Pi

Ask a class of students to name a mathematical constant and you will get $e$ and $\pi$ before anything else. $\pi$ is the more immediately tangible of the two: it is the number you meet the moment a wheel starts turning, the number in the formula for the area of a disc, the number that keeps appearing in wave equations, probability densities, and Fourier series. It is also, at the time of writing, the most thoroughly computed number in existence — trillions of its digits have been produced and checked, and the work continues.

This page treats $\pi$ the way it deserves to be treated: carefully. It starts with the geometric reason the constant exists at all, walks through the history of how humans pinned down ever better values, and then confronts the harder question — what is $\pi$, mathematically? You will meet three standard definitions (by circumference, by area, and by infinite series), the classic formulas that carry it into analysis, and a set of genuinely usable methods for computing it. Five exercises with full reference answers close the page.

The Story of Pi: From Antiquity to the Present

The Mystery of the Circle

A circle looks simple, but the moment you ask for its circumference you run into something remarkable. Take a circle of radius 1, and one of radius 1000, and one of radius $10^{-9}$. Measure the circumference of each and divide by the diameter. The three numbers agree to as many digits as you can measure, and they agree because of a simple scaling argument: any circle is the image of any other circle under a dilation, and a dilation multiplies the circumference and the diameter by the same factor, so their ratio is untouched.

That invariant ratio is $\pi$:

$$\pi = \frac{C}{d}$$

where $C$ is the circumference and $d$ is the diameter. It is a number, not a shape — the same number for every circle ever drawn, which is exactly what makes it useful.

Pi Through History

Two entries deserve comment. Zu Chongzhi’s $\frac{355}{113}$ is still the best rational approximation to $\pi$ with a small denominator, and it was known in China well before it reached Europe. And the symbol itself has a quiet history: William Jones used $\pi$ in 1706 for the ratio of circumference to diameter, and Euler’s adoption in the 1730s is what turned a local usage into a world standard. The name comes from the same Greek word as the letter $\pi$ — periphery.

Understanding Pi in Depth

Definition and Common Approximations

How good are the usual approximations? The table below gives the absolute error of each.

The key observation is that $\frac{355}{113}$ is extraordinarily good: its denominator is only 113, and it agrees with $\pi$ to seven significant figures, far better than $\frac{22}{7}$ despite a denominator roughly sixteen times larger. It is a lucky approximation, and it is the one most textbooks still teach.

Three Ways to Define Pi

1. The circumference definition. $\pi = \frac{C}{d}$, that is, $C = \pi d = 2\pi r$. This is the definition above: the ratio of circumference to diameter. It is the most intuitive one and the one that goes back to Archimedes.

2. The area definition. $\pi = \frac{S}{r^2}$, that is, $S = \pi r^2$. A different-looking statement, but it produces the same number. The honest caveat is that this equality is a theorem, not a free choice: it took real work to prove that the constant in the area formula equals the constant in the circumference formula. Historically, this is exactly what the method of exhaustion achieved — squeezing the circle between inscribed and circumscribed polygons and comparing the resulting areas with the same polygons’ areas.

3. Series definitions. Starting in the 17th and 18th centuries, $\pi$ began to be defined by infinite series and infinite products, which is what made it computable to arbitrary precision. The most famous are:

$$\frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots = \sum_{n=0}^{\infty} \frac{(-1)^n}{2n+1}$$ $$\frac{\pi}{2} = \frac{2}{1} \cdot \frac{2}{3} \cdot \frac{4}{3} \cdot \frac{4}{5} \cdot \frac{6}{5} \cdot \frac{6}{7} \cdots = \prod_{n=1}^{\infty} \frac{(2n)^2}{(2n-1)(2n+1)}$$ $$\frac{2}{\pi} = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{2+\sqrt{2}}}{2} \cdot \frac{\sqrt{2+\sqrt{2+\sqrt{2}}}}{2} \cdots$$

The first is the Leibniz series, the second the Wallis product, the third Viète’s formula. All three define $\pi$ purely by limits of rational numbers, with no reference to circles at all.

Classic Formulas

• Area of a circle: $S = \pi r^2$
• Circumference of a circle: $C = 2\pi r$
• Euler’s formula: $e^{i\pi} + 1 = 0$
• Leibniz series: $\frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots$
• Gauss integral: $\int_{-\infty}^{+\infty} e^{-x^2}\,dx = \sqrt{\pi}$
• The Basel problem: $\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}$, solved by Euler in 1735
• Stirling’s formula: $n! \sim \sqrt{2\pi n}\left(\frac{n}{e}\right)^n$
• The standard normal density: $\frac{1}{\sqrt{2\pi}} e^{-x^2/2}$

Euler’s formula is the one that gets called the most beautiful equation in mathematics, and it earns the title by tying five fundamental constants — $e$, $i$, $\pi$, $1$ and $0$ — into a single identity. The Basel identity is arguably more surprising: it says that the sum of the reciprocal squares of the positive integers is a rational multiple of $\pi^2$, a fact that links an apparently arithmetic series to geometry.

Interesting Facts

• Pi Day falls on 14 March, because the date reads as 3.14. Pi Approximation Day, 22 July, plays on the fraction $\frac{22}{7}$.
• The decimal expansion shows no detectable pattern — no known periodicity, no known normality proof. It is important to be precise here: the absence of a known pattern is not a proof that no deeper pattern exists.
• More than one hundred trillion digits of $\pi$ have been computed and verified with modern hardware, and the records keep moving.
• The memorisation record is 70,000 digits, set by Rajveer Sodhi in 2022 and recognised by the Guinness World Records. Earlier claims of “hundreds of thousands” or “over a million” digits are not supported by any recognised record.
• $\pi$ appears in probability, number theory, physics and engineering — in the Gaussian density, in Buffon’s needle, in the zeta function, in wave equations, and in the resonant frequencies of almost every vibrating system.

Mathematical Meaning and Properties

$\pi$ is irrational, and it is transcendental. Irrationality means that no ratio of two integers equals $\pi$; transcendence means that $\pi$ is not a root of any non-zero polynomial with rational coefficients, which is a strictly stronger statement. Lambert proved irrationality in 1761, using a continued fraction for $\tan x$:

$$\tan x = \frac{x}{1 - \frac{x^2}{3 - \frac{x^2}{5 - \frac{x^2}{7 - \cdots}}}}$$

If $\pi$ were rational, then $\frac{\pi}{4}$ would be a nonzero rational number, and Lambert’s theorem says that the tangent of a nonzero rational is irrational. But $\tan\frac{\pi}{4} = 1$ is rational, a contradiction. Lindemann pushed further in 1882, using Euler’s formula: if $\pi$ were algebraic, then $i\pi$ would be a nonzero algebraic number, and the Lindemann–Weierstrass theorem says that $e^{\alpha}$ is transcendental for every nonzero algebraic $\alpha$. Yet $e^{i\pi} = -1$ is algebraic. Again a contradiction.

$\pi$ and the trigonometric functions are inseparable. The period of $\sin$ and $\cos$ is $2\pi$; the key values $\sin\frac{\pi}{6} = \frac{1}{2}$ and $\cos\frac{\pi}{3} = \frac{1}{2}$ are what let you build exact triangles; and the entire machinery of Fourier analysis — decomposing a periodic function into sines and cosines — is built on the period $2\pi$.

$\pi$ is a bridge between fields. It appears in the area of a disc (geometry), in $\int e^{-x^2}dx = \sqrt{\pi}$ (analysis), in the normal density and in Buffon’s needle (probability), in $\zeta(2) = \frac{\pi^2}{6}$ (number theory), and in the decay rate of a Fourier transform (physics).

The continued fraction of $\pi$ is a memorable object. It begins

$$\pi = [3;\, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, \ldots]$$

The enormous partial quotient $292$ is what makes $\frac{355}{113}$ such a good approximation: after the convergent $3 + \frac{1}{7 + \frac{1}{15 + \frac{1}{1}}}$ (which is exactly $\frac{355}{113}$), the next partial quotient promises a further improvement of about a factor of $292$.

---

Exercises

Exercise 1

Write the definition of $\pi$ and list the common approximations of it.

Exercise 2

Decide whether $\pi$ is rational, and justify your answer.

Exercise 3

Write down a classical series for $\pi$ and explain where it comes from.

Exercise 4

Which of the following numbers is irrational?

(A) $\sqrt{2}$

(B) $\pi$

(C) $e$

(D) All of the above

Exercise 5

Write the area and circumference formulas for a circle.

---

Summary

$\pi$ is the constant that measures the circle, and the surprising thing about it is how far it travels: the same number governs the area of a disc, the period of a sine wave, the density of the normal distribution, and the value of the zeta function at 2. It can be defined geometrically by the ratio of circumference to diameter, geometrically by the ratio of area to radius squared, or purely analytically by infinite series such as the Leibniz, Wallis, and Viète formulas. It is irrational (Lambert, 1761) and transcendental (Lindemann, 1882), so its decimal expansion never repeats and never terminates. And it is computable: modern algorithms such as the Chudnovsky series add roughly fourteen correct digits per term, which is how the record for computed digits has been pushed past one hundred trillion.

Symbols Used in This Article