The Origins of
Numerical Analysis

Samarkand, July 1424. In a city that Ulugh Beg, grandson of Timur, has filled with scholars, a mathematician from Kashan named Jamshīd al-Kāshī finishes a treatise on a single number: the ratio of a circle’s circumference to its diameter. He does not begin with his own answer. He begins by grading everyone else’s.

Archimedes, he writes, proved that the ratio lies between 3 10/71 and 3 1/7 — two bounds that differ by 1/497 of the diameter. On a circle the size of the Earth’s, that uncertainty is about five farsakh, some thirty kilometres. Abū al-Wafāʾ al-Būzjānī in Baghdad, using polygons of 720 sides, had brought it down to about a thousand cubits. Abū Rayḥān al-Bīrūnī, with 180 sides, to about one farsakh. And both of them, al-Kāshī adds, had made a slip in computing a chord. (He was right about al-Bīrūnī. Modern scholars have found that the slip he blamed on Abū al-Wafāʾ was in a later report of his work.)

Then he sets his own target, before he computes a single digit.

Before the story, the map. Press play and watch where these ideas were made and how they moved. Gold arcs are books and methods travelling: base-60 fractions from Babylon to Alexandria, Greek mathematics into Arabic, Indian numerals to Baghdad, Arabic numerals to Pisa. The turquoise arcs are people. Notice how many of the great names in this story were born far to the east — in Khwarazm, Buzjan, Karaj, Nishapur, Tus and Kashan — and travelled to wherever the work was paid for.

Every number in this course is written in a place-value system: the same digit means different amounts in different positions. The first people to do that were the scribes of southern Mesopotamia, in what is now Iraq, nearly four thousand years ago. They counted in sixties. A “1” followed by a “24” could mean 1 + 24/60 — they wrote no point, and the scale came from context — and they wrote fractions as easily as whole numbers, with a stylus pressed into wet clay. We still count their way whenever we split an hour into sixty minutes and a minute into sixty seconds.

Sixty has a great virtue for computing: it divides by 2, 3, 4, 5, 6, 10, 12, 15, 20 and 30. So one tenth, in base 60, is simply 0;6 — six sixtieths, finished. Remember that. At the end of this prologue, the same one tenth will refuse to be written down at all, and in 1991 that refusal will cost lives.

In the Yale Babylonian Collection there is a small, round clay tablet, catalogued YBC 7289, about the size of a palm. Historians think it was a student’s rough work, held in one hand. On it is a square with its two diagonals. One side is marked 30. Along the diagonal are four numbers: 1;24,51,10 — and below them 42;25,35, the length of the diagonal. The first number is the square root of 2, to about six decimal places: wrong by less than one part in two million.

Around 250 BCE, in Syracuse on the island of Sicily, Archimedes did something new with an approximation: he trapped it. In a short work, Measurement of a Circle, he drew regular polygons inside and outside a circle, doubled their sides from 6 to 96, and proved that π lies between 3 10/71 and 3 1/7. He did not claim to know π. He claimed to know an interval that contains it — and that is the idea of an error bound, which you will use in every unit of this course. Seventeen centuries later, al-Kāshī opened his treatise by quoting exactly this result.

In Alexandria, in the first century CE, Heron needed the square root of 720 to find the area of a triangle. He took 27, whose square is close to 729; divided 720 by 27 to get 26 2/3; and averaged the two to get 26 5/6. If you want it closer, he wrote, repeat the step with the new number. That is an iteration: a rule that takes an approximation and returns a better one.

And around 150 CE Ptolemy, also in Alexandria, built the great trigonometric table of antiquity — a table of chords for every half degree, written in Babylonian base-60 fractions — and took π as 3;8,30, which is 3.14167. For the next thirteen centuries, astronomers and mathematicians from Spain to India computed in sixties. Much of the work in this prologue is, at heart, the work of making better tables.

In 499, a 23-year-old astronomer named Aryabhata finished a short book of verses near Pataliputra on the Ganges. In it he tabulated the half-chord, the ancestor of our sine, at 24 equal steps through a quarter circle — and he built the table from its differences, the amount each entry grows by, the way you will build difference tables in Unit 7. He also gave π as 62,832/20,000 = 3.1416 and called it, carefully, “approximate”.

In 628, at Bhinmal in what is now Rajasthan, Brahmagupta wrote the first rules for computing with zero as a number in its own right: what zero added to a number gives, what a number taken from itself leaves. Combined with the Indian decimal place-value numerals, it gave the world the digits you write today. In 665 he did something just as important for this course: he showed how to read between the lines of a table.

The Abbasid caliph al-Mansur founded Baghdad in 762. Within about a decade an embassy from India brought a Sanskrit astronomical text to his court, and with it the Indian numerals — the nine digits and the zero — that Europe would later call “Arabic”. His great-grandson al-Maʾmūn (reigned 813–833) made learning a policy. In the House of Wisdom (بيت الحكمة), scholars paid by the state translated Greek, Persian and Sanskrit mathematics into Arabic, and then went further than any of their sources. The first of them, for this course, was a man from Khwarazm, south of the Aral Sea.

Al-Khwārizmī could solve any quadratic equation by completing a square. The cubic — an equation with x³ in it — would not yield to the same trick, and it occupied the best minds of Persia for two centuries. The story of how they attacked it is, in miniature, the story of numerical analysis: when you cannot find a formula, find a construction; when the construction will not give digits, find an iteration; and before you start, find out whether there is any root to look for.

Ulugh Beg, grandson of Timur, governed Samarkand and preferred mathematics to war. Between 1417 and 1420 he built a madrasa on the Registan square; between 1424 and 1429, an observatory on a hill outside the city, its great meridian arc cut into the rock. He wanted new astronomical tables, and tables need computation: sines and tangents, to more places than anyone had ever computed. To lead that work he brought to Samarkand the man this prologue began with.

We know an unusual amount about al-Kāshī, because letters he wrote in Persian to his father in Kashan survive. In them he praises the prince’s mathematics and is less generous about his colleagues: of the Samarkand scholars, only Qāḍī Zāda al-Rūmī earns his respect. He had reason to be confident. In his years in Samarkand he wrote three works that each, on its own, would have made him famous.

This prologue follows one river, but there were others, and two of them run straight into this course. In China, the Nine Chapters on the Mathematical Art, compiled by the first century CE and commented on by Liu Hui in 263, solves systems of linear equations by laying their coefficients out in columns on a counting board and eliminating them, column against column. It is the method you will learn in Unit 3 as Gaussian elimination — some 1,700 years before Gauss.

In 1145, in Segovia, Robert of Chester translated al-Khwārizmī’s al-Jabr into Latin. In 1202 a young man from Pisa whose father had taken him to the customs house at Bugia, on the coast of what is now Algeria, published the Liber Abaci, which taught Europe the Hindu–Arabic numerals he had learned there. We know him as Fibonacci. Then the centres moved. Simon Stevin made decimals popular in 1585; John Napier published logarithms in 1614, giving astronomers a way to multiply by adding.

In 1669 Isaac Newton described his method for roots — the Babylonian averaging rule, it turns out, is its special case for square roots — and Joseph Raphson published a cleaner version in 1690. Leonhard Euler stepped a differential equation forward in small steps in 1768. Carl Friedrich Gauss recovered the lost asteroid Ceres in 1801 and published his method of least squares in 1809. Carl Runge (1895) and Wilhelm Kutta (1901) gave the methods for differential equations that still fly spacecraft.

Then came the machines, and with them an old question in a new form. In 1947 John von Neumann and Herman Goldstine asked whether the rounding errors of an electronic computer would ruin the solution of a large linear system. In 1948 Alan Turing studied the same question and named the quantity that decides it: the condition number. In 1985 the IEEE 754 standard, led by William Kahan, fixed exactly how a computer stores a number, rounds it and reports a failure. That standard is the subject of Lesson 1-1. It is al-Kāshī’s question — how large an error can I afford, and can I prove I stayed inside it? — asked a billion times a second.

You will meet their work in this course without their names attached. Here are the names.

Here is the thread that runs through four thousand years. The scribe of YBC 7289 wrote √2 in sixties and stopped after three places. Al-Uqlīdisī and al-Kāshī wrote fractions in tens. Each of them knew that the digits they stopped at were not the whole number, and each of them chose where to stop.

A computer counts in twos, and it chooses where to stop for you — a fixed number of binary places, every time. Most of the time you never notice. But some very ordinary numbers have no end in base 2, however harmless they look in base 10. One of them is one tenth.