Words, gestures and symbols - African counting methods

Africa has a rich variety of traditional counting methods

MANY hundreds of well-structured numeration systems were invented in Africa south of the Sahara whose peoples, like those elsewhere in the world, learnt through the ages that it is very difficult to count and calculate if one uses a completely new, different word or symbol for each quantity--that is, for each number. These systems include spoken numeration systems, gesture counting systems, and symbolic systems that use body parts or objects to represent numbers.

The most common way to avoid having to invent completely new words for different numbers has been to compose new number words out of existing ones by using the arithmetical relationships between the numbers concerned. This principle can be seen in many African spoken numeration systems.

In the Makhwa language spoken in northern Mozambique, for example, the words thanu (5) and nloko (10) are dominant in the composition of number words, and constitute the bases of the system of numeration. The expression for 6 is thanu na moza (5 plus 1), and 7 is thanu na pili, (5 plus 2). To express 20, people say miloko mili (tens two or 10 times 2), and 30 is miloko miraru (tens three).

The most common bases in Africa are 10, 5 and 20. Some languages such as Nyungwe, which is spoken in Mozambique, use only base 10. Others like Balante in Guinea-Bissau use 5 and 20. Verbal numeration in the Bete language of Cote d'Ivoire uses three bases: 5, 10 and 20. Fifty-six, for instance, is expressed as golosso-ya-kogbo-gbeplo, that is "20 times two plus 10 (and) 5 (and) 1". The Bambara of Mali and Guinea have a 10-20 system in which the word for 20, mugan, means "one person", while the word for 40, debe, means "mat", referring to a mat on which husband and wife sleep together--and jointly they have 40 digits.

The Bulanda (West Africa) use 6 as a base so that 7 is expressed as 6 + 1, 8 as 6 + 2, and so on. The Adele count koro (6), koroke (6 + 1 = 7), nye (8) and nyeki (8 + 1 = 9). Among the Huku of Uganda the number words for 13, 14, 15 are formed by the addition of 1, 2 or 3 to twelve. Thirteen, for instance, is expressed as bakumba igimo (12 plus 1). The decimal alternatives, 10 + 3, 10 + 4 and 10 + 5, were also known.

One advantage of using a low number such as 5 as the basis of a spoken numeration system is that it may facilitate oral or mental calculation where the answer has not been memorized. For instance, 7 + 8 would be (5 + 2) plus (5 + 3). As 2 + 3 = 5, one finds as answer 5 + 5 + 5, 10 + 5, or 5 multiplied by 3.

* The duplicative principle

A particular case of the use of addition to compose number words is the situation where both numbers are equal or where one of the two is equal to the other plus one. For instance, the Mbai count from 6 to 9 in the following way: mutu muta (3 + 3), sa do muta (4 + 3), soso (4 + 4), and sa dio mi (4 + 5). The Sango of northern Zaire express 7 as na na-thatu (4 + 3), 8 as mnana (4 + 4) and 9 as sano na-na (5 + 4). One possible reason for using the duplicative principle to compose the number words between 6 and 9 is that it may make it easier to do mental arithmetic, in particular duplication operations. For instance, to obtain the double of 7, one has to add, if one has not memorized the answer, 4 + 3 and 4 + 3. As 4 + 3 + 3 = 10, the answer becomes 10 + 4. In sub-Saharan Africa, there is a strong tradition of mental calculation, and oral and mental multiplication often were (and sometimes still are) based on repeated duplication.

In several African languages subtraction, as well as the additive and multiplicative principles, has been used to form number words. In the Yoruba language of Nigeria, for example, 16 is expressed as eerin din logun meaning "four until one arrives at twenty". The Luba-Hemba people of Zaire express seven as habulwa mwanda ("lacking one until eight"), and nine as habulwa likumi ("lacking one until ten").

Spoken numeration systems may vary greatly within relatively small geographic regions. In Guinea-Bissau, for example, the Bijago have a pure decimal system; the Balante use a five-twenty system; the Manjaco use a decimal system with exceptional composite number words such as 6 + 1 for 7 and 8 + 1 for 9; and the Felup use a ten-twenty system in which the duplicative principle is also employed in forms like 7 as 4 + 3 and 8 as 4 + 4.

Some number words are adjectives while others are nouns. In this situation, number word structures may appear that do not correspond directly to an addition, multiplication or subtraction. For instance, in the Tschwa language of central Mozambique 60 is expressed as thlanu wa makuma ni ginwe, that is "five times ten plus one more (ten)".

To denote relatively large numbers, completely new words are often used or words that express a relationship with the base of the numeration system. The Bangongo of Zaire say kama (100), lobombo (1,000), njuku (10,000), lukuli (100,000) and losenene (1,000,000), while the Ziba of Tanzania say tsikumi (100), lukumi (1,000), and kukumi (10,000), the three latter terms being clearly related to kumi (10).

* Counting by gestures

Gesture counting was common among many African peoples. The Yao of Malawi and Mozambique represent 1, 2, 3 and 4 by pointing with the thumb of their right hand at 1, 2, 3 or 4 extended fingers of their left hand. Five is indicated by making a fist with the left hand. Six, 7, 8 and 9 are indicated by joining 1, 2, 3 or 4 extended fingers of the right hand to the left fist. Ten is represented by raising the fingers of both hands and joining the hands. The Makonde of northern Mozambique start counting on their right hand with the help of the index finger of the left hand. Five is indicated by making a fist with the right hand. For 6 to 9, the representation is symmetrical to that of 1 to 4, that is, right and left hands change roles, with the index finger of the right hand pointing at the fingers of the left. Ten is represented by joining two fists.

The Shambaa of Tanzania and Kenya practise a method of gesture counting that uses the duplicative principle. They indicate 6 by extending the three outer fingers of each hand, spread out; 7 by showing 4 on the right hand and 3 on the left, and 8 by showing 4 on each hand.

To express numbers greater than 10, the Sotho of Lesotho employ different individuals to indicate the hundreds, tens and units. To represent 368, for example, the first person raises 3 fingers of the left hand to represent 300, the second one raises the thumb of the right hand to express 6 tens, and the third one raises three fingers of the right hand to express 8 units. This is actually a positional system, since it depends on the position of each man whether he indicates units, tens, hundreds, thousands, and so on.

The use of fingers and hands to count may explain the choice of five and ten as bases for verbal numeration systems. The use of bases may also have been stimulated by practices used to accelerate counting. For instance, Makonde basket makers count the plant strands of the bottom of their likalala-baskets by fours instead of counting one by one.

* Tally devices

Tally devices were commonly used in Africa south of the Sahara. In Mozambique Chuabo boys use the following counting technique when they are playing football. The two halves of the leaf of a coconut tree obtained after taking off its vein serve as a tally device for each team. The halves are called mulobuo. When a team scores a goal, a fold is made in its mulobuo, and at the end of the match, the scorer compares the lengths of each one or counts the folds, in order to see who has won.

Among the Tswa, also from Mozambique, trees are used to record the age of children. After the birth of a child a cut is made on a trunk of a tree, and each year a new cut is added until the person is old enough to count for him or herself. Cuts on tally sticks are also used when counting the number of animals in a herd, each cut corresponding to one animal.

Among the Makonde, knotted strings were used. A man who was setting out on an eleven-day journey would tie eleven knots in a string and would say to his wife, "This knot" (touching the first) "is today, when I am starting; tomorrow" (touching the second knot) "I shall be on the road, and I shall be walking the whole of the second and third days, but here" (seizing the fifth knot) "I shall reach the end of the journey. I shall stay there the sixth day, and start for home on the seventh. Do not forget, wife, to undo a knot every day, and on the tenth day you will have to cook food for me; for see, this is the eleventh day when I shall come back." Pregnant women used to tie a knot in a string at each full moon so that they would know when they were due to give birth. In order to register a person's age, two strings were used. A knot was tied in the first string at each full moon. When twelve knots were tied, a knot was tied in a second string to mark the completion of the first year, and so on.

* Figures in the sand

A variety of numeration systems exist in Africa that are "written" in one way or another. The eastern Bushongo (Zaire) counted simultaneously by threes and tens. For each three objects, they marked in the sand three short parallel strokes with three fingers of a hand. After completing three groups of three strokes, a longer stroke was marked for the next object to indicate that ten more objects had been counted.

The Fulani or Fulbe, a semi-nomadic pastoral people of Niger and northern Nigeria, place sticks in front of their houses to indicate the number of cows or goats they possess. One hundred animals are represented by two short sticks placed on the ground in the form of a V. Two crossed sticks, X, symbolize 50 animals. Four sticks in a "vertical" position represent 4. Two sticks in a "horizontal" and three in a "vertical" position indicate 23 animals. The following was found in front of the house of a rich cattle owner: VVVVVVXII, showing that he had 652 cows.

The Akan peoples (Cote d'Ivoire, Ghana, Togo) used figurines in stone, metal or simply vegetable seeds as coins. The weight of each figurine was agreed to represent the monetary value that corresponded to a certain quantity of gold dust of the same weight. The figurines show animals, knots, stools, sandals, drums and in some cases have diverse geometric forms such as step pyramids, stars or cubes. Many display graphic signs representing numbers. Although in the languages spoken by the Akan peoples only base ten is used, base five is also found on the money weights. The Agni, one group belonging to the Akan people, used a series of units of money weights with a binary structure, each new unit being double the previous one.

PAULUS GERDES is a Mozambican mathematician who is rector of his country's Higher Pedagogical Institute and chairman of the International Commission on the History of Mathematics in Africa (AMUCHMA). His published works include Ethnomathematics and Education in Africa.

MARCOS CHERINDA, Mozambican mathematician, is a lecturer at Mozambique's Higher Pedagogical Institute, where he specializes in ethnomathematics. He is the co-author, with Paulus Gerdes, of Famous Theorems of Geometry.

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