Reema’s Curiosity and Early Counting

3.1 Reema's Curiosity

One lazy afternoon, Reema was flipping through an old book when — whoosh! — a piece of paper slipped out and floated to the floor. She picked it up and stared at the strange symbols all over it. "What is this?" she wondered.

She ran to her father, holding the paper as if it were a secret treasure. He looked at it and smiled. "Around 4000 years ago, there flourished a civilisation in a region called Mesopotamia^?, in the western part of Asia, containing a major part of the present-day Iraq and a few other neighbouring countries. This is one of the ways they wrote their numbers!"

Reema's eyes lit up. "Seriously? These strange symbols were numbers?" Her curiosity was sparked, and questions started swirling in her head: Since when have humans been counting? How did the Mesopotamians have written numbers 3000 years ago? Once when people were writing numbers in the modern form?

Sensing her curiosity, her father started telling her how the idea of number and number representation evolved over the course of time, across geographies, to finally reach its modern efficient form.

Counting in the Stone Age

Humans had the need to count even as early as the Stone Age. They were counting to determine the quantity of food they had, the number of animals in their livestock, details regarding trades of goods, the number of offerings given in rituals, etc. They also wanted to keep track of the passing days, to know and predict when important events such as the new moon, full moon, or onset of a season would occur.

However, when they said or wrote down such numbers, they didn't make use of the numbers that we use today. The structure of the modern oral and written numbers that we use today had its origin thousands of years ago in India. Ancient Indian texts, such as the Yajurveda Samhita, mentioned names of numbers based on powers of 10, almost as we say them orally today. For example, they listed names for the numbers eka (one), dasha (ten), shata (hundred), sahasra (thousand), ayuta (ten thousand), etc., all the way up to \(10^{12}\) and beyond.

The Indian number system was transmitted to the Arab world by around 800 CE. It was popularised in the Arab world by the great Persian mathematician Al-Khwarizmi (after whom the word 'algorithm' is named) through his book On the Calculation with Hindu Numerals (c. 825), and by the noted philosopher Al-Kindi through his work On the Use of the Hindu Numerals (c. 830).

From the Arab world, the Hindu numerals were transmitted to Europe and to parts of Africa by around 1100 CE. Though Al-Khwarizmi's work on calculation with Hindu numerals was translated into Latin, it was the Italian mathematician Fibonacci who around the year 1200 really made the case to Europe to adopt the Indian numerals. However, the Roman numerals were so ingrained in European thinking and writing at the time that the Indian numerals did not gain widespread use for several more centuries. But eventually, during the European Renaissance and by the 17th century, not adopting them became impossible or it would impede scientific progress.

Their use then spread to every continent, and are now used in every corner of the world. Because European scholars learned the Indian numerals from the Arab world, they called them 'Arabic numerals' to reflect their European perspective. On the other hand, as noted above, Arab scholars, such as Al-Khwarizmi and Al-Kindi, called them 'Hindu numerals'. During the period of European colonisation, the European term Arabic numerals became widely used. However, in recent years, this mistake is being corrected in many textbooks and documents around the world, including in Europe. The most commonly used terminologies for the numbers we use today are 'Hindu numerals', 'Indian numerals', and the traditional 'Hindu-Arabic numerals'.

3.2 The Mechanism of Counting

Imagine that we are living in the Stone Age, say, around ten thousand years ago. Suppose we have a herd of cows. Here are some natural questions that we might ask about our herd —

• Q1. How do we ensure that all cows have returned safely after grazing?
• Q2. Do we have fewer cows than our neighbour?
• Q3. If there are fewer, how many more cows would we need so that we have the same number of cows as our neighbour?

We need to tackle these questions without the use of the number names or written numbers of the Hindu number system. How do we do it? Here are some possible methods.

Method 1 — One-to-One Correspondence with Sticks

We could tackle the questions by using pebbles, sticks, or any object that is available in abundance. Let us choose sticks. For every cow in the herd, we could keep a stick. The final collection of sticks tells us the number of cows, which can be used to check if any cows have gone missing.

This way of associating each cow with a stick^?, such that no two sticks are associated or mapped to the same stick is called a one-to-one mapping. This mapping can then be used to come up with a way to represent numbers, as shown below.

Method 2 — Standard Sequence of Sounds/Names

Instead of objects, we could use a standard sequence of sounds or names. For example, we could use the sounds of any language. While counting, we could make a one-to-one mapping^? between the objects and the letters: that is, associate each object to be counted with a letter, following the letter-order. This mapping can then be used to come up with a way of verbally representing numbers.

For example, we get the following number representation if we use English letters 'a' to 'z':

Recognition at a Glance — The Limit of Perception

Up to what group size could you immediately see the number of objects without counting? Most humans find it difficult to count groups having 5 or more objects in a single glance. This limit of perception could have prompted people to use tally marks to replace what amounted to, say, 5 marks, with a new symbol, as seen in the system shown in Table 1 of the textbook.