Math

# Global Calculations: A History of Pi

By the time March comes around each year, many math teachers commemorate the number $$\pi$$ because of the resemblance between the date 3/14 and the number 3.14. (If you teach high school, you might want to check out our article 11 Pi Day Activities for High School Students.) But it doesn't just have to be in March! The number $$\pi$$ has a long and rich history that can tie into lessons all year round.

By high school, students have likely encountered plenty of $$\pi$$ already and may even be equipped to learn about some of its history. Below, you can explore some of the global highlights. This history includes some mathematical concepts intended for an older student audience, such as limits and infinite series.

## The History of Pi

There is no simple, single origin of the number. While we use the Greek letter $$\pi$$ because of the influence of its Greek origins, the number simply represents a universal idea: the ratio between the circumference and diameter of a circle. For as long as mathematicians around the globe have been thinking about circles, they have been discovering new ways to approximate and calculate $$\pi$$.

The number $$\pi$$ has been studied, calculated, and thought about around the world dating back as far as 3000 BC. Take a look at the mathematicians and thinkers who have helped advance our knowledge of the number today. Note that ca. stands for circa, meaning approximately.

### ca. 3000 BC

The first known people to hunt for $$\pi$$ were Babylonians and Egyptians, around 5000 years ago. The Egyptian pyramids of Cheops and Sneferu at Gizeh both have a ratio of half the perimeter to the height equal to $$3\frac{1}{7}$$. This ratio is possibly an early attempt at calculating $$\pi$$, or the ratio between the perimeter of a circle and its diameter.

### ca. 1850 BC

A famous early example of documented evidence exists in the Rhind Papyrus written by an Egyptian scribe named Ahmes. In the papyrus, Ahmes attempts to calculate the surface area of a hemisphere—a calculation that involves circles and, thus, $$\pi$$—and implies that $$\pi=\left(\frac{16}{9}\right)^2$$. While not exactly accurate (that gives a value of about 3.1605), it is strikingly close given the time.

### ca. 440 BC

An exact calculation of the area of a circle would reveal the value of $$\pi$$ because the area $$A$$ of a circle with radius $$r$$ is given by $$A=\pi r^2$$. The Greek mathematician Antiphon took the revolutionary step of inscribing polygons of ever-increasing number of sides inside and circumscribing a circle, in effect discovering limits, one of the tenets of calculus.

Around 265 AD, Chinese mathematician Liu Hui independently discovered the same idea as Antiphon and used an efficient method involving polygons with thousands of sides. Hui correctly determined the first four digits after the decimal point of $$\pi$$ (3.1415). Around 200 years later, the Chinese mathematician Zu Chongzhi calculated $$\pi\approx\frac{355}{113}$$ using Liu Hui's algorithm applied to a 12,288-sided polygon, the most accurate approximation for nearly a millennium (3.141592920…).

In the centuries that followed, Indian mathematicians made many notable advancements in calculating $$\pi$$. Around 499 AD, the Indian astronomer and mathematician Aryabhata—the earliest Indian mathematician whose work is known to modern scholars—used $$\frac{62,832}{20,000}$$, or exactly 3.1416, in his Aryabhatiya. In 628 AD, another Indian astronomer and mathematician, Brahmagupta, tried the inscribed polygon method up to 96 sides and made the hypothesis that $$\pi=\sqrt{10}$$.

Arabic mathematician Muhammad al-Khwarizmi used a variety of values trying to calculate $$\pi$$, including $$3\frac{1}{7}$$, $$\sqrt{10}$$, and $$\frac{62,832}{20,000}$$, claiming "It is an approximation not a proof [...] and no one [...] knows the true circumference of the circle." In the same way that modern students learn that 3.14 or $$\frac{22}{7}$$ work fine for most calculations, al-Khwarizmi acknowledged that just using $$3\frac{1}{7}$$, while not the precise value, is "faster and simpler."

Historically, the first exact formula for $$\pi$$ used infinite series and was not available until around 1400. Medieval Indian mathematician-astronomer Madhava discovered the series, whose discovery remained unknown in the West until relatively recently. Though almost all of Madhava's original work is lost, he is referenced often in later mathematical works and represents early steps away from the finite processes of algebra into considerations of the infinite. He discovered that one can calculate $$\pi$$ using the following infinite series, now known as the Madhava-Leibniz or Madhava-Gregory-Leibniz series, crediting other mathematicians who independently discovered the series centuries later: $$\frac{\pi}{4}=1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\cdots$$.

### 1424

In 1424, Persian astronomer and mathematician Jamshid al-Kashi calculated $$\pi$$ using a polygon with $$3\cdot 2^{28}$$ sides. Al-Kashi generated a number able to calculate the size of the universe within the "width of a horse's hair," in effect setting the world record for 180 years. While it is true that the decimal representation of $$\pi$$ has an infinite number of digits, in truth, modern-day NASA would only need around 16 digits of $$\pi$$ to be able to calculate precise distances for orbiting spacecrafts—a benchmark achieved centuries before NASA even existed!

### 2002

Knowing more digits of $$\pi$$ is no longer very important to mathematics. However, it does have meaning to computer scientists! Being able to calculate $$\pi$$ to high precision is often used as a benchmark for the processing power of computers, along with a way to showcase human ingenuity. Japanese mathematician Yasumasa Kanada set multiple records for computing $$\pi$$ between 1995 and 2002, determining over 1 trillion decimal places.

### 2021

$$\pi$$ is still making headlines! In 2019, Japanese computer scientist Emma Haruka Iwao set a world record when Iwao and her team calculated over 31.4 trillion digits of $$\pi$$. And as recently as August 2021, researchers from the University of Applied Sciences of the Grisons in Switzerland had a supercomputer running calculations for 108 days to break Iwao's record and calculate a mind-boggling 62.8 trillion digits.

Looking for lessons that make use of $$\pi$$? Our post 11 Pi Day Activities for High School Students does not only have to be for Pi Day!

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