African Mathematics

Summary: Throughout African history, mathematics has been used in the arts, in engineering and business dealings, and in games.

As in all societies, mathematics has always been a part of the cultures and daily life of people in Africa. One difficulty of studying African mathematics is that for much of the history of Africa, the societies were non-literate, relying on oral traditions to pass their stories to the coming generations.

The wet tropical and subtropical climates of most African civilizations destroyed whatever records may have been kept—or at least hid them from the eyes of future probing historians. Hence, when assertions like the first African mathematical achievement are proclaimed, there should be a caveat that this is the first “that we know of,” for similar earlier achievements may well have been lost to history. The African Mathematical Union hosts a Commission on History of Mathematics in Africa—and readily recognizes the difficulty of its charge when even details of the social, political, and military history of precolonial Africa remain difficult to find. Discovering the history of African mathematics is an even greater challenge. Hence, much mathematics history in Africa remains speculative, based on general understandings of how mathematics works in other societies past and present, and fitted into the growing framework of bits and pieces of the history of Africa and African society.

Modern Western mathematics (now used around the world) has indeed come from the developments in the European academy, but it is only the formalized structures of pure theoretical mathematics and their applications in science, industry, and technology that grew from this theoretical work. However, mathematical thinking is much broader than the tightly logical structures of academic mathematics. Everyone who thinks about counting, arranging, or designing—anyone who makes strategic plans for achieving a goal—is thinking in mathematical terms. These examples of mathematics have occurred in Africa as much as anywhere else in the world.

Development of African Mathematics

Before recorded history, Africans herded their animals, planted and harvested crops, and built homes and other structures. All these activities required mathematics. Farming required finding the best time to plant and the appropriate time for harvest. Over time, it is likely that this led to formal or informal calendars, so the farmers would be prepared to do their tasks at the right time. They applied measurements and design as they laid out their fields, including sorting out boundary disputes with neighbors. Anthropologists have even studied the variations in the arrangements of fields in farming communities. When the time for harvesting came, several other mathematical issues arose. Initially, there would be a need for containers and storage bins for the produce, requiring geometrical design.

Later, business mathematics would be used in the markets—even those using barter systems—to determine the comparative values of the products, the gains and losses, and the purchases of other products. Some societies developed currencies—a famous example is the use of strings and bundles of cowry shells by the Yorubas. This probably contributed to the complex numeration system of the Yoruba language, which can handle very large numbers. It has even been suggested that the use of higher numbers came as a result of inflation requiring higher prices. Also, the use of strings and bundles easily flows into the grouping used in place-value of counting systems.

Village life also measured the times of human life, from the diurnal movement of the sun and language of timekeeping, to the much longer periods of milestones of maturity—birth, initiation as an adult, old age, and death. These time markers sometimes went beyond the individual and family, such that entire age cohorts measured time and followed the appropriate customs of their ages together. Kinship relations sometimes were built into mathematical structures, attempting to avoid disputes and maintain a smoothly functioning society.

Over the past one to two millennia, villages grew and coalesced into larger units. As societies grew beyond the size of villages, the mathematics correspondingly grew. The savanna of west Africa saw Songhai, ancient Mali and Ghana, and the Hausa States. The Swahili civilization grew along the coast of the Indian Ocean in east Africa. Also, trading links reached to increasingly distant targets across the Sahara and along distant stretches of ocean coastline. Although few records have survived, it is acknowledged that large governmental and trading organizations required complex record keeping and accounting. A trader would certainly want to keep careful records of items being traded to avoid being cheated by faraway customers. Governments had to handle administrative and logistical details of the equivalent of civil servants and the king’s retinue, and, especially, of armies. Longer trade routes required the design of stronger boats for coastal travel and navigational skills for caravan travel across empty desert landscapes. Also, the needs for currencies grew far beyond those of local markets, as traders had to convert the prices of the sellers to those of the buyers and still control costs and profits.

Reaching out from local roots also put Africans in contact with others—and often, the reverse happened as outside groups came into Africa. Either way, this led to a mixing of culture and a growth of experience. Mathematical ideas jumped from culture to culture, contributing a growth of power and sophistication of mathematics. It is reported that when king Mansa Musa of Mali accepted Islam and traveled across the Sahara to make the hajj pilgrimage to Mecca in 1324–1325 c.e., he brought so much of the golden riches of his empire that he upset the economy of Egypt as he passed through! The flow of the Arabs into both west Africa and east Africa brought the intellectual riches of Islamic mathematics. Even in the terminology of counting words, Arabic influence can be seen in the words for the decade numbers (20, 30, 40, and so on) in both the Hausa language of west Africa and Swahili of east Africa. Arab mathematics, which would later also make fundamental contributions to European mathematics, was taught in Qur’anic schools, and scholarly centers were established in various place including Timbuktu and Mombasa. One of the few documented examples of precolonial history of mathematics in west Africa was the work of Muhammad ibn Muhammad, who worked in Katsina—now in northern Nigeria—in the early 1700s. Interestingly, part of his work became controversial—his calculations of “magic squares,” which some of the Islamic authorities considered as flirting with the occult. The astronomical calculations required to maintain the calendar of Islamic festivals led to a growth of formalized geometry and trigonometry.

Mathematics in Egypt

In addition to the mathematics of subsistence, daily life, government, and trade, there was also considerable mathematics used in the arts and recreation. Probably the most famous and spectacular mathematics of the arts and architecture on the African continent is the mathematics of early Egypt. Beyond the famous hieroglyphic mathematics of ancient Egyptian numerals and the arithmetic of the problems found in rolls of papyrus, the mathematics of Egyptian architecture reached the level of “wonders of the world.” Notably, the famous pyramids are built with precise lengths, angles, and alignments. They fit into near-perfect geometrical shapes—all the more impressive given their massive size and the belief they were actually constructed by uneducated laborers working under the supervision of masters of labor. The mathematical history questions remain: Who did the design work? How were the designs communicated to the individual laborers?

Mathematics in Sub-Saharan Africa

In sub-Saharan Africa such spectacular wonders are not often seen, but the mathematics of the arts remains impressive. Other architectural examples include the massive structure of the Zimbabwe fortress as well as decorative design in chiefs’ palaces and public structures throughout the continent. Walls are often decorated with geometrical patterns—some to be washed off for new work when a new king would arrive.

On a smaller scale, many parts of Africa are known for their textile designs. Sierra Leone has intricate tie-and-dye patterns in cloth. Akan weavers in Ghana produce long strips of woven kente cloth in bright colors of red, blue, green, and gold, and then align them side by side to create broad sheets used as toga-like robes in traditional dress. Okenne weavers also make cloth, often with metallic threads giving a shiny appearance to the design. All of these patterns require mathematics in their design—especially considerations of symmetry. Tie-and-dye requires careful planning of the ties so that the resulting dye pattern reflects the design pattern. Kente and Okenne cloth show symmetry both along the initial woven strips and also across the strips in the full cloth of the robe.

The sculptures from many parts of Africa contributed to some of the designs of modern Western art. They show much use of symmetry, scale distortion, and even repetitive fractal-like patterns. Similarly, African music and dance, especially from west Africa, show mathematically complex rhythm structures in drumming and in the use of a variety of plucked and strummed musical instruments. Like African art, the music of Africa has contributed much to Western music, especially via the music the African slaves brought to the Americas, which formed the roots of jazz.

Beyond the arts, recreational mathematics is seen in numerous African games and pastimes. The best example is the many varieties of the mancala games (known under various names in different countries), which involve sharing seeds into pits in a game board, trying to capture the seeds of the opponent. There are many variations of the rules but all require a careful strategy of play and mathematical problem solving. Some game experts have listed mancala among the great games of the world.

Bibliography

Gerdes, Paulus. Geometry From Africa: Mathematical and Educational Explorations. Washington, DC: Mathematical Association of America, 1999.

Zaslavsky, Claudia. Africa Counts: Number and Pattern in African Culture. 3rd ed. Chicago: Lawrence Hill Books, 1999.