# "Episodes in Multicultural Mathematics" workshop.

## Jain large numbers

Historic Indian culture was fascinated with large numbers and extra-large numbers were continually associated with gods.  In a religious text from before 500 BCE there was a story of an army of 10^10 + 10^14 + 10^20 + 10^24+ 10^30 + 10^34 + 10^40 + 10^44 + 10^52 + 10^57 + 10^62 + 5 men.  (With names for each of these powers of ten - compared to the Greeks who had no words for numbers above a "myriad" [10^4].)

The Jains are an ascetic and nonviolent religious response to the excesses of Vedic India.  They had theological interests in the vastness of time and space.  They considered infinite and finite (but quite large) subdivisions of time.  How quite large?  They considered a "purvis" to be 756x10^11 days, and a shirsa prakelika to be ... (8,400,000)^28 purvis.  Here are some other tales of large numbers ...
• a yojana 10 km
• a raijju is the distance traveled by a god in six months if he covers a hundred thousand yojana in each blink of his eye
• a palya is the time it will take to empty a cubic vessel of side one yojana filled with the wool of newborn lambs if one strand is removed every century
• consider a trough whose diameter is that of the earth.  Fill it up with white mustard seeds counting one after another.  Similarly fill up with mustard seeds other troughs of the sizes of the various lands and seas.  Infinity is bigger.

Unsurprisingly, infinite was a common them for them.  They, in fact, considered different sizes of infinity which they called enumerable, innumerable, and infinite, previewing transfinite cardinals by nearly 2000 years.

## Mayan calendars

The Mayans also had a fascination with large numbers (although not quite that large), and their use in calendars (which were much more mathematically well-constructed than any have been in the West, surely than they were during the first millennium CE, when the Mayans were at their peak).  The Mayans kept careful astronomical observations and devised their numeration system around their calendar and astronomical needs.

One commentary

## Incan quipus

The Incans wanted a way to deliver recorded information across great distance before they had developed writing.  They did so by tying a sequence of knots into linked strings.  These were easily transported by runners who delivered them across difficult mountainous terrain.  These quipus are from 1400-1550 CE.

## Egypt

There were over a million people living in Cairo of ancient Egypt, often working together as a complicated society.  The papyri we have sometimes concern the computations necessary to maintain their many projects.  For example, the Reisner papyrus appears to be a set of books for a construction site.  They include a list of employees, as well as calculations of volumes and areas.  A typical computation is determining the number of workmen needed to excavate a tomb.  These documents are from 2000 - 1500 BCE.

Here are some views of the Rhind Papyrus - to give you a sense of what this relic actually is.
The Moscow Papyrus
Here is a little bit from the Moscow Papyrus.
Some problems from the Rhind and Moscow - translated.  More detailed versionMore sources.
The Berlin Papyrus

## Babylon

The Mesopotamian culture was one of the first agricultural societies.  They kept records of this by marking in mud with a stylus and then baking into tablets.  These have the advantage of preserving very well.  Many of their problems were about canals.  They also seem to have an interest in math problems for curiosity sake as well.  The works we have are from a range of time from 2500 to 1600 BCE.  Suffice to say - very long ago.

Babylonian quadratic solution on copy of YBC 6967  Details of original solution.

Babylonian square root of 2 on YBC 7289 (schematic)

## China

The Nine Chapters of the Mathematical Art is the canonical ancient Chinese textbook (held in the same esteem as Euclid's Elements).  It has many problems and sections on diverse practical and instructional topics.  It also has many editions through history with a tradition of several different commentators making additions.  This work is from about 200 BCE to 200 CE.  The "Pythagorean theorem" is well known in many cultures, here we see it was familiar in China in about 500 BCE.

Written Chinese numerationImage of Zhoubi suanjing
An image of nine chaptersText from nine chapters.
Contents of nine chapters (excuse the Wikipedia link - I do have this in a print source but this way I don't need to scan it here).
Yang Hui's triangle

## India

The Indians were also familiar with the "Pythagorean theorem" at least two hundred years before Pythagoras (~600 BCE).  Much of their geometry work was used for temple building in ancient India.  Indian work with trigonometry in about 500 CE is motivated by astronomy and is mostly computational (as is most early trigonometry).  The Lilivati from Bhaskara represents an overview of mathematical achievements at the time throughout their history (~1150 CE).

## Islamic

Early Islamic culture in the middle-east was a varied cultural mixing ground, with regular travelers and commerce from Europe, Africa, and the far East.  It lead to great wealth and a respect for diversity, which included an attempt to gather and develop ideas from each culture they contacted.   Inspired by numeration from  India, and commercial record-keeping needs, in the 9th century, Algebra is developed independently from the European influence in Geometry.  Islamic geometry was also often algebraic, but also included trigonometry (largely for needs in astronomy).  One catalyst for geometric work came from creating architectural art - which was highly geometric due to a prohibition of depiction of living objects.

Some from al-jabr.
Early decimal point
Abu'l Wafa's finger reckoning was an influence on things like this.
al-Haytham on geometry.
Great mosque - detail.
Friday mosque
al-Khayyami on the cubic
al-Mun'im's arithmetic triangle
al-Biruni's qibla problem

## Africa

There is much left to learn.  For me one big area is sub-Saharan Africa.