrt{3}$
Proof of Irrationality by Contradiction
To prove a number is irrational, a common method is proof by contradiction. Let's look at the proof for $\sqrt{2}$:
Assume $\sqrt{2}$ is rational:
$\sqrt{2} = \frac{p}{q}$
where $p$ and $q$ are coprime integers. Then,
$\begin{aligned}
( \sqrt{2} )^2 &= \left( \frac{p}{q} \right)^2 \\
2 &= \frac{p^2}{q^2} \\
2q^2 &= p^2
\end{aligned}$
This implies $p^2$ is even, hence $p$ must be even. Let $p = 2k$:
$\begin{aligned}
2q^2 &= (2k)^2 \\
2q^2 &= 4k^2 \\
q^2 &= 2k^2
\end{aligned}$
Similarly, $q^2$ must be even, implying $q$ is even. But this contradicts the assumption that $p$ and $q$ are coprime. Hence, $\sqrt{2}$ is irrational.
Conclusion
Understanding the properties of irrational numbers provides insight into the broader field of real numbers and numerical analysis. Such knowledge is essential for advanced mathematical concepts and realworld applications.