Three decimals. All start with 1.4 or 3.14. All look, at a glance, like "just numbers." But if you let them keep going, digit by digit, they tell very different stories. One repeats forever. One stops. One never settles.

This article runs them side by side so you can see the difference.

The three suspects

  • \dfrac{22}{7} — a rational number, a simple fraction, often quoted as "pi."
  • \pi — the actual ratio of a circle's circumference to its diameter, irrational.
  • \sqrt{2} — the diagonal of a unit square, also irrational.

You have probably been told that 22/7 \approx \pi, and that is true for the first two decimal places. But they are different kinds of number. One is a ratio of integers. The other is not. That difference is invisible at two decimal places, but becomes obvious once you print enough digits.

What each one does after the first few digits

Here are the three decimals printed out. Scroll down each column. Look for patterns.

The decimal expansions of 22 over 7, pi, and root two compared Three vertical columns showing the first twenty or so decimal digits of twenty-two over seven, pi, and square root of two. The twenty-two over seven column shows a repeating block of six digits one four two eight five seven after the three. The pi and square root of two columns show no repeating pattern. Above each column is a label showing the number, its classification as rational or irrational, and a short comment on what its digits do. 22 / 7 rational repeats every 6 digits π irrational no repeating block, ever √2 irrational no repeating block, ever 3 . 1 4 2 8 5 7 1 4 2 8 5 7 1 4 2 8 5 7 1 4 2 8 5 7 1 4 2 8 5 7 … the block 142857 repeats forever 3 . 1 4 1 5 9 2 6 5 3 5 8 9 7 9 3 2 3 8 4 6 2 6 4 3 3 8 3 2 7 9 … digits never settle into a pattern 1 . 4 1 4 2 1 3 5 6 2 3 7 3 0 9 5 0 4 8 8 0 1 6 8 8 7 2 4 2 0 9 … digits never settle into a pattern at 3 decimals: 22/7 ≈ 3.142, π ≈ 3.141 — already different 22/7 − π ≈ 0.00126 — big enough that engineers care
Three decimals, row by row. The $22/7$ column shows the same block of six digits — $142857$ — repeating over and over. That is the signature of a rational number. The $\pi$ and $\sqrt{2}$ columns do not have any repeating block, no matter how far down you scroll. That is the signature of an irrational number.

Now run the ticker interactively — drag the slider to reveal one digit at a time for each number. The "22/7 vs π gap" readout shows how far apart the two values are at the current number of decimal places.

Drag the slider to reveal digits one at a time. The blue bar is $|22/7 - \pi|$ at the current precision on a log scale; it stays at zero up to $n=2$ (both show $3.14$) and jumps once $22/7$ and $\pi$ diverge.

Why the slider-controlled precision matters: 22/7 agrees with \pi to about two decimal places, which is why 22/7 became the classroom shortcut. But 22/7 - \pi \approx 0.00126. If you use 22/7 to compute the circumference of a circular road 1\,\text{km} across, you are off by about 4\,\text{m}. Close enough for your homework. Not close enough for ISRO.

Why the ticker looks different

The rule is simple, and it follows from long division.

  • A decimal terminates (the ticker stops) if and only if the fraction in lowest terms has a denominator of the form 2^a \cdot 5^b. Example: 3/8 = 0.375, which terminates because 8 = 2^3.

  • A decimal repeats (the ticker falls into a cycle) if and only if the fraction is rational but the denominator has some prime factor other than 2 or 5. Example: 1/7 = 0.\overline{142857}, which has period 6, and 22/7 = 3.\overline{142857}, which has the same repeating block because the "3." in front comes from the integer part.

  • A decimal never terminates and never repeats if and only if the number is irrational.

Why these are the only three options for a decimal: long division at each step produces a remainder between 0 and q-1. If the remainder ever becomes 0, the decimal terminates. If not, there are only q-1 non-zero remainders, so one of them must repeat — and from that point on the digits cycle. An irrational number cannot arise from any long division at all, so neither termination nor repetition ever happens.

One more test: the trailing-zero trick for π

Here is a way to prove \pi isn't 22/7 without looking up a reference table.

Compute 22/7 by hand.

22 \div 7 = 3 \text{ remainder } 1 \quad\Rightarrow\quad 3.\,?\,?\,?\dots

Carry the 1, append a 0, divide:

10 \div 7 = 1 \text{ remainder } 3 \quad\Rightarrow\quad 3.1\dots

Carry 3, append 0:

30 \div 7 = 4 \text{ remainder } 2 \quad\Rightarrow\quad 3.14\dots

Carry 2, append 0:

20 \div 7 = 2 \text{ remainder } 6 \quad\Rightarrow\quad 3.142\dots

So 22/7 starts 3.142\,857\,142\,857\dots. But \pi = 3.141\,592\dots. The third decimal already disagrees — 22/7 says 2, \pi says 1. They were never the same number.

What this tells you about rationals and irrationals

The ticker is a diagnostic. Any time you are handed a decimal, you can ask:

  1. Does it eventually terminate? → rational, and the denominator (in lowest terms) is a product of 2s and 5s.
  2. Does it eventually fall into a repeating block? → rational, and the length of the block divides the multiplicative order of 10 modulo the denominator.
  3. Does it do neither? → irrational.

That last case is what \pi and \sqrt{2} look like. Running the ticker on them is a visual proof that no fraction — no matter how many digits you let it have — will ever be either of these numbers. Each of them has carved out a permanent spot on the number line that no rational can occupy.

Will 355/113 ever repeat? And does it match π better?

355/113 is another famous approximation for \pi, better than 22/7. Two quick questions.

First: does 355/113 have a repeating decimal? Yes — 113 is prime and is neither 2 nor 5, so every fraction with denominator 113 has an eventually repeating decimal, never terminating. The period divides \phi(113) = 112, and in fact it is exactly 112: 355/113 repeats with a block of 112 digits.

Second: how many decimals of \pi does 355/113 match?

\tfrac{355}{113} = 3.14159\,29203\,5\dots
\pi = 3.14159\,26535\,8\dots

They agree to six decimal places — much better than 22/7's two. But they still disagree at the seventh. 355/113 is rational; \pi is not; they are different numbers. No fraction will ever match \pi to all decimal places at once, because a match to infinitely many places would make \pi rational, which it is not.

This satellite sits inside Number Systems.