Скачать или смотреть Number Systems | Class 9 - NCERT | Maths | Chapter 1 |Pi Paradox:Why is π Irrational if its a Ratio?

Welcome to one of the most fascinating questions in mathematics! We are taught that π is defined as the ratio of a circle's circumference to its diameter (π=c/d). We are also taught that any number that can be written as a ratio is a rational number. Yet, we know π is famously irrational. How can both of these things be true? This video resolves this beautiful contradiction.

The key lies in the precise definition of a rational number: it is a ratio of two integers. The formula π=c/d involves physical measurements, not necessarily integers. The resolution to the paradox is this:
In any circle, it is impossible for both the circumference (c) and the diameter (d) to be integers (or even rational numbers).
If you construct a circle with a rational diameter (say, 1 meter), its circumference will be an irrational number (π meters).
Conversely, if you could somehow create a circle with a rational circumference (say, 3 meters), its diameter would have to be an irrational number (3/π meters).
Therefore, the ratio c/d always involves at least one irrational number, meaning it doesn't fit the "integer over integer" definition of a rational number.
This problem goes beyond simple calculation and touches the very philosophy of what numbers are. At Padho.ai, we love these moments of deep inquiry. Our AI-native platform is designed to encourage your curiosity. Our 24/7 AI mentor is the perfect companion to help you explore these paradoxes and build a true, lasting understanding of mathematical concepts.

➡️ Ponder this paradox on Padho.ai: https://learn.padho.ai/courses/MATH10...
➡️ Full Playlist for Number Systems (Chapter 1):    • Why "Every Rational Number is a Whole Numb...  

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