Chapter 1: Real Numbers

Explain · page 6: 1.1.3 Decimal Representation of Irrational Numbers

When you write certain numbers as decimals, the digits after the decimal point go on forever without ever forming a repeating pattern. These endless, non-repeating numbers are called irrational numbers [غیر ناطق اعداد].

Here are the key points you need to remember:

• Non-terminating [غیر مختتم]: The digits after the decimal point never stop or come to an end (like in $\sqrt{2} = 1.41421356...$).
• Non-recurring [غیر تکراری]: The digits never repeat in a fixed block, unlike rational decimals such as $0.333...$ or $0.444...$.
• Key examples: Famous mathematical constants like $\pi \approx 3.14159...$, Euler's number $e \approx 2.71828...$, and square roots of numbers that are not perfect squares (like $\sqrt{2}$) are all irrational decimals.

Daily Life Example

Imagine standing at a busy bazaar in Multan listening to the random background noise of honking horns and chatter. No matter how long you listen, the noise never stops (non-terminating), and it never repeats the exact same sequence of sounds twice (non-recurring). In contrast, a recorded shop announcement playing the exact same $5$-second loop over and over is like a recurring, rational decimal!

Common Mistake

The Mistake: Thinking that every endless decimal is an irrational number. The Correction: An endless decimal is only irrational if it does not repeat. If an endless decimal repeats a pattern (for example, $0.444...$ or $3.555...$), it is a rational number [ناطق عدد]. To be irrational, it must be both non-terminating and non-recurring.

Since these decimal digits never stop and never repeat, you can never write down the complete decimal value of an irrational number on paper—you can only write an approximation.

In short: An irrational number in decimal form is a number whose digits continue forever after the decimal point without ever repeating a set pattern.