No, pi does not belong to q.
Understanding Rational Numbers (q)
Rational numbers, often denoted by the symbol q, are numbers that can be expressed as a fraction, where the numerator (p) and the denominator (q) are integers, and the denominator is not zero (q ≠ 0). Examples of rational numbers include:
- 1/2
- 3
- -7/4
- 0
What is Pi (π)?
Pi (π) is a mathematical constant representing the ratio of a circle's circumference to its diameter. It's approximately 3.14159 but is an irrational number, meaning its decimal representation goes on infinitely without repeating.
Why Pi is Not Rational
According to the reference:
π is not expressible as pq for some integers p , q with q≠0 , though there are some good approximations of that form. So it is not rational and is irrational.28-Jun-2015
This means that pi cannot be written as a fraction of two integers. Though we can approximate pi with fractions (like 22/7), these are only approximations and not the exact value of pi. Because it cannot be expressed as a fraction in the defined form, pi does not belong to the set of rational numbers (q). It belongs to the set of irrational numbers.
Key Differences Summarized
Here's a table to highlight the differences:
| Feature | Rational Numbers (q) | Irrational Numbers (like π) |
|---|---|---|
| Definition | Expressible as p/q | Not expressible as p/q |
| Decimal Form | Terminating or repeating | Non-terminating and non-repeating |
| Example | 1/2, -3, 0.75 | π, √2, e |
Conclusion
Therefore, since π cannot be expressed as a fraction of two integers, it is not a member of the set of rational numbers (q). It is an irrational number.
