by DK
Double base
Around the 2nd century CE, abaci had become a common tool in China. The Chinese abacus, or suanpan, matched the design of the Roman version, but rather than use pebbles set in a metal frame, it employed wooden counters on rods—the template for modern abaci. Whether the Roman or Chinese abaci came first is unclear, but their similarities may be a coincidence, inspired by the way people count using the five fingers of one hand. Both abaci have two decks—the lower deck counting to five, and the upper deck counting the fives.
By the second millennium CE, the suanpan and its counting methods were becoming widespread across Asia. In the 1300s, it was exported to Japan, where it was called the soroban. This was slowly refined and by the 1900s, the soroban was a 1:4 abacus (with 1 upper bead on each rod, and 4 lower beads).
A female personification of Arithmetic judges a contest between the Roman mathematician Boëthius, who uses numbers, and the Greek Pythagoras, who uses a counting board.
The Soroban Championship
Japanese schoolchildren still use the soroban (Japanese abacus) in mathematics lessons as a way of developing mental arithmetic skills. The soroban is also used for far more complex calculations. Expert soroban users can usually do such calculations more quickly than someone punching the values into an electronic calculator.
Every year, the best abacists from across Japan take part in the Soroban Championship. They are tested on their speed and accuracy in a knockout system similar to a spelling bee. One of the highlights of the event is Flash Anzan™, a feat of mental arithmetic in which the players imagine operating an abacus to add 15 three-digit numbers—no physical abacus is allowed. The contestants watch the numbers appear on a big screen, flashing by faster with each round. The 2017 world record for Flash Anzan was 15 numbers added together in 1.68 seconds.
See also: Positional numbers • Pythagoras • Zero • Decimals • Calculus
IN CONTEXT
KEY FIGURE
Archimedes (c. 287–c. 212 BCE)
FIELD
Number theory
BEFORE
c. 1650 BCE The Rhind papyrus, written by Middle Kingdom Egyptian scribes as a mathematics guide, includes estimates of the value of π.
AFTER
5th century CE In China, Zu Chongzhi calculates π to seven decimal places.
1671 Scottish mathematician James Gregory develops the arctangent method for computing π. Gottfried Leibniz makes the same discovery in Germany three years later.
2019 In Japan, Emma Haruka Iwao uses a cloud computing service to calculate π to more than 31 trillion decimal places.
The fact that pi (π)—the ratio of the circumference of a circle to its diameter, roughly given as 3.141—cannot be expressed exactly as a decimal no matter how many decimal places are calculated has fascinated mathematicians for centuries. Welsh mathematician William Jones was the first to use the Greek letter π to represent the number in 1706, but its importance for calculating the circumference and area of a circle and the volume of a sphere has been understood for millennia.
Pi is not merely the ubiquitous factor in high school geometry problems; it is stitched across the whole tapestry of mathematics.
Robert Kanigel
American science writer
Ancient texts
Determining pi’s exact value is not straightforward and the quest continues to find pi’s decimal representation to as many places as possible. Two of the earliest estimates for π are given in the ancient Egyptian documents known as the Rhind and Moscow papyri. The Rhind papyrus, thought to have been intended for trainee scribes, describes how to calculate the volumes of cylinders and pyramids and also the area of a circle. The method used to find the area of a circle was to find the area of a square with sides that are 8⁄9 of the circle’s diameter. Using this method implies that π is approximately 3.1605 calculated to four decimal places, which is just 0.6 per cent greater than the most accurate known value of π.
In ancient Babylon, the area of a circle was found by multiplying the square of the circumference by 1⁄12, implying that the value of π was 3. This value appears in the Bible (1 Kings 7:23): “And he made the Sea of cast bronze, ten cubits from one brim to the other; it was completely round. Its height was five cubits, and a line of thirty cubits measured its circumference.”
In c. 250 BCE, the Greek scholar Archimedes developed an algorithm for determining the value of π based on constructing regular polygons that exactly fit within (inscribed), or enclosed (circumscribed), a circle. He calculated upper and lower limits for π by using Pythagoras’s theorem—that the area of the square of the hypoteneuse (the side opposite the right angle) in a right-angled triangle is equal to the sum of the areas of the squares of the other two sides—to establish the relationship between the lengths of the sides of regular polygons when the number of sides was doubled. This enabled him to extend his algorithm to 96-sided polygons. Determining the area of a circle using a polygon with many sides had been proposed at least 200 years before Archimedes, but he was the first person to consider polygons that were both inscribed and circumscribed.
ARCHIMEDES
Born in c. 287 BCE in Syracuse, Sicily, the Greek polymath Archimedes excelled as a mathematician and engineer, and is also remembered for his “eureka” moment, when he realized that the volume of water displaced by an object is equal to the volume of that object. Among his claimed inventions is the Archimedes’ screw, a revolving screw-shaped blade in a cylinder, which pushes water up a gradient.
In mathematics, he used practical approaches to establish the ratio of the volumes of a cylinder, sphere, and cone with the same maximum radius and height to be 3:2:1. Many consider Archimedes to be a pioneer of calculus, which was not developed until the 1600s. He was killed by a Roman soldier during the Siege of Syracuse in 212 BCE, despite orders that his life be spared.
Key works
c. 250 BCE On the Measurement of a Circle
c. 225 BCE On the Sphere and the Cylinder
c. 225 BCE On Spirals
Squaring the circle
Another method for estimating π, “squaring the circle,” was a popular challenge for mathematicians in ancient Greece. It involved constructing a square with the same area as a given circle. Using only a pair of compasses and a straight edge, the Greeks would superimpose a square on a circle and then use their knowledge of the area of a square to approximate to the area of a circle. The Greeks were not successful with this method, and in the 1800s, squaring the circle was proved to be impossible, due to π’s irrational nature. This is why attempts to achieve an impossible task are sometimes known as “squaring the circle.”
Another way mathematicians have attempted to square the circle is to slice it into sections and rearrange them into a rectangular shape. The area of the rectangle is r × 1⁄2(2πr) = r × πr × πr² (where r is the radius of the circle and 2πr is its diameter). The area of a circle is also πr². The smaller the segments used, the closer the shape is to a rectangle.
Although polygons had long been used to estimate the circumference of circles, Archimedes was the first to use inscribed (inside the circle) and circumscribed (outside the circle) regular polygons to find upper and lower limits for π.
The works of Archimedes are, without exception, works of mathematical exposition.
Thomas L. Heath
Historian and mathematician
The quest spreads
More than 300 years after the death of Archimedes, Ptolemy (c. 100–170 cE) determined π to be 3:8:30 (base-60), that is, 3 + 8⁄60 + 30⁄3,600 = 3.1416, which is just 0.007 percent greater than the closest known value of π. In China, 3 was often used as the value of π, until became common from the 2nd century CE. The latter is 2.1 percent greater than π. In the 3rd century, Wang Fau stated that a circle with a circumference of 142 had a diameter of 45—that is 142⁄45 = 3.15, just 1.4 percent more than π—while Liu Hui used a 3,072-sided polygon to estimate π as 3.1416. In the 5th century, Zu Ch
ongzhi and his son used a 24,576-sided polygon to calculate π as 355⁄113 = 3.14159292, a level of accuracy (to seven decimal places) not achieved in Europe until the 1500s.
In India, the mathematician–astronomer Aryabhata included a method for obtaining π in his Aryabhatiyam astronomical treatise of 499 CE: “Add 4 to 100, multiply by 8, and then add 62,000. By this rule the calculation of the circumference of a circle with a diameter of 20,000 can be approached.” This works out as [8(100 + 4) + 62,000] ÷ 20,000 = 62,832 ÷ 20,000 = 3.1416.
Brahmagupta (c. 598–668 CE) derived square root approximations of π using regular polygons with 12, 24, 48, and 96 sides: , , , and respectively. Having established that π2 = 9.8696 to four decimal places, he simplified these calculations to π = . During the 9th century, Arab mathematician al-Khwarizmi used 31⁄7, , and 62,832⁄20,000 as values for π, attributing the first value to Greece and the other two to India. English cleric Adelard of Bath translated al-Khwarizmi’s work in the 12th century, renewing an interest in the search for π in Europe. In 1220, Leonardo of Pisa (Fibonacci), who popularized Hindu-Arabic numerals in his book Liber Abaci (The Book of Calculation), 1202, computed π to be 864⁄275 = 3.141, a small improvement on Archimedes’s approximation, but not as accurate as the calculations of Ptolemy, Zu Chongzhi, or Aryabhata. Two centuries later, Italian polymath Leonardo da Vinci (1452–1519) proposed making a rectangle whose length was the same as a circle’s circumference and whose height was half its radius to determine the area of the circle.
Archimedes’ method used in ancient Greece for calculating π was still being used in the late 16th century. In 1579, French mathematician François Viète used 393 regular polygons each with 216 sides to calculate π to 10 decimal places. In 1593, Flemish mathematician Adriaan van Roomen (Romanus) used a polygon with 230 sides to compute π to 17 decimal places; three years later, German–Dutch professor of mathematics Ludolph van Ceulen calculated π to 35 decimal places.
The development of arctangent series by Scottish astronomer–mathematician James Gregory in 1671, and independently by Gottfried Leibniz in 1674, provided a new approach for finding π. An arctangent (arctan) series is a way of determining the angles in a triangle from knowledge of the length of its sides, and involves radian measure, where a full turn is 2π radians (equivalent to 360°).
Unfortunately, hundreds of terms are needed to compute π to even a few decimal places using this series. Many mathematicians attempted to find more efficient methods to calculate π using arctan, including Leonhard Euler in the 1700s. Then, in 1841, British mathematician William Rutherford computed 208 digits of π using arctan series.
The advent of calculators and electronic computers in the 1900s made finding the digits of π much easier. In 1949, 2,037 digits of π were calculated in 70 hours. Four years later, it took around 13 minutes to compute 3,089 digits. In 1961, American mathematicians Daniel Shanks and John Wrench used arctan series to compute 100,625 digits in under eight hours. In 1973, French mathematicians Jean Guillaud and Martin Bouyer achieved 1 million decimal places, and in 1989, a billion decimal places were computed by Ukrainian–American brothers David and Gregory Chudnovsky.
In 2016, Peter Trueb, a Swiss particle physicist, used the y-cruncher software to calculate π to 22.4 trillion digits. A new world record was set when computer scientist Emma Haruka Iwao calculated π to more than 31 trillion decimal places in March 2019.
By arranging the segments of a circle in a near-rectangular shape, it can be shown that the area of a circle is πr2. The height of the “rectangle” is approximately equal to the radius r of the circle, and the width is half of the circumference (half of 2πr, which is πr).
There is no end with pi. I would love to try with more digits.
Emma Haruka Iwao
Japanese computer scientist
The perimeter to height ratio of the Great Pyramid of Giza, in Egypt, is almost exactly π, which might suggest that ancient Egyptian architects were aware of the number.
Applying pi
Astrophysicists use π in their calculations to determine the orbital paths and characteristics of planetary bodies such as Saturn.
Space scientists constantly use π in their calculations. For example, the length of orbits at different altitudes above a planet’s surface can be worked out by using the basic principle that if the diameter of a circle is known, its circumference can be calculated by multiplying by π. In 2015, NASA scientists applied this method to compute the time it took the spacecraft Dawn to orbit Ceres, a dwarf planet in the asteroid belt between Mars and Jupiter.
When scientists at NASA's Jet Propulsion Laboratory in California wanted to know how much hydrogen might be available beneath the surface of Europa, one of Jupiter's moons, they estimated the hydrogen produced in a given unit area by first calculating Europa’s surface area, which is 4πr2, as it is for any sphere. Since they knew Europa’s radius, calculating its surface area was easy.
It is also possible to work out the distance traveled during one rotation of Earth by a person standing at a point on its surface using π, providing the latitude of the person’s position is known.
See also: The Rhind papyrus • Irrational numbers • Euclid’s Elements • Eratosthenes’ sieve • Zu Chongzhi • Calculus • Euler’s number • Buffon’s needle experiment
IN CONTEXT
KEY FIGURE
Eratosthenes (c. 276–c. 194 BCE)
FIELD
Number theory
BEFORE
c. 1500 BCE The Babylonians distinguish between prime and composite numbers.
c. 300 BCE In Elements (Book IX proposition 20), Euclid proves that there are infinitely many prime numbers.
AFTER
Early 1800s Carl Friedrich Gauss and French mathematician Adrien-Marie Legendre independently produce a conjecture about the density of primes.
1859 Bernhard Riemann states a hypothesis about the distribution of prime numbers. The hypothesis has been used to prove many other theories about prime numbers, but it has not yet been proved.
In addition to calculating Earth’s circumference and the distances from Earth to the Moon and Sun, the Greek polymath Eratosthenes devised a method for finding prime numbers. Such numbers, divisible only by 1 and themselves, had intrigued mathematicians for centuries. By inventing his “sieve” to eliminate nonprimes—using a number grid and crossing off multiples of 2, 3, 5, and above—Eratosthenes made prime numbers considerably more accessible.
Prime numbers have exactly two factors: 1 and the number itself. The Greeks understood the importance of primes as the building blocks of all positive integers. In his Elements, Euclid stated many properties of both composite numbers (integers above one that can be made by multiplying other integers) and primes. These included the fact that every integer can be written as a product of prime numbers or is itself a prime. A few decades later, Eratosthenes developed his method, which can be extended to uncover all primes. Using a number grid for 1 to 100 (see right), it is clear that 1 is not a prime number as its only factor is 1. The first prime number—and also the only even prime—is 2. As all other even numbers are divisible by 2, they cannot be primes, so all other primes must be odd. The next prime, 3, has only two factors, so all the other multiples of 3 cannot be primes. The number 4 (2 × 2) has already had its multiples removed, since they are all even. The next prime is 5, so all other multiples of 5 cannot be prime. The number 6 and all its multiples have been removed from the list of potential primes, as they are even multiples of 3. The next prime is 7, and removing its multiples eliminates 49, 77, and 91. All the multiples of 9 have gone, as they are multiples of 3, and all the multiples of 10 have been removed, because they are the even multiples of 5. The multiples of 11 up to 100 have already been removed, and so on for all successive numbers. There are only 25 prime numbers up to 100—starting with 2, 3, 5, 7, and 11, and ending with 97—all identified by simply removing every multiple of 2, 3, 5, and 7.
Eratosthenes’ method star
ts with a table of consecutive numbers. First, 1 is crossed out. Then all multiples of 2 are crossed out except 2 itself. The same is then done for multiples of 3, 5, and 7. Multiples of any number higher than 7 are already crossed out, since 8, 9, and 10 are composites of 2, 3, and 5.
The search continues
Prime numbers attracted the attention of mathematicians from the 1600s onward, when figures such as Pierre de Fermat, Marin Mersenne, Leonhard Euler, and Carl Friedrich Gauss probed further into their properties.
Even in the age of computers, determining whether a large number is prime remains highly challenging. Public key cryptography—the use of two large primes to encrypt a message—is the basis of all internet security. If hackers ever do figure out a simple way of determining the prime factorization of very large numbers, a new system will need to be devised.
ERATOSTHENES
Born around 276 BCE in Cyrene, a Greek city in Libya, Eratosthenes studied in Athens and became a mathematician, astronomer, geographer, music theorist, literary critic, and poet. He was the chief librarian at the Library of Alexandria, the greatest academic institution of the ancient world. He is known as the father of geography for founding and naming the subject as an academic discipline and developing much of the geographical language used today.
Eratosthenes also recognized that Earth is a sphere and calculated its circumference by comparing the angles of elevation of the Sun at noon at Aswan in southern Egypt and at Alexandria in the north of the country. In addition, he produced the first world map that featured meridian lines, the Equator, and even polar zones. He died around 194 BCE.