In school you learned \pi \approx \tfrac{22}{7}. Then you typed both into your calculator and got 3.142857\dots and 3.141592\dots and thought — wait, those aren't the same. So which is it? Is \pi equal to \tfrac{22}{7} or not? And if not, why did every teacher let you use \tfrac{22}{7} in class?
The short answer
\pi is not \tfrac{22}{7}. The two numbers are genuinely different, and the difference matters.
They agree for the first two decimal places (3.14) and then they part ways forever. The gap is
About one part in two and a half thousand. Small enough to be invisible in a rough calculation, large enough to ruin a spacecraft trajectory.
Why: \tfrac{22}{7} is a rational number — a ratio of two integers — so by the decimal-classification theorem, its decimal expansion must eventually repeat. And it does: \tfrac{22}{7} = 3.\overline{142857}, with the six-digit block repeating forever. \pi is irrational — not expressible as p/q — so its decimals never settle into a repeating pattern. Two decimals that disagree at the fourth digit cannot be the same number.
Why your calculator seems to confirm \pi = \tfrac{22}{7}
It doesn't — you were reading it wrong. A typical calculator displays eight to ten digits. If you computed \tfrac{22}{7} you saw 3.142857143. If you pressed the π key you saw 3.141592654. Those are different numbers. They only look close because the first three characters (3.14) are identical and students stop reading after that.
Try this: compute \tfrac{22}{7} - \pi directly on your calculator. It will give you something like 0.001264489. That is the answer that should have been obvious all along — but the screen is small and the mind is trusting.
Why \tfrac{22}{7} was ever used
\tfrac{22}{7} is a very good rational approximation to \pi, and historically it is old — Archimedes (about 250 BCE) proved that \pi lies between \tfrac{223}{71} and \tfrac{22}{7} by inscribing and circumscribing polygons around a circle. The upper bound \tfrac{22}{7} stuck because it is easy to remember and easy to compute with by hand.
For a circle of radius 1\text{ m}, using \tfrac{22}{7} instead of \pi gives a circumference of 6.2857\text{ m} instead of the true 6.2832\text{ m}. The error is 2.5\text{ mm}. For a classroom problem about the circumference of a bicycle tyre, 2.5\text{ mm} in a 2\text{ m} wheel is invisible. For a precision-engineered gear, it is catastrophic.
Indian school textbooks used \tfrac{22}{7} for decades because it gave nice cancellations on exam papers — problems were designed so the 7 in the denominator would cancel with a radius of 7\text{ cm} or 14\text{ cm}, leaving whole-number answers. It was pedagogical convenience, not mathematical truth.
The picture: where they part
Better approximations
If \tfrac{22}{7} is not exactly \pi, you might ask — is there any fraction that is? The answer, by definition, is no. \pi is irrational, so no fraction p/q equals \pi. But some fractions are shockingly close.
- \tfrac{22}{7} = 3.14285\dots — agrees with \pi to 2 decimals (error \sim 10^{-3}).
- \tfrac{333}{106} = 3.14150\dots — agrees to 4 decimals (error \sim 10^{-5}).
- \tfrac{355}{113} = 3.1415929\dots — agrees to 6 decimals (error \sim 10^{-7}).
That last one, \tfrac{355}{113}, was known to the Chinese mathematician Zu Chongzhi around 480 CE and is astonishingly accurate for such a small denominator. But none of them equal \pi. Every rational approximation to \pi eventually diverges — that is what "irrational" means.
The misconception, stated and answered
The misconception: "\pi = \tfrac{22}{7}, and any calculator will show this."
Why it feels true: Indian school problems almost always use \tfrac{22}{7}. The calculator screen is short. Both numbers start with 3.14. A student who has never been asked "where do these two numbers differ?" has no reason to notice.
The counterexample: Compute \tfrac{22}{7} - \pi. You get a nonzero answer, \approx 0.00126. Two numbers whose difference is not zero are not equal. Full stop.
The correct view: \tfrac{22}{7} is a rational approximation to \pi that is convenient for hand computation and accurate enough for everyday problems. \pi itself is an irrational number — its decimal expansion is infinite and non-repeating, and no fraction can pin it down exactly. When a physics or engineering problem demands real precision, you use \pi (via π on a calculator or a high-precision stored value), never \tfrac{22}{7}.
Two circles, one mistake
A satellite dish has radius r = 1.4\text{ m}. Compute its circumference with \tfrac{22}{7} and with \pi.
Using \tfrac{22}{7}: C = 2 \cdot \tfrac{22}{7} \cdot 1.4 = 8.8\text{ m} exactly.
Using \pi: C = 2\pi \cdot 1.4 = 8.79646\dots\text{ m}.
The difference is \approx 3.5\text{ mm} on a dish that is almost 9\text{ m} around. For a cooking pot, ignore it. For a dish whose signal has to focus precisely on a feedhorn, you just misaligned the receiver by more than the wavelength of the signal you are trying to collect. The same formula, the same radius — one answer works, the other breaks an antenna.
Takeaway
\pi is not \tfrac{22}{7}. It was never \tfrac{22}{7}. The equality sign you saw in school was a pedagogical shortcut, and it hides the fact that \pi is irrational — a genuinely different species of number. When you see "\pi = \tfrac{22}{7}" in a textbook, mentally rewrite it as "\pi \approx \tfrac{22}{7}" and remember that the approximation is good to about three decimal places and no further.
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