Why (22/7) is a rational number and (π) is irrational number. please explain.

Edit: How can you say that $22/7=\pi$, when one number if rational and the other is irrational?


closed as unclear what you're asking by JMP, SchrodingersCat, Davide Giraudo, user147263, Alexander Konovalov Nov 26 '15 at 21:17

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

  • 16
    $\begingroup$ If you don't know why 22/7 is a rational number, you are not going to understand why $\pi$ is an irrational number. $\endgroup$ – Gerry Myerson Nov 26 '15 at 11:48
  • $\begingroup$ Welcome to the site. I've made an answer to your question. Does this clear up the confusion? Otherwise you can elaborate on your question in your original post by editing it. $\endgroup$ – Mankind Nov 26 '15 at 11:56

My guess is that you are thinking of the approximation $22/7 \approx \pi$. The two numbers $\pi$ and $22/7$ are not equal, so there's no contradiction in one being rational and the other irrational.

Sometimes you use $22/7$ as a number "fairly close" to $\pi$, but:

$$\pi = 3.14159265...$$ while $$22/7 = 3.14285714...$$

  • 2
    $\begingroup$ Just to add on to your excellent answer (for the OP's benefit), 22/7 is only one rational approximation to $\pi$, albeit the one most commonly taught in schools. In fact, it's possible to get arbitrarily close to $\pi$ using successively more accurate rational approximations (but never actually equalling $\pi$). One of the "simple" rational approximations is especially interesting: the Chinese "Milü", which is 355/113, which has almost incredible accuracy for such a simple fraction. It's fairly easy to remember: think of the sequence 1,1,3,3,5,5 & take the last 3 digits divided by the first 3. $\endgroup$ – Deepak Nov 26 '15 at 12:04
  • $\begingroup$ Thanks for answer. Now i understand why π is irrational number $\endgroup$ – user293607 Nov 26 '15 at 12:05
  • $\begingroup$ In fact, both $\frac{22}{7}$ and $\frac{355}{113}$ are the second and fourth rational approximations of $\pi$ in a sequence of approximations called convergents of $\pi$, and they are in a sense that can be made precise "as good as possible". One can compute them quickly using truncations of the continued fraction $[3; 7, 15, 1, 292, 1, 1, \ldots]$ of $\pi$, and the largeness of the entry $292$ quantifies just how remarkably good the mentioned Chinese approximation (which dates to the 5th Century AD) is. mathworld.wolfram.com/PiContinuedFraction.html $\endgroup$ – Travis Willse Nov 26 '15 at 12:17

Not the answer you're looking for? Browse other questions tagged or ask your own question.