Write π on a calculator:
3.141592653589793...
Now divide 22 by 7:
3.142857142857143...
They are surprisingly close.
So two questions appear at once:
Why is such a simple fraction so good?
And if it is so good, why can we never write π = 22/7?
In the previous article, we established that every circle shares the same circumference-to-diameter ratio, π. Now the question changes from why the constant is universal to how efficiently we can represent that constant with simpler numbers.
Quick Answer
22/7 is a very good approximation to π, but it is not π.
Historically, Archimedes used 22/7 as an upper bound:
223/71 < π < 22/7
Modern continued fractions give another reason 22/7 stands out: it is one of the early convergents of π, meaning it is an unusually efficient rational approximation for such a small denominator.
Accuracy asks how small the error is. Equality asks whether the error is exactly zero. Those are different questions.
The Structural Question — How Good Is “Good”?
A fraction can be judged in at least two ways.
First: How close is it?
Second: How simple is it?
A fraction with a huge denominator can often get very close to π. That is not surprising.
What makes 22/7 interesting is that the denominator is only 7.
So the deeper question is:
How much accuracy do we get for such a small denominator?
This gives us a useful mental model:
approximation quality = small error for a limited complexity budget
How Close Is 22/7?
Compare:
π ≈ 3.141592653589793
22/7 ≈ 3.142857142857143
The absolute error is:
|22/7 − π| ≈ 0.0012644893
The relative error is about:
0.04025%
Here is a physical way to feel that number.
For a circle with diameter 1 meter, using 22/7 instead of π makes the calculated circumference too long by about:
1.26 millimeters
That is small enough to make 22/7 useful for rough mental calculation, but it is not zero.
Archimedes Used 22/7 as a Fence, Not an Identity
This historical point matters because 22/7 is often taught so casually that the distinction disappears.
Archimedes' result was:
223/71 < π < 22/7
In decimals:
3.140845... < π < 3.142857...
The true value is trapped between two rational numbers.
So 22/7 is not “Archimedes' value of π.” It is the upper wall of a proven interval.
22/7 was above π. Archimedes' achievement was knowing which side of the fence it belonged on.
Why Does Denominator 7 Stand Out?
Suppose we restrict ourselves to small denominators.
For each denominator q, choose the nearest integer numerator p to πq. Then compare the error.
| Fraction | Decimal value | Absolute error |
|---|---|---|
| 3/1 | 3.000000 | 0.141593 |
| 13/4 | 3.250000 | 0.108407 |
| 16/5 | 3.200000 | 0.058407 |
| 19/6 | 3.166667 | 0.025074 |
| 22/7 | 3.142857 | 0.001264 |
The jump at denominator 7 is dramatic.
Even more striking: if we search by this simple “nearest numerator” rule, no denominator from 8 through 56 gives a smaller absolute error than 22/7. A new record finally appears at 179/57.
So 22/7 is not merely close. It is unusually efficient for its size.
Figure 1. 22/7 stands out among small fractions and also appears as an early convergent of π's continued fraction.
A Modern Explanation — Continued Fractions
A continued fraction rewrites a number as a nested sequence of integer parts and reciprocals.
For π, the simple continued fraction begins:
π = [3; 7, 15, 1, 292, ...]
Written out just far enough to see the structure:
π = 3 + 1/(7 + 1/(15 + ...))
Now compare that with:
22/7 = 3 + 1/7
This is the key picture.
22/7 is what we get when we stop the continued fraction after the 7.
The part we ignore is a small positive correction added after that 7:
7 → 7 + 1/(15 + ...)
Because the next coefficient, 15, is fairly large, that correction is small. So replacing the corrected denominator by exactly 7 changes the value only a little.
And because the true denominator is slightly larger than 7, its reciprocal is slightly smaller than 1/7. That immediately tells us:
π < 22/7
This is a modern number-theory explanation of both the closeness and the direction of the error.
It is not Archimedes' historical method. His proof was geometric.
What Does “Convergent” Mean?
If we stop a continued fraction at different stages, we get fractions called convergents.
For π, the first few are:
3, 22/7, 333/106, 355/113, ...
Convergents are famous because they tend to give especially strong rational approximations relative to the size of their denominators.
That is why 22/7 appears naturally. It is not a random lucky fraction pulled from nowhere.
Try It — Let Small Fractions Compete with Python
This code searches denominator by denominator and prints a fraction only when it sets a new record for absolute error.
import math
best_error = float("inf")
for q in range(1, 120):
p = round(math.pi * q)
value = p / q
error = abs(value - math.pi)
if error < best_error:
print(
f"{p}/{q}",
"value =", value,
"error =", error
)
best_error = errorThe early record-breakers include:
3/1 → 13/4 → 16/5 → 19/6 → 22/7
Much later, other fractions finally beat 22/7. By denominator 106, the convergent 333/106 is much better.
Try a denominator budget.
Change the loop to stop at 8, 20, 57, and 120.
At what budget does 22/7 stop being the record-holder?
Why Can 22/7 Never Be Exactly π?
22/7 is a ratio of two integers.
That makes it a rational number.
π is irrational. It cannot be written exactly as one integer divided by another.
Johann Heinrich Lambert gave the first rigorous proof of π's irrationality in the eighteenth century.
Therefore:
no fraction p/q can equal π exactly
Not 22/7.
Not 355/113.
Not a fraction with a billion-digit numerator and denominator.
A rational approximation may be astonishingly accurate. Its error can be tiny. But if the target is irrational, the error cannot be exactly zero.
Useful and Exact Are Different Standards
This distinction appears far beyond π.
Engineering may ask:
Is the approximation accurate enough for this job?
Pure mathematics may ask:
Is the value exactly identical?
Those are both legitimate questions, but they have different pass conditions.
The same tradeoff appears in modern numerical work. Computers often store or calculate with finite precision, so engineers and scientists routinely ask whether a simpler or rounded representation is accurate enough for the task. The reusable idea is not “always use more digits,” but match the accuracy to the required purpose and cost.
| Question | What matters? |
|---|---|
| Is 22/7 useful? | Whether its error is small enough for the purpose |
| Is 22/7 equal to π? | Whether the error is exactly zero |
A number can be good enough for a calculation and still be mathematically different.
Where This Article Stops
We now understand why a simple fraction can be extraordinarily close to an irrational number.
But the story of π began with a geometric trick: replace a curve by straight pieces, measure the pieces, then refine.
That raises a broader question that is not only about circles:
How can straight lines measure a curved path?
Words to Keep
rational number
A number that can be written exactly as a ratio of two integers.
irrational number
A number that cannot be written exactly as a ratio of two integers.
approximation
A value close to the target but with some nonzero error.
continued fraction
A nested fraction representation that reveals especially effective rational approximations.
convergent
A fraction obtained by stopping a continued fraction after a finite number of terms.
One Sentence to Keep
22/7 is unusually accurate for such a small denominator, but π is irrational, so “very close” can never become “exactly equal.”
What Should We Ask Next?
We have now separated three ideas: a universal constant, a useful approximation to that constant, and exact equality.
Archimedes used straight-sided polygons to learn about a curved circle.
Now we can generalize that idea beyond π itself.
Next: How Can Straight Lines Help Us Measure Curves?
Previous: Why Is Pi the Same for Every Circle?
Related: How Did Archimedes Calculate Pi Without a Calculator?
Sources & Further Reading
- MacTutor History of Mathematics, “A History of Pi” — Archimedes' bounds `223/71 < π < 22/7` and the historical warning that 22/7 was not claimed as the exact value.
- Eric W. Weisstein, “Pi Continued Fraction” — Wolfram MathWorld — π's continued fraction and early convergents `3, 22/7, 333/106, 355/113, ...`.
- Eric W. Weisstein, “Regular Continued Fraction” — Wolfram MathWorld — general background on continued fractions, convergents, and their approximation behavior.
- MacTutor, “Johann Heinrich Lambert” — historical account of Lambert's rigorous proof that π is irrational.
- Thomas L. Heath (ed.), The Works of Archimedes — historical edition containing Measurement of a Circle.
Historical note: continued fractions, denominator-budget searches, decimal errors, and Python are modern explanatory tools. They are not presented as Archimedes' historical method for obtaining his bound.